MatthewF.Dixon IgorHalperin PaulBilokon

Machine Learning in Finance From Theory toPractice

Machine Learning in Finance

Matthew F. Dixon • Igor Halperin • Paul Bilokon

Machine Learning in Finance From Theory to Practice

Matthew F. Dixon Department of Applied Mathematics Illinois Institute of Technology Chicago, IL, USA

Igor Halperin Tandon School of Engineering New York University Brooklyn, NY, USA

Paul Bilokon Department of Mathematics Imperial College London London, UK

Additional material to this book can be downloaded from http://mypages.iit.edu/~mdixon7/ book/ML_Finance_Codes-Book.zip ISBN 978-3-030-41067-4 ISBN 978-3-030-41068-1 (eBook) https://doi.org/10.1007/978-3-030-41068-1 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. —Arthur Conan Doyle

Introduction

Machine learning in finance sits at the intersection of a number of emergent and established disciplines including pattern recognition, financial econometrics, statistical computing, probabilistic programming, and dynamic programming. With the trend towards increasing computational resources and larger datasets, machine learning has grown into a central computational engineering field, with an emphasis placed on plug-and-play algorithms made available through open-source machine learning toolkits. Algorithm focused areas of finance, such as algorithmic trading have been the primary adopters of this technology. But outside of engineering-based research groups and business activities, much of the field remains a mystery. A key barrier to understanding machine learning for non-engineering students and practitioners is the absence of the well-established theories and concepts that financial time series analysis equips us with. These serve as the basis for the development of financial modeling intuition and scientific reasoning. Moreover, machine learning is heavily entrenched in engineering ontology, which makes developments in the field somewhat intellectually inaccessible for students, academics, and finance practitioners from the quantitative disciplines such as mathematics, statistics, physics, and economics. Consequently, there is a great deal of misconception and limited understanding of the capacity of this field. While machine learning techniques are often effective, they remain poorly understood and are often mathematically indefensible. How do we place key concepts in the field of machine learning in the context of more foundational theory in time series analysis, econometrics, and mathematical statistics? Under which simplifying conditions are advanced machine learning techniques such as deep neural networks mathematically equivalent to well-known statistical models such as linear regression? How should we reason about the perceived benefits of using advanced machine learning methods over more traditional econometrics methods, for different financial applications? What theory supports the application of machine learning to problems in financial modeling? How does reinforcement learning provide a model-free approach to the Black–Scholes–Merton model for derivative pricing? How does Q-learning generalize discrete-time stochastic control problems in finance?

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This book is written for advanced graduate students and academics in financial econometrics, management science, and applied statistics, in addition to quants and data scientists in the field of quantitative finance. We present machine learning as a non-linear extension of various topics in quantitative economics such as financial econometrics and dynamic programming, with an emphasis on novel algorithmic representations of data, regularization, and techniques for controlling the bias-variance tradeoff leading to improved out-of-sample forecasting. The book is presented in three parts, each part covering theory and applications. The first part presents supervised learning for cross-sectional data from both a Bayesian and frequentist perspective. The more advanced material places a firm emphasis on neural networks, including deep learning, as well as Gaussian processes, with examples in investment management and derivatives. The second part covers supervised learning for time series data, arguably the most common data type used in finance with examples in trading, stochastic volatility, and fixed income modeling. Finally, the third part covers reinforcement learning and its applications in trading, investment, and wealth management. We provide Python code examples to support the readers’ understanding of the methodologies and applications. As a bridge to research in this emergent field, we present the frontiers of machine learning in finance from a researcher’s perspective, highlighting how many wellknown concepts in statistical physics are likely to emerge as research topics for machine learning in finance.

Prerequisites This book is targeted at graduate students in data science, mathematical finance, financial engineering, and operations research seeking a career in quantitative finance, data science, analytics, and fintech. Students are expected to have completed upper section undergraduate courses in linear algebra, multivariate calculus, advanced probability theory and stochastic processes, statistics for time series (econometrics), and gained some basic introduction to numerical optimization and computational mathematics. Students shall find the later chapters of this book, on reinforcement learning, more accessible with some background in investment science. Students should also have prior experience with Python programming and, ideally, taken a course in computational finance and introductory machine learning. The material in this book is more mathematical and less engineering focused than most courses on machine learning, and for this reason we recommend reviewing the recent book, Linear Algebra and Learning from Data by Gilbert Strang as background reading.

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Advantages of the Book Readers will find this book useful as a bridge from well-established foundational topics in financial econometrics to applications of machine learning in finance. Statistical machine learning is presented as a non-parametric extension of financial econometrics and quantitative finance, with an emphasis on novel algorithmic representations of data, regularization, and model averaging to improve out-of-sample forecasting. The key distinguishing feature from classical financial econometrics and dynamic programming is the absence of an assumption on the data generation process. This has important implications for modeling and performance assessment which are emphasized with examples throughout the book. Some of the main contributions of the book are as follows: • The textbook market is saturated with excellent books on machine learning. However, few present the topic from the prospective of financial econometrics and cast fundamental concepts in machine learning into canonical modeling and decision frameworks already well established in finance such as financial time series analysis, investment science, and financial risk management. Only through the integration of these disciplines can we develop an intuition into how machine learning theory informs the practice of financial modeling. • Machine learning is entrenched in engineering ontology, which makes developments in the field somewhat intellectually inaccessible for students, academics, and finance practitioners from quantitative disciplines such as mathematics, statistics, physics, and economics. Moreover, financial econometrics has not kept pace with this transformative field, and there is a need to reconcile various modeling concepts between these disciplines. This textbook is built around powerful mathematical ideas that shall serve as the basis for a graduate course for students with prior training in probability and advanced statistics, linear algebra, times series analysis, and Python programming. • This book provides financial market motivated and compact theoretical treatment of financial modeling with machine learning for the benefit of regulators, wealth managers, federal research agencies, and professionals in other heavily regulated business functions in finance who seek a more theoretical exposition to allay concerns about the “black-box” nature of machine learning. • Reinforcement learning is presented as a model-free framework for stochastic control problems in finance, covering portfolio optimization, derivative pricing, and wealth management applications without assuming a data generation process. We also provide a model-free approach to problems in market microstructure, such as optimal execution, with Q-learning. Furthermore, our book is the first to present on methods of inverse reinforcement learning. • Multiple-choice questions, numerical examples, and more than 80 end-ofchapter exercises are used throughout the book to reinforce key technical concepts.

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• This book provides Python codes demonstrating the application of machine learning to algorithmic trading and financial modeling in risk management and equity research. These codes make use of powerful open-source software toolkits such as Google’s TensorFlow and Pandas, a data processing environment for Python.

Overview of the Book Chapter 1 Chapter 1 provides the industry context for machine learning in finance, discussing the critical events that have shaped the finance industry’s need for machine learning and the unique barriers to adoption. The finance industry has adopted machine learning to varying degrees of sophistication. How it has been adopted is heavily fragmented by the academic disciplines underpinning the applications. We view some key mathematical examples that demonstrate the nature of machine learning and how it is used in practice, with the focus on building intuition for more technical expositions in later chapters. In particular, we begin to address many finance practitioner’s concerns that neural networks are a “black-box” by showing how they are related to existing well-established techniques such as linear regression, logistic regression, and autoregressive time series models. Such arguments are developed further in later chapters.

Chapter 2 Chapter 2 introduces probabilistic modeling and reviews foundational concepts in Bayesian econometrics such as Bayesian inference, model selection, online learning, and Bayesian model averaging. We develop more versatile representations of complex data with probabilistic graphical models such as mixture models.

Chapter 3 Chapter 3 introduces Bayesian regression and shows how it extends many of the concepts in the previous chapter. We develop kernel-based machine learning methods—specifically Gaussian process regression, an important class of Bayesian machine learning methods—and demonstrate their application to “surrogate” models of derivative prices. This chapter also provides a natural point from which to

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develop intuition for the role and functional form of regularization in a frequentist setting—the subject of subsequent chapters.

Chapter 4 Chapter 4 provides a more in-depth description of supervised learning, deep learning, and neural networks—presenting the foundational mathematical and statistical learning concepts and explaining how they relate to real-world examples in trading, risk management, and investment management. These applications present challenges for forecasting and model design and are presented as a reoccurring theme throughout the book. This chapter moves towards a more engineering style exposition of neural networks, applying concepts in the previous chapters to elucidate various model design choices.

Chapter 5 Chapter 5 presents a method for interpreting neural networks which imposes minimal restrictions on the neural network design. The chapter demonstrates techniques for interpreting a feedforward network, including how to rank the importance of the features. In particular, an example demonstrating how to apply interpretability analysis to deep learning models for factor modeling is also presented.

Chapter 6 Chapter 6 provides an overview of the most important modeling concepts in financial econometrics. Such methods form the conceptual basis and performance baseline for more advanced neural network architectures presented in the next chapter. In fact, each type of architecture is a generalization of many of the models presented here. This chapter is especially useful for students from an engineering or science background, with little exposure to econometrics and time series analysis.

Chapter 7 Chapter 7 presents a powerful class of probabilistic models for financial data. Many of these models overcome some of the severe stationarity limitations of the frequentist models in the previous chapters. The fitting procedure demonstrated is also different—the use of Kalman filtering algorithms for state-space models rather

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than maximum likelihood estimation or Bayesian inference. Simple examples of hidden Markov models and particle filters in finance and various algorithms are presented.

Chapter 8 Chapter 8 presents various neural network models for financial time series analysis, providing examples of how they relate to well-known techniques in financial econometrics. Recurrent neural networks (RNNs) are presented as non-linear time series models and generalize classical linear time series models such as AR(p). They provide a powerful approach for prediction in financial time series and generalize to non-stationary data. The chapter also presents convolution neural networks for filtering time series data and exploiting different scales in the data. Finally, this chapter demonstrates how autoencoders are used to compress information and generalize principal component analysis.

Chapter 9 Chapter 9 introduces Markov decision processes and the classical methods of dynamic programming, before building familiarity with the ideas of reinforcement learning and other approximate methods for solving MDPs. After describing Bellman optimality and iterative value and policy updates before moving to Q-learning, the chapter quickly advances towards a more engineering style exposition of the topic, covering key computational concepts such as greediness, batch learning, and Q-learning. Through a number of mini-case studies, the chapter provides insight into how RL is applied to optimization problems in asset management and trading. These examples are each supported with Python notebooks.

Chapter 10 Chapter 10 considers real-world applications of reinforcement learning in finance, as well as further advances the theory presented in the previous chapter. We start with one of the most common problems of quantitative finance, which is the problem of optimal portfolio trading in discrete time. Many practical problems of trading or risk management amount to different forms of dynamic portfolio optimization, with different optimization criteria, portfolio composition, and constraints. The chapter introduces a reinforcement learning approach to option pricing that generalizes the classical Black–Scholes model to a data-driven approach using Q-learning. It then presents a probabilistic extension of Q-learning called G-learning and shows how it

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can be used for dynamic portfolio optimization. For certain specifications of reward functions, G-learning is semi-analytically tractable and amounts to a probabilistic version of linear quadratic regulators (LQRs). Detailed analyses of such cases are presented and we show their solutions with examples from problems of dynamic portfolio optimization and wealth management.

Chapter 11 Chapter 11 provides an overview of the most popular methods of inverse reinforcement learning (IRL) and imitation learning (IL). These methods solve the problem of optimal control in a data-driven way, similarly to reinforcement learning, however with the critical difference that now rewards are not observed. The problem is rather to learn the reward function from the observed behavior of an agent. As behavioral data without rewards are widely available, the problem of learning from such data is certainly very interesting. The chapter provides a moderate-level description of the most promising IRL methods, equips the reader with sufficient knowledge to understand and follow the current literature on IRL, and presents examples that use simple simulated environments to see how these methods perform when we know the “ground truth" rewards. We then present use cases for IRL in quantitative finance that include applications to trading strategy identification, sentiment-based trading, option pricing, inference of portfolio investors, and market modeling.

Chapter 12 Chapter 12 takes us forward to emerging research topics in quantitative finance and machine learning. Among many interesting emerging topics, we focus here on two broad themes. The first one deals with unification of supervised learning and reinforcement learning as two tasks of perception-action cycles of agents. We outline some recent research ideas in the literature including in particular information theory-based versions of reinforcement learning and discuss their relevance for financial applications. We explain why these ideas might have interesting practical implications for RL financial models, where feature selection could be done within the general task of optimization of a long-term objective, rather than outside of it, as is usually performed in “alpha-research.” The second topic presented in this chapter deals with using methods of reinforcement learning to construct models of market dynamics. We also introduce some advanced physics-based approaches for computations for such RL-inspired market models.

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Source Code Many of the chapters are accompanied by Python notebooks to illustrate some of the main concepts and demonstrate application of machine learning methods. Each notebook is lightly annotated. Many of these notebooks use TensorFlow. We recommend loading these notebooks, together with any accompanying Python source files and data, in Google Colab. Please see the appendices of each chapter accompanied by notebooks, and the README.md in the subfolder of each chapter, for further instructions and details.

Scope We recognize that the field of machine learning is developing rapidly and to keep abreast of the research in this field is a challenging pursuit. Machine learning is an umbrella term for a number of methodology classes, including supervised learning, unsupervised learning, and reinforcement learning. This book focuses on supervised learning and reinforcement learning because these are the areas with the most overlap with econometrics, predictive modeling, and optimal control in finance. Supervised machine learning can be categorized as generative and discriminative. Our focus is on discriminative learners which attempt to partition the input space, either directly, through affine transformations or through projections onto a manifold. Neural networks have been shown to provide a universal approximation to a wide class of functions. Moreover, they can be shown to reduce to other wellknown statistical techniques and are adaptable to time series data. Extending time series models, a number of chapters in this book are devoted to an introduction to reinforcement learning (RL) and inverse reinforcement learning (IRL) that deal with problems of optimal control of such time series and show how many classical financial problems such as portfolio optimization, option pricing, and wealth management can naturally be posed as problems for RL and IRL. We present simple RL methods that can be applied for these problems, as well as explain how neural networks can be used in these applications. There are already several excellent textbooks covering other classical machine learning methods, and we instead choose to focus on how to cast machine learning into various financial modeling and decision frameworks. We emphasize that much of this material is not unique to neural networks, but comparisons of alternative supervised learning approaches, such as random forests, are beyond the scope of this book.

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Multiple-Choice Questions Multiple-choice questions are included after introducing a key concept. The correct answers to all questions are provided at the end of each chapter with selected, partial, explanations to some of the more challenging material.

Exercises The exercises that appear at the end of every chapter form an important component of the book. Each exercise has been chosen to reinforce concepts explained in the text, to stimulate the application of machine learning in finance, and to gently bridge material in other chapters. It is graded according to difficulty ranging from (*), which denotes a simple exercise which might take a few minutes to complete, through to (***), which denotes a significantly more complex exercise. Unless specified otherwise, all equations referenced in each exercise correspond to those in the corresponding chapter.

Instructor Materials The book is supplemented by a separate Instructor’s Manual which provides worked solutions to the end of chapter questions. Full explanations for the solutions to the multiple-choice questions are also provided. The manual provides additional notes and example code solutions for some of the programming exercises in the later chapters.

Acknowledgements This book is dedicated to the late Mark Davis (Imperial College) who was an inspiration in the field of mathematical finance and engineering, and formative in our careers. Peter Carr, Chair of the Department of Financial Engineering at NYU Tandon, has been instrumental in supporting the growth of the field of machine learning in finance. Through providing speaker engagements and machine learning instructorship positions in the MS in Algorithmic Finance Program, the authors have been able to write research papers and identify the key areas required by a text book. Miquel Alonso (AIFI), Agostino Capponi (Columbia), Rama Cont (Oxford), Kay Giesecke (Stanford), Ali Hirsa (Columbia), Sebastian Jaimungal (University of Toronto), Gary Kazantsev (Bloomberg), Morton Lane (UIUC), Jörg Osterrieder (ZHAW) have established various academic and joint academic-industry workshops

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and community meetings to proliferate the field and serve as input for this book. At the same time, there has been growing support for the development of a book in London, where several SIAM/LMS workshops and practitioner special interest groups, such as the Thalesians, have identified a number of compelling financial applications. The material has grown from courses and invited lectures at NYU, UIUC, Illinois Tech, Imperial College and the 2019 Bootcamp on Machine Learning in Finance at the Fields Institute, Toronto. Along the way, we have been fortunate to receive the support of Tomasz Bielecki (Illinois Tech), Igor Cialenco (Illinois Tech), Ali Hirsa (Columbia University), and Brian Peterson (DV Trading). Special thanks to research collaborators and colleagues Kay Giesecke (Stanford University), Diego Klabjan (NWU), Nick Polson (Chicago Booth), and Harvey Stein (Bloomberg), all of whom have shaped our understanding of the emerging field of machine learning in finance and the many practical challenges. We are indebted to Sri Krishnamurthy (QuantUniversity), Saeed Amen (Cuemacro), Tyler Ward (Google), and Nicole Königstein for their valuable input on this book. We acknowledge the support of a number of Illinois Tech graduate students who have contributed to the source code examples and exercises: Xiwen Jing, Bo Wang, and Siliang Xong. Special thanks to Swaminathan Sethuraman for his support of the code development, to Volod Chernat and George Gvishiani who provided support and code development for the course taught at NYU and Coursera. Finally, we would like to thank the students and especially the organisers of the MSc Finance and Mathematics course at Imperial College, where many of the ideas presented in this book have been tested: Damiano Brigo, Antoine (Jack) Jacquier, Mikko Pakkanen, and Rula Murtada. We would also like to thank Blanka Horvath for many useful suggestions. Chicago, IL, USA Brooklyn, NY, USA London, UK December 2019

Matthew F. Dixon Igor Halperin Paul Bilokon

Contents

Part I Machine Learning with Cross-Sectional Data 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Big Data—Big Compute in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fintech . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Machine Learning and Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Statistical Modeling vs. Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Modeling Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Financial Econometrics and Machine Learning . . . . . . . . . . . . . . . 3.3 Over-fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Examples of Supervised Machine Learning in Practice . . . . . . . . . . . . . . 5.1 Algorithmic Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 High-Frequency Trade Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Mortgage Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 6 8 11 14 16 16 18 21 22 28 29 32 34 40 41 44

2

Probabilistic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Bayesian vs. Frequentist Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Frequentist Inference from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Assessing the Quality of Our Estimator: Bias and Variance . . . . . . . . . 5 The Bias–Variance Tradeoff (Dilemma) for Estimators . . . . . . . . . . . . . . 6 Bayesian Inference from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 A More Informative Prior: The Beta Distribution . . . . . . . . . . . . . 6.2 Sequential Bayesian updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 51 53 55 56 60 61

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6.3

Practical Implications of Choosing a Classical or Bayesian Estimation Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Bayesian Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Model Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Model Selection When There Are Many Models . . . . . . . . . . . . . 7.4 Occam’s Razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Probabilistic Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

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Bayesian Regression and Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Bayesian Inference with Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bayesian Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schur Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Gaussian Process Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Gaussian Processes in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gaussian Processes Regression and Prediction . . . . . . . . . . . . . . . 3.3 Hyperparameter Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Computational Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Massively Scalable Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Structured Kernel Interpolation (SKI) . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Kernel Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Example: Pricing and Greeking with Single-GPs. . . . . . . . . . . . . . . . . . . . . 5.1 Greeking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mesh-Free GPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Massively Scalable GPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Multi-response Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Multi-Output Gaussian Process Regression and Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Programming Related Questions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 86 88 89 91 92 93 94 96 96 97 97 98 101 101 103 103 104 105 106 107 108

Feedforward Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Feedforward Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Geometric Interpretation of Feedforward Networks . . . . . . . . . . 2.3 Probabilistic Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 112 114 117

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2.4 Function Approximation with Deep Learning* . . . . . . . . . . . . . . . 2.5 VC Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 When Is a Neural Network a Spline?* . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Why Deep Networks? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Convexity and Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Similarity of MLPs with Other Supervised Learners . . . . . . . . . 4 Training, Validation, and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Stochastic Gradient Descent (SGD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Back-Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Bayesian Neural Networks* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Programming Related Questions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 120 124 127 132 138 140 142 143 146 149 153 153 156 164

Interpretability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background on Interpretability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Explanatory Power of Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Multiple Hidden Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Example: Step Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Interaction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Example: Friedman Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Bounds on the Variance of the Jacobian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Chernoff Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simulated Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Factor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Non-linear Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Fundamental Factor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Programming Related Questions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 168 168 169 170 170 170 171 172 174 174 177 177 178 183 184 184 188

Part II Sequential Learning 6

Sequence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Autoregressive Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Autoregressive Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Partial Autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 192 192 194 195 195 197

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2.6 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Moving Average Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Exponential Smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fitting Time Series Models: The Box–Jenkins Approach . . . . . . . . . . . . 3.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Transformation to Ensure Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Predicting Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Time Series Cross-Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Principal Component Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 200 201 202 204 205 205 206 206 208 210 210 213 213 215 216 217 218 220

7

Probabilistic Sequence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hidden Markov Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Viterbi Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Particle Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Sequential Importance Resampling (SIR) . . . . . . . . . . . . . . . . . . . . . 3.2 Multinomial Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Application: Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . 4 Point Calibration of Stochastic Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Bayesian Calibration of Stochastic Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 221 222 224 227 227 228 229 230 231 233 235 235 237

8

Advanced Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Recurrent Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 RNN Memory: Partial Autocovariance . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Generalized Recurrent Neural Networks (GRNNs) . . . . . . . . . . . 3 Gated Recurrent Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 α-RNNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Neural Network Exponential Smoothing . . . . . . . . . . . . . . . . . . . . . . 3.3 Long Short-Term Memory (LSTM) . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 239 240 244 245 246 248 249 249 251 254

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4

Python Notebook Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Bitcoin Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Predicting from the Limit Order Book. . . . . . . . . . . . . . . . . . . . . . . . . 5 Convolutional Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Weighted Moving Average Smoothers . . . . . . . . . . . . . . . . . . . . . . . . 5.2 2D Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Dilated Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Python Notebooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Autoencoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Linear Autoencoders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Equivalence of Linear Autoencoders and PCA . . . . . . . . . . . . . . . 6.3 Deep Autoencoders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Programming Related Questions* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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255 256 256 257 258 261 263 264 265 266 267 268 270 271 272 273 275

Part III Sequential Data with Decision-Making 9

Introduction to Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Elements of Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rewards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Value and Policy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Observable Versus Partially Observable Environments . . . . . . . 3 Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Decision Policies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Value Functions and Bellman Equations . . . . . . . . . . . . . . . . . . . . . . 3.3 Optimal Policy and Bellman Optimality. . . . . . . . . . . . . . . . . . . . . . . 4 Dynamic Programming Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Policy Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Policy Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Value Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Reinforcement Learning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Policy-Based Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Temporal Difference Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 SARSA and Q-Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Stochastic Approximations and Batch-Mode Q-learning . . . . . 5.6 Q-learning in a Continuous Space: Function Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Batch-Mode Q-Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Least Squares Policy Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Deep Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 279 284 284 286 286 289 291 293 296 299 300 302 303 306 307 309 311 313 316 323 327 331 335

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6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10

Applications of Reinforcement Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The QLBS Model for Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Discrete-Time Black–Scholes–Merton Model . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hedge Portfolio Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Optimal Hedging Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Option Pricing in Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Hedging and Pricing in the BS Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The QLBS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bellman Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Optimal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 DP Solution: Monte Carlo Implementation . . . . . . . . . . . . . . . . . . . 4.5 RL Solution for QLBS: Fitted Q Iteration . . . . . . . . . . . . . . . . . . . . . 4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Option Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Possible Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 G-Learning for Stock Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Investment Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Terminal Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Asset Returns Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Signal Dynamics and State Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 One-Period Rewards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Multi-period Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Stochastic Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Reference Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Bellman Optimality Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Entropy-Regularized Bellman Optimality Equation . . . . . . . . . . 5.12 G-Function: An Entropy-Regularized Q-Function . . . . . . . . . . . . 5.13 G-Learning and F-Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Portfolio Dynamics with Market Impact . . . . . . . . . . . . . . . . . . . . . . 5.15 Zero Friction Limit: LQR with Entropy Regularization . . . . . . 5.16 Non-zero Market Impact: Non-linear Dynamics . . . . . . . . . . . . . . 6 RL for Wealth Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Merton Consumption Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Portfolio Optimization for a Defined Contribution Retirement Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 G-Learning for Retirement Plan Optimization . . . . . . . . . . . . . . . . 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

347 347 349 352 352 354 356 359 360 361 362 365 368 370 373 375 379 380 380 381 382 383 383 384 386 386 388 388 389 391 393 395 396 400 401 401 405 408 413 413

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8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 11

Inverse Reinforcement Learning and Imitation Learning . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Inverse Reinforcement Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 RL Versus IRL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 What Are the Criteria for Success in IRL? . . . . . . . . . . . . . . . . . . . . 2.3 Can a Truly Portable Reward Function Be Learned with IRL?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Maximum Entropy Inverse Reinforcement Learning . . . . . . . . . . . . . . . . . 3.1 Maximum Entropy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Maximum Causal Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 G-Learning and Soft Q-Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Maximum Entropy IRL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Estimating the Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Example: MaxEnt IRL for Inference of Customer Preferences . . . . . . 4.1 IRL and the Problem of Customer Choice. . . . . . . . . . . . . . . . . . . . . 4.2 Customer Utility Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Maximum Entropy IRL for Customer Utility . . . . . . . . . . . . . . . . . 4.4 How Much Data Is Needed? IRL and Observational Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Counterfactual Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Finite-Sample Properties of MLE Estimators . . . . . . . . . . . . . . . . . 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Adversarial Imitation Learning and IRL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Imitation Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 GAIL: Generative Adversarial Imitation Learning. . . . . . . . . . . . 5.3 GAIL as an Art of Bypassing RL in IRL . . . . . . . . . . . . . . . . . . . . . . 5.4 Practical Regularization in GAIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Adversarial Training in GAIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Other Adversarial Approaches* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 f-Divergence Training* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Wasserstein GAN*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Least Squares GAN* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Beyond GAIL: AIRL, f-MAX, FAIRL, RS-GAIL, etc.* . . . . . . . . . . . . . 6.1 AIRL: Adversarial Inverse Reinforcement Learning . . . . . . . . . 6.2 Forward KL or Backward KL?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 f-MAX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Forward KL: FAIRL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Risk-Sensitive GAIL (RS-GAIL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Gaussian Process Inverse Reinforcement Learning. . . . . . . . . . . . . . . . . . . 7.1 Bayesian IRL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Gaussian Process IRL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419 419 423 425 426 427 428 430 433 436 438 442 443 444 445 446 450 452 454 455 457 457 459 461 464 466 468 468 469 471 471 472 474 476 477 479 481 481 482 483

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Can IRL Surpass the Teacher? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 IRL from Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Learning Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 T-REX: Trajectory-Ranked Reward EXtrapolation . . . . . . . . . . . 8.4 D-REX: Disturbance-Based Reward EXtrapolation . . . . . . . . . . 9 Let Us Try It Out: IRL for Financial Cliff Walking . . . . . . . . . . . . . . . . . . 9.1 Max-Causal Entropy IRL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 IRL from Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 T-REX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Financial Applications of IRL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Algorithmic Trading Strategy Identification. . . . . . . . . . . . . . . . . . . 10.2 Inverse Reinforcement Learning for Option Pricing . . . . . . . . . . 10.3 IRL of a Portfolio Investor with G-Learning . . . . . . . . . . . . . . . . . . 10.4 IRL and Reward Learning for Sentiment-Based Trading Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 IRL and the “Invisible Hand” Inference . . . . . . . . . . . . . . . . . . . . . . . 11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

484 485 487 488 490 490 491 492 493 494 495 495 497 499

Frontiers of Machine Learning and Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Market Dynamics, IRL, and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 “Quantum Equilibrium–Disequilibrium” (QED) Model . . . . . . 2.2 The Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The GBM Model as the Langevin Equation . . . . . . . . . . . . . . . . . . . 2.4 The QED Model as the Langevin Equation . . . . . . . . . . . . . . . . . . . 2.5 Insights for Financial Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Insights for Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Physics and Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Hierarchical Representations in Deep Learning and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Tensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bounded-Rational Agents in a Non-equilibrium Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A “Grand Unification” of Machine Learning? . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Perception-Action Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Information Theory Meets Reinforcement Learning. . . . . . . . . . 4.3 Reinforcement Learning Meets Supervised Learning: Predictron, MuZero, and Other New Ideas . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

519 519 521 522 523 524 525 527 528 529

504 505 512 513 515

529 530 534 535 537 538 539 540

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

About the Authors

Matthew F. Dixon is an Assistant Professor of Applied Math at the Illinois Institute of Technology. His research in computational methods for finance is funded by Intel. Matthew began his career in structured credit trading at Lehman Brothers in London before pursuing academics and consulting for financial institutions in quantitative trading and risk modeling. He holds a Ph.D. in Applied Mathematics from Imperial College (2007) and has held postdoctoral and visiting professor appointments at Stanford University and UC Davis, respectively. He has published over 20 peer-reviewed publications on machine learning and financial modeling, has been cited in Bloomberg Markets and the Financial Times as an AI in fintech expert, and is a frequently invited speaker in Silicon Valley and on Wall Street. He has published R packages, served as a Google Summer of Code mentor, and is the co-founder of the Thalesians Ltd. Igor Halperin is a Research Professor in Financial Engineering at NYU and an AI Research Associate at Fidelity Investments. He was previously an Executive Director of Quantitative Research at JPMorgan for nearly 15 years. Igor holds a Ph.D. in Theoretical Physics from Tel Aviv University (1994). Prior to joining the financial industry, he held postdoctoral positions in theoretical physics at the Technion and the University of British Columbia. Paul Bilokon is CEO and Founder of Thalesians Ltd. and an expert in electronic and algorithmic trading across multiple asset classes, having helped build such businesses at Deutsche Bank and Citigroup. Before focusing on electronic trading, Paul worked on derivatives and has served in quantitative roles at Nomura, Lehman Brothers, and Morgan Stanley. Paul has been educated at Christ Church College, Oxford, and Imperial College. Apart from mathematical and computational finance, his academic interests include machine learning and mathematical logic.

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Part I

Machine Learning with Cross-Sectional Data

Chapter 1

Introduction

This chapter introduces the industry context for machine learning in finance, discussing the critical events that have shaped the finance industry’s need for machine learning and the unique barriers to adoption. The finance industry has adopted machine learning to varying degrees of sophistication. How it has been adopted is heavily fragmented by the academic disciplines underpinning the applications. We view some key mathematical examples that demonstrate the nature of machine learning and how it is used in practice, with the focus on building intuition for more technical expositions in later chapters. In particular, we begin to address many finance practitioner’s concerns that neural networks are a “black-box” by showing how they are related to existing well-established techniques such as linear regression, logistic regression, and autoregressive time series models. Such arguments are developed further in later chapters. This chapter also introduces reinforcement learning for finance and is followed by more in-depth case studies highlighting the design concepts and practical challenges of applying machine learning in practice.

1 Background In 1955, John McCarthy, then a young Assistant Professor of Mathematics, at Dartmouth College in Hanover, New Hampshire, submitted a proposal with Marvin Minsky, Nathaniel Rochester, and Claude Shannon for the Dartmouth Summer Research Project on Artificial Intelligence (McCarthy et al. 1955). These organizers were joined in the summer of 1956 by Trenchard More, Oliver Selfridge, Herbert Simon, Ray Solomonoff, among others. The stated goal was ambitious: “The study is to proceed on the basis of the conjecture that every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it. An attempt will be made to © Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_1

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find how to make machines use language, form abstractions and concepts, solve kinds of problems now reserved for humans, and improve themselves.” Thus the field of artificial intelligence, or AI, was born. Since this time, AI has perpetually strived to outperform humans on various judgment tasks (Pinar Saygin et al. 2000). The most fundamental metric for this success is the Turing test—a test of a machine’s ability to exhibit intelligent behavior equivalent to, or indistinguishable from, that of a human (Turing 1995). In recent years, a pattern of success in AI has emerged—one in which machines outperform in the presence of a large number of decision variables, usually with the best solution being found through evaluating an exponential number of candidates in a constrained high-dimensional space. Deep learning models, in particular, have proven remarkably successful in a wide field of applications (DeepMind 2016; Kubota 2017; Esteva et al. 2017) including image processing (Simonyan and Zisserman 2014), learning in games (DeepMind 2017), neuroscience (Poggio 2016), energy conservation (DeepMind 2016), skin cancer diagnostics (Kubota 2017; Esteva et al. 2017). One popular account of this reasoning points to humans’ perceived inability to process large amounts of information and make decisions beyond a few key variables. But this view, even if fractionally representative of the field, does no justice to AI or human learning. Humans are not being replaced any time soon. The median estimate for human intelligence in terms of gigaflops is about 104 times more than the machine that ran alpha-go. Of course, this figure is caveated on the important question of whether the human mind is even a Turing machine.

1.1 Big Data—Big Compute in Finance The growth of machine-readable data to record and communicate activities throughout the financial system combined with persistent growth in computing power and storage capacity has significant implications for every corner of financial modeling. Since the financial crises of 2007–2008, regulatory supervisors have reoriented towards “data-driven” regulation, a prominent example of which is the collection and analysis of detailed contractual terms for the bank loan and trading book stresstesting programs in the USA and Europe, instigated by the crisis (Flood et al. 2016). “Alternative data”—which refers to data and information outside of the usual scope of securities pricing, company fundamentals, or macroeconomic indicators— is playing an increasingly important role for asset managers, traders, and decision makers. Social media is now ranked as one of the top categories of alternative data currently used by hedge funds. Trading firms are hiring experts in machine learning with the ability to apply natural language processing (NLP) to financial news and other unstructured documents such as earnings announcement reports and SEC 10K reports. Data vendors such as Bloomberg, Thomson Reuters, and RavenPack are providing processed news sentiment data tailored for systematic trading models.

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In de Prado (2019), some of the properties of these new, alternative datasets are explored: (a) many of these datasets are unstructured, non-numerical, and/or noncategorical, like news articles, voice recordings, or satellite images; (b) they tend to be high-dimensional (e.g., credit card transactions) and the number of variables may greatly exceed the number of observations; (c) such datasets are often sparse, containing NaNs (not-a-numbers); (d) they may implicitly contain information about networks of agents. Furthermore, de Prado (2019) explains why classical econometric methods fail on such datasets. These methods are often based on linear algebra, which fail when the number of variables exceeds the number of observations. Geometric objects, such as covariance matrices, fail to recognize the topological relationships that characterize networks. On the other hand, machine learning techniques offer the numerical power and functional flexibility needed to identify complex patterns in a high-dimensional space offering a significant improvement over econometric methods. The “black-box” view of ML is dismissed in de Prado (2019) as a misconception. Recent advances in ML make it applicable to the evaluation of plausibility of scientific theories; determination of the relative informational variables (usually referred to as features in ML) for explanatory and/or predictive purposes; causal inference; and visualization of large, high-dimensional, complex datasets. Advances in ML remedy the shortcomings of econometric methods in goal setting, outlier detection, feature extraction, regression, and classification when it comes to modern, complex alternative datasets. For example, in the presence of p features there may be up to 2p − p − 1 multiplicative interaction effects. For two features there is only one such interaction effect, x1 x2 . For three features, there are x1 x2 , x1 x3 , x2 x3 , x1 x2 x3 . For as few as ten features, there are 1,013 multiplicative interaction effects. Unlike ML algorithms, econometric models do not “learn” the structure of the data. The model specification may easily miss some of the interaction effects. The consequences of missing an interaction effect, e.g. fitting yt = x1,t + x2,t + t instead of yt = x1,t + x2,t + x1,t x2,t + t , can be dramatic. A machine learning algorithm, such as a decision tree, will recursively partition a dataset with complex patterns into subsets with simple patterns, which can then be fit independently with simple linear specifications. Unlike the classical linear regression, this algorithm “learns” about the existence of the x1,t x2,t effect, yielding much better out-of-sample results. There is a draw towards more empirically driven modeling in asset pricing research—using ever richer sets of firm characteristics and “factors” to describe and understand differences in expected returns across assets and model the dynamics of the aggregate market equity risk premium (Gu et al. 2018). For example, Harvey et al. (2016) study 316 “factors,” which include firm characteristics and common factors, for describing stock return behavior. Measurement of an asset’s risk premium is fundamentally a problem of prediction—the risk premium is the conditional expectation of a future realized excess return. Methodologies that can reliably attribute excess returns to tradable anomalies are highly prized. Machine learning provides a non-linear empirical approach for modeling realized security

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1 Introduction

returns from firm characteristics. Dixon and Polson (2019) review the formulation of asset pricing models for measuring asset risk premia and cast neural networks in canonical asset pricing frameworks.

1.2 Fintech The rise of data and machine learning has led to a “fintech” industry, covering digital innovations and technology-enabled business model innovations in the financial sector (Philippon 2016). Examples of innovations that are central to fintech today include cryptocurrencies and the blockchain, new digital advisory and trading systems, peer-to-peer lending, equity crowdfunding, and mobile payment systems. Behavioral prediction is often a critical aspect of product design and risk management needed for consumer-facing business models; consumers or economic agents are presented with well-defined choices but have unknown economic needs and limitations, and in many cases do not behave in a strictly economically rational fashion. Therefore it is necessary to treat parts of the system as a black-box that operates under rules that cannot be known in advance.

1.2.1

Robo-Advisors

Robo-advisors are financial advisors that provide financial advice or portfolio management services with minimal human intervention. The focus has been on portfolio management rather than on estate and retirement planning, although there are exceptions, such as Blooom. Some limit investors to the ETFs selected by the service, others are more flexible. Examples include Betterment, Wealthfront, WiseBanyan, FutureAdvisor (working with Fidelity and TD Ameritrade), Blooom, Motif Investing, and Personal Capital. The degree of sophistication and the utilization of machine learning are on the rise among robo-advisors.

1.2.2

Fraud Detection

In 2011 fraud cost the financial industry approximately $80 billion annually (Consumer Reports, June 2011). According to PwC’s Global Economic Crime Survey 2016, 46% of respondents in the Financial Services industry reported being victims of economic crime in the last 24 months—a small increase from 45% reported in 2014. 16% of those that reported experiencing economic crime had suffered more than 100 incidents, with 6% suffering more than 1,000. According to the survey, the top 5 types of economic crime are asset misappropriation (60%, down from 67% in 2014), cybercrime (49%, up from 39% in 2014), bribery and corruption (18%, down from 20% in 2014), money laundering (24%, as in 2014), and accounting fraud (18%, down from 21% in 2014). Detecting economic crimes is

1 Background

7

one of the oldest successful applications of machine learning in the financial services industry. See Gottlieb et al. (2006) for a straightforward overview of some of the classical methods: logistic regression, naïve Bayes, and support vector machines. The rise of electronic trading has led to new kinds of financial fraud and market manipulation. Some exchanges are investigating the use of deep learning to counter spoofing.

1.2.3

Cryptocurrencies

Blockchain technology, first implemented by Satoshi Nakamoto in 2009 as a core component of Bitcoin, is a distributed public ledger recording transactions. Its usage allows secure peer-to-peer communication by linking blocks containing hash pointers to a previous block, a timestamp, and transaction data. Bitcoin is a decentralized digital currency (cryptocurrency) which leverages the blockchain to store transactions in a distributed manner in order to mitigate against flaws in the financial industry. In contrast to existing financial networks, blockchain based cryptocurrencies expose the entire transaction graph to the public. This openness allows, for example, the most significant agents to be immediately located (pseudonymously) on the network. By processing all financial interactions, we can model the network with a high-fidelity graph, as illustrated in Fig. 1.1 so that it is possible to characterize how the flow of information in the network evolves over time. This novel data representation permits a new form of financial econometrics—with the emphasis on the topological network structures in the microstructure rather than solely the covariance of historical time series of prices. The role of users, entities, and their interactions in formation and dynamics of cryptocurrency risk investment, financial predictive analytics and, more generally, in re-shaping the modern financial world is a novel area of research (Dyhrberg 2016; Gomber et al. 2017; Sovbetov 2018).

a1 a2

t1

a6

a9 t3

a7

a10 a11 a12

a3 a4 a5

t2

t4

a8

a13

Time

Fig. 1.1 A transaction–address graph representation of the Bitcoin network. Addresses are represented by circles, transactions with rectangles, and edges indicate a transfer of coins. Blocks order transactions in time, whereas each transaction with its input and output nodes represents an immutable decision that is encoded as a subgraph on the Bitcoin network. Source: Akcora et al. (2018)

8

1 Introduction

2 Machine Learning and Prediction With each passing year, finance becomes increasingly reliant on computational methods. At the same time, the growth of machine-readable data to monitor, record, and communicate activities throughout the financial system has significant implications for how we approach the topic of modeling. One of the reasons that AI and the set of computer algorithms for learning, referred to as “machine learning,” have been successful is a result of a number of factors beyond computer hardware and software advances. Machines are able to model complex and high-dimensional data generation processes, sweep through millions of model configurations, and then robustly evaluate and correct the models in response to new information (Dhar 2013). By continuously updating and hosting a number of competing models, they prevent any one model leading us into a data gathering silo effective only for that market view. Structurally, the adoption of ML has even shifted our behavior—the way we reason, experiment, and shape our perspectives from data using ML has led to empirically driven trading and investment decision processes. Machine learning is a broad area, covering various classes of algorithms for pattern recognition and decision-making. In supervised learning, we are given labeled data, i.e. pairs (x1 , y1 ), . . . , (xn , yn ), x1 , . . . , xn ∈ X, y1 , . . . , yn ∈ Y , and the goal is to learn the relationship between X and Y . Each observation xi is referred to as a feature vector and yi is the label or response. In unsupervised learning, we are given unlabeled data, x1 , x2 , . . . , xn and our goal is to retrieve exploratory information about the data, perhaps grouping similar observations or capturing some hidden patterns. Unsupervised learning includes cluster analysis algorithms such as hierarchical clustering, k-means clustering, self-organizing maps, Gaussian mixture, and hidden Markov models and is commonly referred to as data mining. In both instances, the data could be financial time series, news documents, SEC documents, and textual information on important events. The third type of machine learning paradigm is reinforcement learning and is an algorithmic approach for enforcing Bellman optimality of a Markov Decision Process—defining a set of states and actions in response to a changing regime so as to maximize some notion of cumulative reward. In contrast to supervised learning, which just considers a single action at each point in time, reinforcement learning is concerned with the optimal sequence of actions. It is therefore a form of dynamic programming that is used for decisions leading to optimal trade execution, portfolio allocation, and liquidation over a given horizon. Supervised learning addresses a fundamental prediction problem: Construct a non-linear predictor, Yˆ (X), of an output, Y , given a high-dimensional input matrix X = (X1 , . . . , XP ) of P variables. Machine learning can be simply viewed as the study and construction of an input–output map of the form Y = F (X) where X = (X1 , . . . , XP ). F (X) is sometimes referred to as the “data-feature” map. The output variable, Y , can be continuous, discrete, or mixed. For example, in a classification problem,

2 Machine Learning and Prediction

9

F : X → G, where G ∈ K := {0, . . . , K − 1}, K is the number of categories and ˆ is the predictor. G Supervised machine learning uses a parameterized1 model g(X|θ ) over independent variables X, to predict the continuous or categorical output Y or G. The model is parameterized by one or more free parameters θ which are fitted to data. Prediction of categorical variables is referred to as classification and is common in pattern recognition. The most common approach to predicting categorical variables is to encode the response G as one or more binary values, then treat the model prediction as continuous.

•

? Multiple Choice Question 1

Select all the following correct statements: 1. Supervised learning involves learning the relationship between input and output variables. 2. Supervised learning requires a human supervisor to prepare labeled training data. 3. Unsupervised learning does not require a human supervisor and is therefore superior to supervised learning. 4. Reinforcement learning can be viewed as a generalization of supervised learning to Markov Decision Processes.

There are two different classes of supervised learning models, discriminative and generative. A discriminative model learns the decision boundary between the classes and implicitly learns the distribution of the output conditional on the input. A generative model explicitly learns the joint distribution of the input and output. An example of the former is a neural network or a decision tree and a restricted Boltzmann machine (RBM) is an example of the latter. Learning the joint distribution has the advantage that by the Bayes’ rule, it can also give the conditional distribution of the output given the input, but also be used for other purposes such as selecting features based on the joint probability. Generative models are typically more difficult to build. This book will mostly focus on discriminative models only, but the distinction should be made clear. A discriminative model predicts the probability of an output given an input. For example, if we are predicting the probability of a label G = k, k ∈ K, then g(x|θ) is a map g : Rp → [0, 1]K and the outputs represent a discrete probability distribution over G referred to as a “one-hot” encoding—a Kvector of zeros with 1 at the kth position: ˆ k := P(G = k | X = x, θ ) = gk (x|θ) G

(1.1)

1 The model is referred to as non-parametric if the parameter space is infinite dimensional and parametric if the parameter space is finite dimensional.

10

1 Introduction

and hence we have that

gk (x|θ) = 1.

(1.2)

k∈K

In particular, when G is dichotomous (K = 2), the second component of the model output is the conditional expected value of G ˆ := G ˆ 1 =g1 (x|θ)=0·P(G = 0 | X=x, θ )+1·P(G = 1 | X=x, θ )=E[G | X=x, θ ]. G (1.3) The conditional variance of G is given by ˆ 2 | X = x, θ ] = g1 (x|θ ) − (g1 (x|θ))2 , σ 2 := E[(G − G)

(1.4)

which is an inverted parabola with a maximum at g1 (x|θ ) = 0.5. The following example illustrates a simple discriminative model which, here, is just based on a set of fixed rules for partitioning the input space. Example 1.1

Model Selection

Suppose G ∈ {A, B, C} and the input X ∈ {0, 1}2 are binary 2-vectors given in Table 1.1. Table 1.1 Sample model data

G A B C C

x (0, 1) (1, 1) (1, 0) (0, 0)

To match the input and output in this case, one could define a parameter-free step function g(x) over {0, 1}2 so that ⎧ ⎪ {1, 0, 0} ⎪ ⎪ ⎪ ⎨{0, 1, 0} g(x) = ⎪ {0, 0, 1} ⎪ ⎪ ⎪ ⎩ {0, 0, 1}

if x = (0, 1) if x = (1, 1) if x = (1, 0)

(1.5)

if x = (0, 0).

The discriminative model g(x), defined in Eq. 1.5, specifies a set of fixed rules which predict the outcome of this experiment with 100% accuracy. Intuitively, it seems clear that such a model is flawed if the actual relation between inputs and outputs is non-deterministic. Clearly, a skilled analyst would typically not build such

2 Machine Learning and Prediction

11

a model. Yet, hard-wired rules such as this are ubiquitous in the finance industry such as rule-based technical analysis and heuristics used for scoring such as credit ratings. If the model is allowed to be general, there is no reason why this particular function should be excluded. Therefore automated systems analyzing datasets such as this may be prone to construct functions like those given in Eq. 1.5 unless measures are taken to prevent it. It is therefore incumbent on the model designer to understand what makes the rules in Eq. 1.5 objectionable, with the goal of using a theoretically sound process to generalize the input–output map to other data. Example 1.2

Model Selection (Continued)

Consider an alternate model for Table 1.1 ⎧ ⎪ {0.9, 0.05, 0.05} ⎪ ⎪ ⎪ ⎨{0.05, 0.9, 0.05} h(x) = ⎪ {0.05, 0.05, 0.9} ⎪ ⎪ ⎪ ⎩ {0.05, 0.05, 0.9}

if x = (0, 1) if x = (1, 1) if x = (1, 0) if x = (0, 0).

If this model were sampled, it would produce the data in Table 1.1 with probability (0.9)4 = 0.6561. We can hardly exclude this model from consideration on the basis of the results in Table 1.1, so which one do we choose? Informally, the heart of the model selection problem is that model g has excessively high confidence about the data, when that confidence is often not warranted. Many other functions, such as h, could have easily generated Table 1.1. Though there is only one model that can produce Table 1.1 with probability 1.0, there is a whole family of models that can produce the table with probability at least 0.66. Many of these plausible models do not assign overwhelming confidence to the results. To determine which model is best on average, we need to introduce another key concept.

2.1 Entropy Model selection in machine learning is based on a quantity known as entropy. Entropy represents the amount of information associated with each event. To illustrate the concept of entropy, let us consider a non-fair coin toss. There are two outcomes, = {H, T }. Let Y be a Bernoulli random variable representing the coin flip with density f (Y = 1) = P(H ) = p and f (Y = 0) = P(T ) = 1 − p. The (binary) entropy of Y under f is

12

1 Introduction

Fig. 1.2 (Left) This figure shows the binary entropy of a biased coin. If the coin is fully biased, then each flip provides no new information as the outcome is already known and hence the entropy is zero. (Right) The concept of entropy was introduced by Claude Shannon2 in 19483 and was originally intended to represent an upper limit on the average length of a lossless compression encoding. Shannon’s entropy is foundational to the mathematical discipline of information theory

H(f ) = −p log2 p − (1 − p)log2 (1 − p) ≤ 1bit.

(1.6)

The reason why base 2 is chosen is so that the upper bound represents the number of bits needed to represent the outcome of the random variable, i.e. {0, 1} and hence 1 bit. The binary entropy for a biased coin is shown in Fig. 1.2. If the coin is fully biased, then each flip provides no new information as the outcome is already known. The maximum amount of information that can be revealed by a coin flip is when the coin is unbiased. Let us now reintroduce our parameterized mass in the setting of the biased coin. Let us consider an i.i.d. discrete random variable Y : → Y ⊂ R and let g(y|θ ) = P(ω ∈ ; Y (ω) = y) denote a parameterized probability mass function for Y . We can measure how different g(y|θ ) is from the true density f (y) using the cross-entropy H(f, g) := −Ef log2 g = f (y) log2 g(y|θ ) ≥ H(f ),

(1.7)

y∈Y

2 Photo:

Jacobs, Konrad [CC BY-SA 2.0 de (https://creativecommons.org/licenses/by-sa/2.0/de/ deed.en)]. 3 C. Shannon, A Mathematical Theory of Communication, The Bell System Technical Journal, Vol. 27, pp. 379-423, 623-656, July, October, 1948.

2 Machine Learning and Prediction

13

Fig. 1.3 A comparison of the true distribution, f , of a biased coin with a parameterized model g of the coin

so that H(f, f ) = H(f ), where H(f ) is the entropy of f : H(f ) := −Ef [log2 f ] = −

f (y)log2 f (y).

(1.8)

y∈Y

If g(y|θ ) is a model of the non-fair coin with g(Y = 1|θ ) = pθ , g(Y = 0|θ ) = 1 − pθ . The cross-entropy is H(f, g) = −p log2 pθ − (1 − p)log2 (1 − pθ ) ≥ −p log2 p − (1 − p)log2 (1 − p). (1.9) Let us suppose that p = 0.7 and pθ = 0.68, as illustrated in Fig. 1.3, then the cross-entropy is H(f, g) = −0.3 log2 (0.32) − 0.7 log2 (0.68) = 0.8826322. Returning to our experiment in Table 1.1, let us consider the cross-entropy of these models which, as you will recall, depends on inputs too. Model g completely characterizes the data in Table 1.1 and we interpret it here as the truth. Model h, however, only summarizes some salient aspects of the data, and there is a large family of tables that would be consistent with model h. In the presence of noise or strong evidence indicating that Table 1.1 was the only possible outcome, we should interpret models like h as a more plausible explanation of the actual underlying phenomenon. Evaluating the cross-entropy between model h and model g, we get − log2 (0.9) for each observation in the table, which gives the negative log-likelihood when summed over all samples. The cross-entropy is at its minimum when h = g, we get − log2 (1.0) = 0. If g were a parameterized model, then clearly minimizing crossentropy or equivalently maximizing log-likelihood gives the maximum likelihood estimate of the parameter. We shall revisit the topic of parameter estimation in Chap. 2.

14

1 Introduction

•

? Multiple Choice Question 2

Select all of the following statements that are correct: 1. Neural network classifiers are a discriminative model which output probabilistic weightings for each category, given an input feature vector. 2. If the data is independent and identically distributed (i.i.d.), then the output of a dichotomous classifier is a conditional probability of a Bernoulli random variable. 3. A θ -parameterized discriminative model for a biased coin dependent on the environment X can be written as {gi (X|θ )}1i=0 . 4. A model of two biased coins, both dependent on the environment X, can be equivalently modeled with either the pair {gi(1) (X|θ )}1i=0 and {gi(2) (X|θ )}1i=0 , or the multi-classifier {gi (X|θ )}3i=0 .

2.2 Neural Networks Neural networks represent the non-linear map F (X) over a high-dimensional input space using hierarchical layers of abstractions. An example of a neural network is a feedforward network—a sequence of L layers4 formed via composition:

•

> Deep Feedforward Networks

A deep feedforward network is a function of the form

(L) (1) Yˆ (X) := FW,b (X) = fW (L) ,b(L) . . . ◦ fW (1) ,b(1) (X), where • fW(l)(l) ,b(l) (X) := σ (l) (W (l) X +b(l) ) is a semi-affine function, where σ (l) is a univariate and continuous non-linear activation function such as max(·, 0) or tanh(·). • W = (W (1) , . . . , W (L) ) and b = (b(1) , . . . , b(L) ) are weight matrices and offsets (a.k.a. biases), respectively.

4 Note that we do

not treat the input as a layer. So there are L − 1 hidden layers and an output layer.

2 Machine Learning and Prediction

(a)

15

(b)

(c)

Fig. 1.4 Examples of neural networks architectures discussed in this book. Source: Van Veen, F. & Leijnen, S. (2019), “The Neural Network Zoo,” Retrieved from https://www.asimovinstitute. org/neural-network-zoo. The input nodes are shown in yellow and represent the input variables, the green nodes are the hidden neurons and present hidden latent variables, the red nodes are the outputs or responses. Blue nodes denote hidden nodes with recurrence or memory. (a) Feedforward. (b) Recurrent. (c) Long short-term memory

An earlier example of a feedforward network architecture is given in Fig. 1.4a. The input nodes are shown in yellow and represent the input variables, the green nodes are the hidden neurons and present hidden latent variables, the red nodes are the outputs or responses. The activation functions are essential for the network to approximate non-linear functions. For example, if there is one hidden layer and σ (1) is the identify function, then Yˆ (X) = W (2) (W (1) X + b(1) ) + b(2) = W (2) W (1) X + W (2) b(1) + b(2) = W X + b (1.10) is just linear regression, i.e. an affine transformation.5 Clearly, if there are no hidden layers, the architecture recovers standard linear regression Y = WX + b and logistic regression φ(W X + b), where φ is a sigmoid or softmax function, when the response is continuous or categorical, respectively. Some of the terminology used here and the details of this model will be described in Chap. 4. The theoretical roots of feedforward neural networks are given by the Kolmogorov–Arnold representation theorem (Arnold 1957; Kolmogorov 1957) of multivariate functions. Remarkably, Hornik et al. (1989) showed how neural networks, with one hidden layer, are universal approximators to non-linear functions. Clearly there are a number of issues in any architecture design and inference of the model parameters (W, b). How many layers? How many neurons Nl in each hidden layer? How to perform “variable selection”? How to avoid over-fitting? The details and considerations given to these important questions will be addressed in Chap. 4.

5 While

the functional form of the map is the same as linear regression, neural networks do not assume a data generation process and hence inference is not identical to ordinary least squares regression.

16

1 Introduction

3 Statistical Modeling vs. Machine Learning Supervised machine learning is often an algorithmic form of statistical model estimation in which the data generation process is treated as an unknown (Breiman 2001). Model selection and inference is automated, with an emphasis on processing large amounts of data to develop robust models. It can be viewed as a highly efficient data compression technique designed to provide predictors in complex settings where relations between input and output variables are non-linear and input space is often high-dimensional. Machine learners balance filtering data with the goal of making accurate and robust decisions, often discrete and as a categorical function of input data. This fundamentally differs from maximum likelihood estimators used in standard statistical models, which assume that the data was generated by the model and typically have difficulty with over-fitting, especially when applied to high-dimensional datasets. Given the complexity of modern datasets, whether they are limit order books or high-dimensional financial time series, it is increasingly questionable whether we can posit inference on the basis of a known data generation process. It is a reasonable assertion, even if an economic interpretation of the data generation process can be given, that the exact form cannot be known all the time. The paradigm that machine learning provides for data analysis therefore is very different from the traditional statistical modeling and testing framework. Traditional fit metrics, such as R 2 , t-values, p-values, and the notion of statistical significance, are replaced by out-of-sample forecasting and understanding the bias–variance tradeoff. Machine learning is data-driven and focuses on finding structure in large datasets. The main tools for variable or predictor selection are regularization and dropout which are discussed in detail in Chap. 4. Table 1.2 contrasts maximum likelihood estimation-based inference with supervised machine learning. The comparison is somewhat exaggerated for ease of explanation. Rather the two approaches should be viewed as opposite ends of a continuum of methods. Linear regression techniques such as LASSO and ridge regression, or hybrids such as Elastic Net, fall somewhere in the middle, providing some combination of the explanatory power of maximum likelihood estimation while retaining out-of-sample predictive performance on high-dimensional datasets.

3.1 Modeling Paradigms Machine learning and statistical methods can be further characterized by whether they are parametric or non-parametric. Parametric models assume some finite set of parameters and attempt to model the response as a function of the input variables and the parameters. Due to the finiteness of the parameter space, they have limited flexibility and cannot capture complex patterns in big data. As a general rule, examples of parametric models include ordinary least squares linear

3 Statistical Modeling vs. Machine Learning

17

Table 1.2 This table contrasts maximum likelihood estimation-based inference with supervised machine learning. The comparison is somewhat exaggerated for ease of explanation; however, the two should be viewed as opposite ends of a continuum of methods. Regularized linear regression techniques such as LASSO and ridge regression, or hybrids such as Elastic Net, provide some combination of the explanatory power of maximum likelihood estimation while retaining out-ofsample predictive performance on high-dimensional datasets Property Goal Data Framework Expressibility Model selection Scalability Robustness Diagnostics

Statistical inference Causal models with explanatory power The data is generated by a model Probabilistic Typically linear Based on information criteria Limited to lower-dimensional data Prone to over-fitting Extensive

Supervised machine learning Prediction performance, often with limited explanatory power The data generation process is unknown Algorithmic and Probabilistic Non-linear Numerical optimization Scales to high-dimensional input data Designed for out-of-sample performance Limited

regression, polynomial regression, mixture models, neural networks, and hidden Markov models. Non-parametric models treat the parameter space as infinite dimensional—this is equivalent to introducing a hidden or latent function. The model structure is, for the most part, not specified a priori and they can grow in complexity with more data. Examples of non-parametric models include kernel methods such as support vector machines and Gaussian processes, the latter will be the focus of Chap. 3. Note that there is a gray area in whether neural networks are parametric or non-parametric and it strictly depends on how they are fitted. For example, it is possible to treat the parameter space in a neural network as infinite dimensional and hence characterize neural networks as non-parametric (see, for example, Philipp and Carbonell (2017)). However, this is an exception rather than the norm. While on the topic of modeling paradigms, it is helpful to further distinguish between probabilistic models, the subject of the next two chapters, and deterministic models, the subject of Chaps. 4, 5, and 8. The former treats the parameters as random and the latter assumes that the parameters are given. Within probabilistic modeling, a particular niche is occupied by the so-called state-space models. In these models one assumes the existence of a certain unobserved, latent, process, whose evolution drives a certain observable process. The evolution of the latent process and the dependence of the observable process on the latent process may be given in stochastic, probabilistic terms, which places the state-space models within the realm of probabilistic modeling. Note, somewhat counter to the terminology, that a deterministic model may produce a probabilistic output, for example, a logistic regression gives the probability that the response is positive given the input variables. The choice of whether to use a probabilistic or deterministic model is discussed further in the next chapter

18

1 Introduction

and falls under the more general and divisive topic of “Bayesian versus frequentist modeling.”

3.2 Financial Econometrics and Machine Learning Machine learning generalizes parametric methods in financial econometrics. A taxonomy of machine learning in econometrics in shown in Fig. 1.5 together with the section references to the material in the first two parts of this book. When the data is a time series, neural networks can be configured with recurrence to build memory into the model. By relaxing the modeling assumptions needed for econometrics techniques, such as ARIMA (Box et al. 1994) and GARCH models (Bollerslev 1986), recurrent neural networks provide a semi-parametric or even non-parametric extension of classical time series methods. That use, however, comes with much caution. Whereas financial econometrics is built on rigorous experimental design methods such as the estimation framework of Box and Jenkins (1976), recurrent neural networks have grown from the computational engineering literature and many engineering studies overlook essential diagnostics such as Dickey–Fuller tests for verifying stationarity of the time series, a critical aspect of financial time series modeling. We take an integrative approach, showing how to cast recurrent neural networks into financial econometrics frameworks such as Box-Jenkins. More formally, if the input–output pairs D = {Xt , Yt }N t=1 are autocorrelated observations of X and Y at times t = 1, . . . , N , then the fundamental prediction problem can be expressed as a sequence prediction problem: construct a non-linear

Fig. 1.5 Overview of how machine learning generalizes parametric econometrics, together with the section references to the material in the first two parts of this book

3 Statistical Modeling vs. Machine Learning

19

times series predictor, Yˆ (Xt ), of an output, Y , using a high-dimensional input matrix of T length sub-sequences Xt : Yˆt = F (Xt ) where Xt := seqT ,0 (Xt ) := (Xt−T +1 , . . . , Xt ),

(1.11)

where Xt−j is a j th lagged observation of Xt , Xt−j = Lj [Xj ], for 0 = 1, . . . , T − 1. Sequence learning, then, is just a composition of a non-linear map and a vectorization of the lagged input variables. If the data is i.i.d., then no sequence is needed (i.e., T = 1), and we recover the standard cross-sectional prediction problem which can be approximated with a feedforward neural network model. Recurrent neural networks (RNNs), shown in Fig. 1.4b, are time series methods or sequence learners which have achieved much success in applications such as natural language understanding, language generation, video processing, and many other tasks (Graves 2012). There are many types of RNNs—we will just concentrate on simple RNN models for brevity of notation. Like multivariate structural (1) autoregressive models, RNNs apply an autoregressive function fW (1) ,b(1) (Xt ) to each input sequence Xt , where T denotes the look back period at each time step— the maximum number of lags. However, rather than directly imposing a linear autocovariance structure, a RNN provides a flexible functional form to directly model the predictor, Yˆ . A simple RNN can be understood as an unfolding of a single hidden layer neural network (a.k.a. Elman network (Elman 1991)) over all time steps in the sequence, (1) j = 0, . . . , T . For each time step, j , this function fW (1) ,b(1) (Xt,j ) generates a hidden state Zt−j from the current input Xt−j and the previous hidden state Zt−j −1 and Xt,j = seqT ,j (Xt ) ⊂ Xt which appears in general form as: response:

(2) Yˆt =fW (2) ,b(2) (Zt ):=σ (2) (W (2) Zt +b(2) ), (1)

hidden states: Zt−j = fW (1) ,b(1) (Xt,j ) := σ (1) (Wz(1) Zt−j −1 + Wx(1) Xt−j + b(1) ), j ∈ {T , . . . , 0}, where σ (1) is an activation function such as tanh(x) and σ (2) is either a softmax function, sigmoid function, or identity function depending on whether the response is categorical, binary, or continuous, respectively. The connections between the extremal inputs Xt and the H hidden units are weighted by the time invariant matrix (1) Wx ∈ RH ×P . The recurrent connections between the H hidden units are weighted (1) by the time invariant matrix Wz ∈ RH ×H . Without such a matrix, the architecture is simply a single layered feedforward network without memory—each independent observation Xt is mapped to an output Yˆt using the same hidden layer. It is important to note that a plain RNN, de-facto, is not a deep network. The recurrent layer has the deceptive appearance of being a deep network when “unfolded,” i.e. viewed as being repeatedly applied to each new input, Xt−j , so (1) (1) that Zt−j = σ (1) (Wz Zt−j −1 + Wx Xt−j ). However the same recurrent weights

20

1 Introduction

remain fixed over all repetitions—there is only one recurrent layer with weights Wz(1) . The amount of memory in the model is equal to the sequence length T . This means that the maximum lagged input that affects the output, Yˆt , is Xt−T . We shall see later in Chap. 8 that RNNs are simply non-linear autoregressive models with exogenous variables (NARX). In the special case of the univariate time series p prediction Xˆ t = F (Xt−1 ), using T = p previous observations {Xt−i }i=1 , only one neuron in the recurrent layer with weight φ and no activation function, a RNN is an AR(p) model with zero drift and geometric weights: Xˆ t = (φ1 L + φ2 L2 + · · · + φp Lp )[Xt ], φi := φ i , with |φ| < 1 to ensure that the model is stationary. The order p can be found through autocorrelation tests of the residual if we make the additional assumption that the error Xt − Xˆ t is Gaussian. Example tests include the Ljung–Box and Lagrange multiplier tests. However, the over-reliance on parametric diagnostic tests should be used with caution since the conditions for satisfying the tests may not be satisfied on complex time series data. Because the weights are time independent, plain RNNs are static time series models and not suited to non-covariance stationary time series data. Additional layers can be added to create deep RNNs by stacking them on top of each other, using the hidden state of the RNN as the input to the next layer. However, RNNs have difficulty in learning long-term dynamics, due in part to the vanishing and exploding gradients that can result from propagating the gradients down through the many unfolded layers of the network. Moreover, RNNs like most methods in supervised machine learning are inherently designed for stationary data. Oftentimes, financial time series data is non-stationary. In Chap. 8, we shall introduce gated recurrent units (GRUs) and long short term memory (LSTM) networks, the latter is shown in Fig. 1.4c as a particular form of recurrent network which provide a solution to this problem by incorporating memory units. In the language of time series modeling, we shall construct dynamic RNNs which are suitable for non-stationary data. More precisely, we shall see that these architecture shall learn when to forget previous hidden states and when to update hidden states given new information. This ability to model hidden states is of central importance in financial time series modeling and applications in trading. Mixture models and hidden Markov models have historically been the primary probabilistic methods used in quantitative finance and econometrics to model regimes and are reviewed in Chap. 2 and Chap. 7 respectively. Readers are encouraged to review Chap. 2, before reading Chap. 7.

•

? Multiple Choice Question 3

Select all the following correct statements: 1. A linear recurrent neural network with a memory of p lags is an autoregressive model AR(p) with non-parametric error.

3 Statistical Modeling vs. Machine Learning

21

2. Recurrent neural networks, as time series models, are guaranteed to be stationary, for any choice of weights. 3. The amount of memory in a shallow recurrent network corresponds to the number of times a single perceptron layer is unfolded. 4. The amount of memory in a deep recurrent network corresponds to the number of perceptron layers.

3.3 Over-fitting Undoubtedly the pivotal concern with machine learning, and especially deep learning, is the propensity for over-fitting given the number of parameters in the model. This is why skill is needed to fit deep neural networks. In frequentist statistics, over-fitting is addressed by penalizing the likelihood function with a penalty term. A common approach is to select models based on Akaike’s information criteria (Akaike 1973), which assumes that the model error is Gaussian. The penalty term is in fact a sample bias correction term to the Kullback– Leibler divergence (the relative Entropy) and is applied post-hoc to the unpenalized maximized loss likelihood. Machine learning methods such as least absolute shrinkage and selection operator (LASSO) and ridge regression more conveniently directly optimize a loss function with a penalty term. Moreover the approach is not restricted to modeling error distributional assumptions. LASSO or L1 regularization favors sparser parameterizations, whereas ridge regression or L2 reduces the magnitude of the parameters. Regularization is arguably the most important aspect of why machine learning methods have been so successful in finance and other distributions. Conversely, its absence is why neural networks fell out-of-favor in the finance industry in the 1990s. Regularization and information criteria are closely related, a key observation which enables us to express model selection in terms of information entropy and hence root our discourse in the works of Shannon (1948), Wiener (1964), Kullback and Leibler (1951). How to choose weights, the concept of regularization for model selection, and cross-validation is discussed in Chap. 4. It turns out that the choice of priors in Bayesian modeling provides a probabilistic analog to LASSO and ridge regression. L2 regularization is equivalent to a Gaussian prior and L1 is an equivalent to a Laplacian prior. Another important feature of Bayesian models is that they have a natural mechanism for prevention of over-fitting built-in. Introductory Bayesian modeling is covered extensively in Chap. 2.

22

1 Introduction

4 Reinforcement Learning Recall that supervised learning is essentially a paradigm for inferring the parameters of a map between input data and an output through minimizing an error over training samples. Performance generalization is achieved through estimating regularization parameters on cross-validation data. Once the weights of a network are learned, they are not updated in response to new data. For this reason, supervised learning can be considered as an “offline” form of learning, i.e. the model is fitted offline. Note that we avoid referring to the model as static since it is possible, under certain types of architectures, to create a dynamical model in which the map between input and output changes over time. For example, as we shall see in Chap. 8, a LSTM maintains a set of hidden state variables which result in a different form of the map over time. In such learning, a “teacher” provides an exact right output for each data point in a training set. This can be viewed as “feedback” from the teacher, which for supervised learning amounts to informing the agent with the correct label each time the agent classifies a new data point in the training dataset. Note that this is opposite to unsupervised learning, where there is no teacher to provide correct answers to a ML algorithm, which can be viewed as a setting with no teacher, and, respectively, no feedback from a teacher. An alternative learning paradigm, referred to as “reinforcement learning,” exists which models a sequence of decisions over state space. The key difference of this setting from supervised learning is feedback from the teacher is somewhat in between of the two extremes of unsupervised learning (no feedback at all) and supervised learning that can be viewed as feedback by providing the right labels. Instead, such partial feedback is provided by “rewards” which encourage a desired behavior, but without explicitly instructing the agent what exactly it should do, as in supervised learning. The simplest way to reason about reinforcement learning is to consider machine learning tasks as a problem of an agent interacting with an environment, as illustrated in Fig. 1.6.

Fig. 1.6 This figure shows a reinforcement learning agent which performs actions at times t0 , . . . , tn . The agent perceives the environment through the state variable St . In order to perform better on its task, feedback on an action at is provided to the agent at the next time step in the form of a reward Rt

4 Reinforcement Learning

23

The agent learns about the environment in order to perform better on its task, which can be formulated as the problem of performing an optimal action. If an action performed by an agent is always the same and does not impact the environment, in this case we simply have a perception task, because learning about the environment helps to improve performance on this task. For example, you might have a model for prediction of mortgage defaults where the action is to compute the default probability for a given mortgage. The agent, in this case, is just a predictive model that produces a number and there is measurement of how the model impacts the environment. For example, if a model at a large mortgage broker predicted that all borrowers will default, it is very likely that this would have an impact on the mortgage market, and consequently future predictions. However, this feedback is ignored as the agent just performs perception tasks, ideally suited for supervised learning. Another example is in trading. Once an action is taken by the strategy there is feedback from the market which is referred to as “market impact.” Such a learner is configured to maximize a long-run utility function under some assumptions about the environment. One simple assumption is to treat the environment as being fully observable and evolving as a first-order Markov process. A Markov Decision Process (MDP) is then the simplest modeling framework that allows us to formalize the problem of reinforcement learning. A task solved by MDPs is the problem of optimal control, which is the problem of choosing action variables over some period of time, in order to maximize some objective function that depends both on the future states and action taken. In a discrete-time setting, the state of the environment St ∈ S is used by the learner (a.k.a. agent) to decide which action at ∈ A(St ) to take at each time step. This decision is made dynamic by updating the probabilities of selecting each action conditioned on St . These conditional probabilities πt (a|s) are referred to as the agent’s policy. The mechanism for updating the policy as a result of its learning is as follows: one time step later and as a consequence of its action, the learner receives a reward defined by a reward function, an immediate reward given the current state St and action taken at . As a result of the dynamic environment and the action of the agent, we transition to a new state St+1 . A reinforcement learning method specifies how to change the policy so as to maximize the total amount of reward received over the long-run. The constructs for reinforcement learning will be formalized in Chap. 9 but we shall informally discuss some of the challenges of reinforcement learning in finance here. Most of the impressive progress reported recently with reinforcement learning by researchers and companies such as Google’s DeepMind or OpenAI, such as playing video games, walking robots, self-driving cars, etc., assumes complete observability, using Markovian dynamics. The much more challenging problem, which is a better setting for finance, is how to formulate reinforcement learning for partially observable environments, where one or more variables are hidden. Another, more modest, challenge exists in how to choose the optimal policy when no environment is fully observable but the dynamic process for how the states evolve over time is unknown. It may be possible, for simple problems, to reason about how the states evolve, perhaps adding constraints on the state-action space. However,

24

1 Introduction

the problem is especially acute in high-dimensional discrete state spaces, arising from, say, discretizing continuous state spaces. Here, it is typically intractable to enumerate all combinations of states and actions and it is hence not possible to exactly solve the optimal control problem. Chapter 9 will present approaches for approximating the optimal control problem. In particular, we will turn to neural networks to approximate an action function known as a “Q-function.” Such an approach is referred to as “Q-Learning” and more recently, with the use of deep learning to approximate the Q-function, is referred to as “Deep Q-Learning.” To fix ideas, we consider a number of examples to illustrate different aspects of the problem formulation and challenge in applying reinforcement learning. We start with arguably the most famous toy problem used to study stochastic optimal control theory, the “multi-armed bandit problem.” This problem is especially helpful in developing our intuition of how an agent must balance the competing goals of exploring different actions versus exploitation of known outcomes. Example 1.3

Multi-armed Bandit Problem

Suppose there is a fixed and finite set of n actions, a.k.a. arms, denoted A. Learning proceeds in rounds, indexed by t = 1, . . . , T . The number of rounds T , a.k.a. the time horizon, is fixed and known in advance. In each round, the agent picks an arm at and observes the reward Rt (at ) for the chosen arm only. For avoidance of doubt, the agent does not observe rewards for other actions that could have been selected. If the goal is to maximize total reward over all rounds, how should the agent choose an arm? Suppose the rewards Rt are independent and identical random variables with distribution ν ∈ [0, 1]n and mean μ. The best action is then the distribution with the maximum mean μ∗ . The difference between the player’s accumulated reward and the maximum the player (a.k.a. the “cumulative regret”) could have obtained had she known all the parameters is R¯ T = T μ∗ − E

Rt .

t∈[T ]

Intuitively, an agent should pick arms that performed well in the past, yet the agent needs to ensure that no good option has been missed. The theoretical origins of reinforcement learning are in stochastic dynamic programming. In this setting, an agent must make a sequence of decisions under uncertainty about the reward. If we can characterize this uncertainty with probability distributions, then the problem is typically much easier to solve. We shall assume that the reader has some familiarity with dynamic programming—the extension to stochastic dynamic programming is a relatively minor conceptual development. Note that Chap. 9 will review pertinent aspects of dynamic programming, including

4 Reinforcement Learning

25

Bellman optimality. The following optimal payoff example will likely just serve as a simple review exercise in dynamic programming, albeit with uncertainty introduced into the problem. As we follow the mechanics of solving the problem, the example exposes the inherent difficulty of relaxing our assumptions about the distribution of the uncertainty. Example 1.4

Uncertain Payoffs

A strategy seeks to allocate $600 across 3 markets and is equally profitable once the position is held, returning 1% of the size of the position over a short trading horizon [t, t + 1]. However, the markets vary in liquidity and there is a lower probability that the larger orders will be filled over the horizon. The amount allocated to each market must be either K = {100, 200, 300}. Strategy M1

M2

M3

Allocation 100 200 300 100 200 300 100 200 300

Fill probability 0.8 0.7 0.6 0.75 0.7 0.65 0.75 0.75 0.6

Strategy M1

M2

M3

Allocation 100 200 300 100 200 300 100 200 300

Return 0.8 1.4 1.8 0.75 1.4 1.95 0.75 1.5 1.8

The optimal allocation problem under uncertainty is a stochastic dynamic programming problem. We can define value functions vi (x) for total allocation amount x for each stage of the problem, corresponding to the markets. We then find the optimal allocation using the backward recursive formulae: v3 (x) = R3 (x), ∀x ∈ K, v2 (x) = max{R2 (k) + v3 (x − k)}, ∀x ∈ K + 200, k∈K

v1 (x) = max{R1 (k) + v2 (x − k)}, x = 600, k∈K

The left-hand side of the table below tabulates the values of R2 + v3 corresponding to the second stage of the backward induction for each pair (M2 , M3 ). (continued)

26

1 Introduction

Example 1.4 R2 + v3 M3 100 200 300

(continued)

M2 100 1.5 2.25 2.55

200 2.15 2.9 3.2

300 2.7 3.45 3.75

M1 100 200 300

(M2∗ , M3∗ ) (300, 200) (200, 200) (100, 200)

v2 3.45 2.9 2.25

R1 0.8 1.4 1.8

R1 + v2 4.25 4.3 4.05

The right-hand side of the above table tabulates the values of R1 + v2 corresponding to the third and final stage of the backward induction for each tuple (M1 , M2∗ , M3∗ ). In the above example, we can see that the allocation {200, 200, 200} maximizes the reward to give v1 (600) = 4.3. While this exercise is a straightforward application of a Bellman optimality recurrence relation, it provides a glimpse of the types of stochastic dynamic programming problems that can be solved with reinforcement learning. In particular, if the fill probabilities are unknown but must be learned over time by observing the outcome over each period [ti , ti+1 ), then the problem above cannot be solved by just using backward recursion. Instead we will move to the framework of reinforcement learning and attempt to learn the best actions given the data. Clearly, in practice, the example is much too simple to be representative of real-world problems in finance—the profits will be unknown and the state space is significantly larger, compounding the need for reinforcement learning. However, it is often very useful to benchmark reinforcement learning on simple stochastic dynamic programming problems with closed-form solutions. In the previous example, we assumed that the problem was static—the variables in the problem did not change over time. This is the so-called static allocation problem and is somewhat idealized. Our next example will provide a glimpse of the types of problems that typically arise in optimal portfolio investment where random variables are dynamic. The example is also seated in more classical finance theory, that of a “Markowitz portfolio” in which the investor seeks to maximize a riskadjusted long-term return and the wealth process is self-financing.6 Example 1.5

Optimal Investment in an Index Portfolio

Let St be a time-t price of a risky asset such as a sector exchange-traded fund (ETF). We assume that our setting is discrete time, and we denote different time steps by integer valued-indices t = 0, . . . , T , so there are T + 1 values on our discrete-time grid. The discrete-time random evolution of the risky asset St is (continued)

6A

wealth process is self-financing if, at each time step, any purchase of an additional quantity of the risky asset is funded from the bank account. Vice versa, any proceeds from a sale of some quantity of the asset go to the bank account.

4 Reinforcement Learning

Example 1.5

27

(continued) St+1 = St (1 + φt ) ,

(1.12)

where φt is a random variable whose probability distribution may depend on the current asset value St . To ensure non-negativity of prices, we assume that φt is bounded from below φt ≥ −1. Consider a wealth process Wt of an investor who starts with an initial wealth W0 = 1 at time t = 0 and, at each period t = 0, . . . , T − 1 allocates a fraction ut = ut (St ) of the total portfolio value to the risky asset, and the remaining fraction 1 − ut is invested in a risk-free bank account that pays a risk-free interest rate rf = 0. We will refer to a set of decision variables for all time T −1 steps as a policy π := {ut }t=0 . The wealth process is self-financing and so the wealth at time t + 1 is given by Wt+1 = (1 − ut ) Wt + ut Wt (1 + φt ) .

(1.13)

This produces the one-step return rt =

Wt+1 − Wt = ut φt . Wt

(1.14)

Note this is a random function of the asset price St . We define one-step rewards Rt for t = 0, . . . , T − 1 as risk-adjusted returns Rt = rt − λVar [rt |St ] = ut φt − λu2t Var [φt |St ] ,

(1.15)

where λ is a risk-aversion parameter.a We now consider the problem of maximization of the following concave function of the control variable ut :

T

2 V (s)= max E Rt St =s = max E ut φt −λut Var [φt |St ] St =s . ut ut t=0 t=0 (1.16) Equation 1.16 defines an optimal investment problem for T − 1 steps faced by an investor whose objective is to optimize risk-adjusted returns over each period. This optimization problem is equivalent to maximizing the long-run

π

T

(continued)

a Note, for avoidance of doubt, that the risk-aversion 1 2 to ensure consistency with the finance literature.

parameter must be scaled by a factor of

28

1 Introduction

Example 1.5

(continued)

returns over the period [0, T ]. For each t = T − 1, T − 2, . . . , 0, the optimality condition for action ut is now obtained by maximization of V π (s) with respect to ut : u∗t =

E [ φt | St ] , 2λVar [φt |St ]

(1.17)

where we allow for short selling in the ETF (i.e., ut < 0) and borrowing of cash ut > 1. This is an example of a stochastic optimal control problem for a portfolio that maximizes its cumulative risk-adjusted return by repeatedly rebalancing between cash and a risky asset. Such problems can be solved using means of dynamic programming or reinforcement learning. In our problem, the dynamic programming solution is given by an analytical expression (1.17). Chapter 9 will present more complex settings including reinforcement learning approaches to optimal control problems, as well as demonstrate how expressions like the optimal action of Eq. 1.17 can be computed in practice.

•

? Multiple Choice Question 4

Select all the following correct statements: 1. The name “Markov processes” first historically appeared as a result of a misspelled name “Mark-Off processes” that was previously used for random processes that describe learning in certain types of video games, but has become a standard terminology since then. 2. The goal of (risk-neutral) reinforcement learning is to maximize the expected total reward by choosing an optimal policy. 3. The goal of (risk-neutral) reinforcement learning is to neutralize risk, i.e. make the variance of the total reward equal zero. 4. The goal of risk-sensitive reinforcement learning is to teach a RL agent to pick action policies that are most prone to risk of failure. Risk-sensitive RL is used, e.g. by venture capitalists and other sponsors of RL research, as a tool to assess the feasibility of new RL projects.

5 Examples of Supervised Machine Learning in Practice The practice of machine learning in finance has grown somewhat commensurately with both theoretical and computational developments in machine learning. Early adopters have been the quantitative hedge funds, including Bridgewater Associates,

5 Examples of Supervised Machine Learning in Practice

29

Renaissance Technologies, WorldQuant, D.E. Shaw, and Two Sigma who have embraced novel machine learning techniques although there are mixed degrees of adoption and a healthy skepticism exists that machine learning is a panacea for quantitative trading. In 2015, Bridgewater Associates announced a new artificial intelligence unit, having hired people from IBM Watson with expertise in deep learning. Anthony Ledford, chief scientist at MAN AHL: “It’s at an early stage. We have set aside a pot of money for test trading. With deep learning, if all goes well, it will go into test trading, as other machine learning approaches have.” Winton Capital Management’s CEO David Harding: “People started saying, ‘There’s an amazing new computing technique that’s going to blow away everything that’s gone before.’ There was also a fashion for genetic algorithms. Well, I can tell you none of those companies exist today—not a sausage of them.” Some qualifications are needed to more accurately assess the extent of adoption. For instance, there is a false line of reasoning that ordinary least squares regression and logistic regression, as well as Bayesian methods, are machine learning techniques. Only if the modeling approach is algorithmic, without positing a data generation process, can the approach be correctly categorized as machine learning. So regularized regression without use of parametric assumptions on the error distribution is an example of machine learning. Unregularized regression with, say, Gaussian error is not a machine learning technique. The functional form of the input–output map is the same in both cases, which is why we emphasize that the functional form of the map is not a sufficient condition for distinguishing ML from statistical methods. With that caveat, we shall view some examples that not only illustrate some of the important practical applications of machine learning prediction in algorithmic trading, high-frequency market making, and mortgage modeling but also provide a brief introduction to applications that will be covered in more detail in later chapters.

5.1 Algorithmic Trading Algorithmic trading is a natural playground for machine learning. The idea behind algorithmic trading is that trading decisions should be based on data, not intuition. Therefore, it should be viable to automate this decision-making process using an algorithm, either specified or learned. The advantages of algorithmic trading include complex market pattern recognition, reduced human produced error, ability to test on historic data, etc. In recent times, as more and more information is being digitized, the feasibility and capacity of algorithmic trading has been expanding drastically. The number of hedge funds, for example, that apply machine learning for algorithmic trading is steadily increasing. Here we provide a simple example of how machine learning techniques can be used to improve traditional algorithmic trading methods, but also provide new trading strategy suggestions. The example here is not intended to be the “best” approach, but rather indicative of more out-of-the-box strategies that machine learning facilitates, with the emphasis on minimizing out-of-sample error by pattern matching through efficient compression across high-dimensional datasets.

30

1 Introduction

Momentum strategies are one of the most well-known algo-trading strategies; In general, strategies that predict prices from historic price data are categorized as momentum strategies. Traditionally momentum strategies are based on certain regression-based econometric models, such as ARIMA or VAR (see Chap. 6). A drawback of these models is that they impose strong linearity which is not consistently plausible for time series of prices. Another caveat is that these models are parametric and thus have strong bias which often causes underfitting. Many machine learning algorithms are both non-linear and semi/non-parametric, and therefore prove complementary to existing econometric models. In this example we build a simple momentum portfolio strategy with a feedforward neural network. We focus on the S&P 500 stock universe, and assume we have daily close prices for all stocks over a ten-year period.7 Problem Formulation The most complex practical aspect of machine learning is how to choose the input (“features”) and output. The type of desired output will determine whether a regressor or classifier is needed, but the general rule is that it must be actionable (i.e., tradable). Suppose our goal is to invest in an equally weighted, long only, stock portfolio only if it beats the S&P 500 index benchmark (which is a reasonable objective for a portfolio manager). We can therefore label the portfolio at every observation t based on the mean directional excess return of the portfolio: Gt =

1 0

1 N 1 N

i i rt+h,t i i rt+h,t

− r˜t+h,t ≥ , − r˜t+h,t < 0,

(1.18)

i where rt+h,t is the return of stock i between times t and t + h, r˜t+h,t is the return of the S&P 500 index in the same period, and is some target next period excess portfolio return. Without loss of generality, we could invest in the universe (N = 500), although this is likely to have adverse practical implications such as excessive transaction costs. We could easily just have restricted the number of stocks to a subset, such as the top decile of performing stocks in the last period. Framed this way, the machine learner is thus informing us when our stock selection strategy will outperform the market. It is largely agnostic to how the stocks are selected, provided the procedure is systematic and based solely on the historic data provided to the classifier. It is further worth noting that the map between the decision to hold the customized portfolio has a non-linear relationship with the past returns of the universe.

To make the problem more concrete, let us set h = 5 days. The algorithmic strategy here is therefore automating the decision to invest in the customized 7 The

question of how much data is needed to train a neural network is a central one, with the immediate concern being insufficient data to avoid over-fitting. The amount of data needed is complex to assess; however, it is partly dependent on the number of edges in the network and can be assessed through bias–variance analysis, as described in Chap. 4.

5 Examples of Supervised Machine Learning in Practice

31

Table 1.3 Training samples for a classification problem Date 2007-01-03 2017-01-04 2017-01-05 ... 2017-12-29 2017-12-30 2017-12-31

X1 0.051 −0.092 0.021

X2 −0.035 0.125 0.063

0.093 0.020 −0.109

−0.023 0.019 0.025

...

X500 0.072 −0.032 −0.058

G 0 0 1

0.045 0.022 −0.092

1 1 1

portfolio or the S&P 500 index every week based on the previous 5-day realized returns of all stocks. To apply machine learning to this decision, the problem translates into finding the weights in the network between past returns and the decision to invest in the equally weighted portfolio. For avoidance of doubt, we emphasize that the interpretation of the optimal weights differs substantially from Markowitz’s mean–variance portfolios, which simply finds the portfolio weights to optimize expected returns for a given risk tolerance. Here, we either invest equal amounts in all stocks of the portfolio or invest the same amount in the S&P 500 index and the weights in the network signify the relevance of past stock returns in the expected excess portfolio return outperforming the market. Data Feature engineering is always important in building models and requires careful consideration. Since the original price data does not meet several machine learning requirements, such as stationarity and i.i.d. distributional properties, one needs to engineer input features to prevent potential “garbage-in-garbage-out” phenomena. In this example, we take a simple approach by using only the 5-day realized returns of all S&P 500 stocks.8 Returns are scale-free and no further standardization is needed. So for each time t, the input features are 1 500 . Xt = rt,t−5 , . . . , rt,t−5

(1.19)

Now we can aggregate the features and labels into a panel indexed by date. Each column is an entry in Eq. 1.19, except for the last column which is the assigned label from Eq. 1.18, based on the realized excess stock returns of the portfolio. An example of the labeled input data (X, G) is shown in Table 1.3. The process by which we train the classifier and evaluate its performance will be described in Chap. 4, but this example illustrates how algo-trading strategies can be crafted around supervised machine learning. Our model problem could be tailored

8 Note

that the composition of the S&P 500 changes over time and so we should interpret a feature as a fixed symbol.

32

1 Introduction

for specific risk-reward and performance reporting metrics such as, for example, Sharpe or information ratios meeting or exceeding a threshold. is typically chosen to be a small value so that the labels are not too imbalanced. As the value is increased, the problem becomes an “outlier prediction problem”— a highly imbalanced classification problem which requires more advanced sampling and interpolation techniques beyond an off-the-shelf classifier. In the next example, we shall turn to another important aspect of machine learning in algorithmic trading, namely execution. How the trades are placed is a significant aspect of algorithmic trading strategy performance, not only to minimize price impact of market taking strategies but also for market making. Here we shall look to transactional data to perfect the execution, an engineering challenge by itself just to process market feeds of tick-by-tick exchange transactions. The example considers a market making application but could be adapted for price impact and other execution considerations in algorithmic trading by moving to a reinforcement learning framework. A common mistake is to assume that building a predictive model will result in a profitable trading strategy. Clearly, the consideration given to reliably evaluating machine learning in the context of trading strategy performance is a critical component of its assessment.

5.2 High-Frequency Trade Execution Modern financial exchanges facilitate the electronic trading of instruments through an instantaneous double auction. At each point in time, the market demand and the supply can be represented by an electronic limit order book, a cross-section of orders to execute at various price levels away from the market price as illustrated in Table 1.4. Electronic market makers will quote on both sides of the market in an attempt to capture the bid–ask spread. Sometimes a large market order, or a succession of smaller markets orders, will consume an entire price level. This is why the market price fluctuates in liquid markets—an effect often referred to by practitioners as a “price-flip.” A market maker can take a loss if only one side of the order is filled as a result of an adverse price movement. Figure 1.7 (left) illustrates a typical mechanism resulting in an adverse price movement. A snapshot of the limit order book at time t, before the arrival of a market order, and after at time t +1 is shown in the left and right panels, respectively. The resting orders placed by the market marker are denoted with the “+” symbol— red denotes a bid and blue denotes an ask quote. A buy market order subsequently arrives and matches the entire resting quantity of best ask quotes. Then at event time t + 1 the limit order book is updated—the market maker’s ask has been filled (blue minus symbol) and the bid now rests away from the inside market. The market marker may systematically be forced to cancel the bid and buy back at a higher price, thus taking a loss.

5 Examples of Supervised Machine Learning in Practice

33

Table 1.4 This table shows a snapshot of the limit order book of S&P 500 e-mini futures (ES). The top half (“sell-side”) shows the ask volumes and prices and the lower half (“buy side”) shows the bid volumes and prices. The quote levels are ranked by the most competitive at the center (the “inside market”), outward to the least competitive prices at the top and bottom of the limit order book. Note that only five bid or ask levels are shown in this example, but the actual book is much deeper

Bid

477 1038 950 1349 1559

Price 2170.25 2170.00 2169.75 2169.50 2169.25 2169.00 2168.75 2168.50 2168.25 2168.00

Ask 1284 1642 1401 1266 290

Fig. 1.7 (Top) A snapshot of the limit order book is taken at time t. Limit orders placed by the market marker are denoted with the “+” symbol—red denotes a bid and blue denotes an ask. A buy market order subsequently arrives and matches the entire resting quantity of best ask quotes. Then at event time t + 1 the limit order book is updated. The market maker’s ask has been filled (blue minus symbol) and the bid rests away from the inside market. (Bottom) A pre-emptive strategy for avoiding adverse price selection is illustrated. The ask is requoted at a higher ask price. In this case, the bid is not replaced and the market maker may capture a tick more than the spread if both orders are filled

Machine learning can be used to predict these price movements (Kearns and Nevmyvaka 2013; Kercheval and Zhang 2015; Sirignano 2016; Dixon et al. 2018; Dixon 2018b,a) and thus to potentially avoid adverse selection. Following Cont and de Larrard (2013) we can treat queue sizes at each price level as input variables. We can additionally include properties of market orders, albeit in a form which our machines deem most relevant to predicting the direction of price movements (a.k.a. feature engineering). In contrast to stochastic modeling, we do not impose conditional distributional assumptions on the independent variables (a.k.a. features) nor assume that price movements are Markovian. Chapter 8 presents a RNN for

34

1 Introduction

mid-price prediction from the limit order book history which is the starting point for the more in-depth study of Dixon (2018b) which includes market orders and demonstrates the superiority of RNNs compared to other time series methods such as Kalman filters. We reiterate that the ability to accurately predict does not imply profitability of the strategy. Complex issues concerning queue position, exchange matching rules, latency, position constraints, and price impact are central considerations for practitioners. The design of profitable strategies goes beyond the scope of this book but the reader is referred to de Prado (2018) for the pitfalls of backtesting and designing algorithms for trading. Dixon (2018a) presents a framework for evaluating the performance of supervised machine learning algorithms which accounts for latency, position constraints, and queue position. However, supervised learning is ultimately not the best machine learning approach as it cannot capture the effect of market impact and is too inflexible to incorporate more complex strategies. Chapter 9 presents examples of reinforcement learning which demonstrate how to capture market impact and also how to flexibly formulate market making strategies.

5.3 Mortgage Modeling Beyond the data rich environment of algorithmic trading, does machine learning have a place in finance? One perspective is that there simply is not sufficient data for some “low-frequency” application areas in finance, especially where traditional models have failed catastrophically. The purpose of this section is to serve as a sobering reminder that long-term forecasting goes far beyond merely selecting the best choice of machine learning algorithm and why there is no substitute for strong domain knowledge and an understanding of the limitations of data. In the USA, a mortgage is a loan collateralized by real-estate. Mortgages are used to securitize financial instruments such as mortgage backed securities and collateralized mortgage obligations. The analysis of such securities is complex and has changed significantly over the last decade in response to the 2007–2008 financial crises (Stein 2012). Unless otherwise specified, a mortgage will be taken to mean a “residential mortgage,” which is a loan with payments due monthly that is collateralized by a single family home. Commercial mortgages do exist, covering office towers, rental apartment buildings, and industrial facilities, but they are different enough to be considered separate classes of financial instruments. Borrowing money to buy a house is one of the most common, and largest balance, loans that an individual borrower is ever likely to commit to. Within the USA alone, mortgages comprise a staggering $15 trillion dollars in debt. This is approximately the same balance as the total federal debt outstanding (Fig. 1.8). Within the USA, mortgages may be repaid (typically without penalty) at will by the borrower. Usually, borrowers use this feature to refinance their loans in favorable interest rate regimes, or to liquidate the loan when selling the underlying house. This

5 Examples of Supervised Machine Learning in Practice

35

Fig. 1.8 Total mortgage debt in the USA compared to total federal debt, millions of dollars, unadjusted for inflation. Source: https://fred.stlouisfed.org/series/MDOAH, https://fred.stlouisfed. org/series/GFDEBTN Table 1.5 At any time, the states of any US style residential mortgage is in one of the several possible states Symbol P C 3 6 9 F R D

Name Paid Current 30-days delinquent 60-days delinquent 90+ delinquent Foreclosure Real-Estate-Owned (REO) Default liquidation

Definition All balances paid, loan is dissolved All payments due have been paid Mortgage is delinquent by one payment Delinquent by 2 payments Delinquent by 3 or more payments Foreclosure has been initiated by the lender The lender has possession of the property Loan is involuntarily liquidated for nonpayment

has the effect of moving a great deal of financial risk off of individual borrowers, and into the financial system. It also drives a lively and well developed industry around modeling the behavior of these loans. The mortgage model description here will generally follow the comprehensive work in Sirignano et al. (2016), with only a few minor deviations. Any US style residential mortgage, in each month, can be in one of the several possible states listed in Table 1.5. Consider this set of K available states to be K = {P , C, 3, 6, 9, F, R, D}. Following the problem formulation in Sirignano et al. (2016), we will refer to the status of loan n at time t as Utn ∈ K, and this will be represented as a probability vector using a standard one-hot encoding. If X = (X1 , . . . , XP ) is the input matrix of P explanatory variables, then we define a probability transition density function g : RP → [0, 1]K×K parameterized by θ so that n = i | Utn = j, Xtn ) = gi,j (Xtn | θ ), ∀i, j ∈ K. P(Ut+1

(1.20)

36

1 Introduction

Note that g(Xtn | θ ) is a time in-homogeneous K × K Markov transition matrix. Also, not all transitions are even conceptually possible—there are non-commutative states. For instance, a transition from C to 6 is not possible since a borrower cannot miss two payments in a single month. Here we will write p(i,j ) := gi,j (Xtn | θ ) for ease of notation and because of the non-commutative state transitions where p(i,j ) = 0, the Markov matrix takes the form: ⎡ 1 p(c,p) ⎢0 p(c,c) ⎢ ⎢0 p ⎢ (c,3) ⎢ ⎢0 0 n g(Xt | θ ) = ⎢ ⎢0 0 ⎢ ⎢0 0 ⎢ ⎣0 0 0 0

p(3,p) p(3,c) p(3,3) p(3,6) 0 0 0 0

p(6,c) p(6,3) p(6,6) p(6,9) p(6,f ) p(6,r) p(6,d)

p(9,c) p(9,3) p(9,6) p(9,9) p(9,f ) p(9,r) p(9,d)

0 0 0 0 0 0

p(f,c) p(f,3) p(f,6) p(f,9) p(f,f ) p(f,r) p(r,r) p(f,d) p(r,d)

⎤ 0 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

Our classifier gi,j (Xtn | θ ) can thus be constructed so that only the probability of transition between the commutative states are outputs and we can apply softmax functions on a subset of the outputs to ensure that j ∈K gi,j (Xtn | θ ) = 1 and hence the transition probabilities in each row sum to one. For the purposes of financial modeling, it is important to realize that both states P and D are loan liquidation terminal states. However, state P is considered to be voluntary loan liquidation (e.g., prepayment due to refinance), whereas state D is considered to be involuntary liquidation (e.g., liquidation via foreclosure and auction). These states are not distinguishable in the mortgage data itself, but rather the driving force behind liquidation must be inferred from the events leading up to the liquidation. One contributor to mortgage model misprediction in the run up to the 2008 financial crisis was that some (but not all) modeling groups considered loans liquidating from deep delinquency (e.g., status 9) to be the transition 9 → P if no losses were incurred. However, behaviorally, these were typically defaults due to financial hardship, and they would have had losses in a more difficult house price regime. They were really 9 → D transitions that just happened to be lossless due to strong house price gains over the life of the mortgage. Considering them to be voluntary prepayments (status P ) resulted in systematic over-prediction of prepayments in the aftermath of major house price drops. The matrix above therefore explicitly excludes this possibility and forces delinquent loan liquidation to be always considered involuntary. The reverse of this problem does not typically exist. In most states it is illegal to force a borrower into liquidation until at least 2 payments have been missed. Therefore, liquidation from C or 3 is always voluntary, and hence C → P and 3 → P . Except in cases of fraud or severe malfeasance, it is almost never economically advantageous for a lender to force liquidation from status 6, but it is not illegal. Therefore the transition 3 → D is typically a data error, but 6 → D is merely very rare.

5 Examples of Supervised Machine Learning in Practice

Example 1.6

37

Parameterized Probability Transitions

If loan n is current in time period t, then P(Utn ) = (0, 1, 0, 0, 0, 0, 0, 0)T .

(1.21)

If we have p(c,p) = 0.05, p(c,c) = 0.9, and p(c,3) = 0.05, then n | Xtn ) = g(Xtn | θ ) · P(Utn ) = (0.05, 0.9, 0.05, 0, 0, 0, 0, 0)T . P(Ut+1 (1.22)

Common mortgage models sometimes use additional states, often ones that are (without additional information) indistinguishable from the states listed above. Table 1.6 describes a few of these. The reason for including these is the same as the reason for breaking out states like REO, status R. It is known on theoretical grounds that some model regressors from Xtn should not be relevant for R. For instance, since the property is now owned by the lender, and the loan itself no longer exists, the interest rate (and rate incentive) of the original loan should no longer have a bearing on the outcome. To avoid over-fitting due to highly colinear variables, these known-useless variables are then excluded from transitions models starting in status R. This is the same reason status T is sometimes broken out, especially for logistic regressions. Without an extra status listed in this way, strong rate disincentives could drive prepayments in the model to (almost) zero, but we know that people die and divorce in all rate regimes, so at least some minimal level of premature loan liquidations must still occur based on demographic factors, not financial ones.

5.3.1

Model Stability

Unlike many other models, mortgage models are designed to accurately predict events a decade or more in the future. Generally, this requires that they be built on regressors that themselves can be accurately predicted, or at least hedged. Therefore, it is common to see regressors like FICO at origination, loan age in months, rate incentive, and loan-to-value (LTV) ratio. Often LTV would be called MTMLTV if it is marked-to-market against projected or realized housing price moves. Of these regressors, original FICO is static over the life of the loan, age is deterministic,

Table 1.6 A brief description of mortgage states Symbol T U

Name Turnover Curtailment

Overlaps with P C

Definition Loan prepaid due to non-financial life event Borrower overpaid to reduce loan principal

38 Table 1.7 Loan originations by year (Freddie Mac, FRM30)

1 Introduction Year 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

Loans originated 976,159 733,567 1,542,025 1,403,515 2,063,488 1,133,015 1,618,748 1,300,559 1,238,814 1,237,823 1,879,477 1,250,484 1,008,731 1,249,486 1,375,423 942,208

Fig. 1.9 Sample mortgage model predicting C → 3 and fit on loans originated in 2001 and observed until 2006, by loan age (in months). The prepayment probabilities are shown on the y-axis

rates can be hedged, and MTMLTV is rapidly driven down by loan amortization and inflation thus eliminating the need to predict it accurately far into the future. Consider the Freddie Mac loan level dataset of 30 year fixed rate mortgages originated through 2014. This includes each monthly observation from each loan present in the dataset. Table 1.7 shows the loan count by year for this dataset. When a model is fit on 1 million observations from loans originated in 2001 and observed until the end of 2006, its C → P probability charted against age is shown in Fig. 1.9. In Fig. 1.9 the curve observed is the actual prepayment probability of the observations with the given age in the test dataset, “Model” is the model prediction,

5 Examples of Supervised Machine Learning in Practice

39

Fig. 1.10 Sample mortgage model predicting C → 3 and fit on loans originated in 2006 and observed until 2015, by loan age (in months). The prepayment probabilities are shown on the yaxis

and “Theoretical” is the response to age by a theoretical loan with all other regressors from Xtn held constant. Two observations are worth noting: 1. The marginal response to age closely matches the model predictions; and 2. The model predictions match actual behavior almost perfectly. This is a regime where prepayment behavior is largely driven by age. When that same model is run on observations from loans originated in 2006 (the peak of housing prices before the crisis), and observed until 2015, Fig. 1.10 is produced. Three observations are warranted from this figure: 1. The observed distribution is significantly different from Fig. 1.9; 2. The model predicted a decline of 25%, but the actual decline was approximately 56%; and 3. Prepayment probabilities are largely indifferent to age. The regime shown here is clearly not driven by age. In order to provide even this level of accuracy, the model had to extrapolate far from any of the available data and “imagine” a regime where loan age is almost irrelevant to prepayment. This model meets with mixed success. This particular one was fit on only 8 regressors, a more complicated model might have done better, but the actual driver of this inaccuracy was a general tightening of lending standards. Moreover, there was no good data series available before the crisis to represent lending standards. This model was reasonably accurate even though almost 15 years separated the start of the fitting data from the end of the projection period, and a lot happened in that time. Mortgage models in particular place a high premium on model stability, and the ability to provide as much accuracy as possible even though the underlying distribution may have changed dramatically from the one that generated the fitting data. Notice also that cross-validation would not help here, as we cannot draw testing data from the distribution we care about, since that distribution comes from the future.

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Most importantly, this model shows that the low-dimensional projections of this (moderately) high-dimensional problem are extremely deceptive. No modeler would have chosen a shape like the model prediction from Fig. 1.9 as function of age. That prediction arises due to the interaction of several variables, interactions that are not interpretable from one-dimensional plots such as this. As we will see in subsequent chapters, such complexities in data are well suited to machine learning, but not without a cost. That cost is understanding the “bias–variance tradeoff” and understanding machine learning with sufficient rigor for its decisions to be defensible.

6 Summary In this chapter, we have identified some of the key elements of supervised machine learning. Supervised machine learning 1. is an algorithmic approach to statistical inference which, crucially, does not depend on a data generation process; 2. estimates a parameterized map between inputs and outputs, with the functional form defined by the methodology such as a neural network or a random forest; 3. automates model selection, using regularization and ensemble averaging techniques to iterate through possible models and arrive at a model with the best out-of-sample performance; and 4. is often well suited to large sample sizes of high-dimensional non-linear covariates. The emphasis on out-of-sample performance, automated model selection, and absence of a pre-determined parametric data generation process is really the key to machine learning being a more robust approach than many parametric, financial econometrics techniques in use today. The key to adoption of machine learning in finance is the ability to run machine learners alongside their parametric counterparts, observing over time the differences and limitations of parametric modeling based on in-sample fitting metrics. Statistical tests must be used to characterize the data and guide the choice of algorithm, such as, for example, tests for stationary. See Dixon and Halperin (2019) for a checklist and brief but rounded discussion of some of the challenges in adopting machine learning in the finance industry. Capacity to readily exploit a wide form of data is their other advantage, but only if that data is sufficiently high quality and adds a new source of information. We close this chapter with a reminder of the failings of forecasting models during the financial crisis of 2008 and emphasize the importance of avoiding siloed data extraction. The application of machine learning requires strong scientific reasoning skills and is not a panacea for commoditized and automated decision-making.

7 Exercises

41

7 Exercises Exercise 1.1**: Market Game Suppose that two players enter into a market game. The rules of the game are as follows: Player 1 is the market maker, and Player 2 is the market taker. In each round, Player 1 is provided with information x, and must choose and declare a value α ∈ (0, 1) that determines how much it will pay out if a binary event G occurs in the round. G ∼ Bernoulli(p), where p = g(x|θ) for some unknown parameter θ . Player 2 then enters the game with a $1 payment and chooses one of the following payoffs: V1 (G, p) =

1 α

with probability p

with probability (1 − p)

or V2 (G, p) =

with probability p

1 (1−α)

with probability (1 − p)

1. Given that α is known to Player 2, state the strategy9 that will give Player 2 an expected payoff, over multiple games, of $1 without knowing p. 2. Suppose now that p is known to both players. In a given round, what is the optimal choice of α for Player 1? 3. Suppose Player 2 knows with complete certainty, that G will be 1 for a particular round, what will be the payoff for Player 2? 4. Suppose Player 2 has complete knowledge in rounds {1, . . . , i} and can reinvest payoffs from earlier rounds into later rounds. Further suppose without loss of generality that G = 1 for each of these rounds. What will be the payoff for Player 2 after i rounds? You may assume that the each game can be played with fractional dollar costs, so that, for example, if Player 2 pays Player 1 $1.5 to enter the game, then the payoff will be 1.5V1 . Exercise 1.2**: Model Comparison Recall Example 1.2. Suppose additional information was added such that it is no longer possible to predict the outcome with 100% probability. Consider Table 1.8 as the results of some experiment. Now if we are presented with x = (1, 0), the result could be B or C. Consider three different models applied to this value of x which encode the value A, B, or C.

strategy refers the choice of weight if Player 2 is to choose a payoff V = wV1 + (1 − w)V2 , i.e. a weighted combination of payoffs V1 and V2 .

9 The

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1 Introduction

Table 1.8 Sample model data

G A B B C C

x (0, 1) (1, 1) (1, 0) (1, 0) (0, 0)

f ((1, 0)) = (0, 1, 0), Predicts B with 100% certainty.

(1.23)

g((1, 0)) = (0, 0, 1), Predicts C with 100% certainty.

(1.24)

h((1, 0)) = (0, 0.5, 0.5), Predicts B or C with 50% certainty.

(1.25)

1. Show that each model has the same total absolute error, over the samples where x = (1, 0). 2. Show that all three models assign the same average probability to the values from Table 1.8 when x = (1, 0). 3. Suppose that the market game in Exercise 1 is now played with models f or g. B or C each triggers two separate payoffs, V1 and V2 , respectively. Show that the losses to Player 1 are unbounded when x = (1, 0) and α = 1 − p. 4. Show also that if the market game in Exercise 1 is now played with model h, the losses to Player 1 are bounded. Exercise 1.3**: Model Comparison Example 1.1 and the associated discussion alluded to the notion that some types of models are more common than others. This exercise will explore that concept briefly. Recall Table 1.1 from Example 1.1: G A B C C

x (0, 1) (1, 1) (1, 0) (0, 0)

For this exercise, consider two models “similar” if they produce the same projections for G when applied to the values of x from Table 1.1 with probability strictly greater than 0.95. In the following subsections, the goal will be to produce sets of mutually dissimilar models that all produce Table 1.1 with a given likelihood. 1. How many similar models produce Table 1.1 with likelihood 1.0? 2. Produce at least 4 dissimilar models that produce Table 1.1 with likelihood 0.9. 3. How many dissimilar models can produce Table 1.1 with likelihood exactly 0.95?

Appendix

43

Exercise 1.4*: Likelihood Estimation When the data is i.i.d., the negative of log-likelihood function (the “error function”) for a binary classifier is the cross-entropy E(θ ) = −

n

Gi ln (g1 (xi | θ )) + (1 − Gi )ln (g0 (xi | θ )).

i=1

Suppose now that there is a probability πi that the class label on a training data point xi has been correctly set. Write down the error function corresponding to the negative log-likelihood. Verify that the error function in the above equation is obtained when πi = 1. Note that this error function renders the model robust to incorrectly labeled data, in contrast to the usual least squares error function. Exercise 1.5**: Optimal Action Derive Eq. 1.17 by setting the derivative of Eq. 1.16 with respect to the timet action ut to zero. Note that Eq. 1.17 gives a non-parametric expression for the optimal action ut in terms of a ratio of two conditional expectations. To be useful in practice, the approach might need some further modification as you will use in the next exercise. Exercise 1.6***: Basis Functions Instead of non-parametric specifications of an optimal action in Eq. 1.17, we can develop a parametric model of optimal action. To this end, assume we have a set of basic functions ψk (S) with k = 1, . . . , K. Here K is the total number of basis functions—the same as the dimension of your model space. We now define the optimal action ut = ut (St ) in terms of coefficients θk of expansion over basis functions k (for example, we could use spline basis functions, Fourier bases, etc.) : ut = ut (St ) =

K

θk (t) k (St ).

k=1

Compute the optimal coefficients θk (t) by substituting the above equation for ut into Eq. 1.16 and maximizing it with respect to a set of weights θk (t) for a t-th time step.

Appendix Answers to Multiple Choice Questions Question 1 Answer: 1, 2. Answer 3 is incorrect. While it is true that unsupervised learning does not require a human supervisor to train the model, it is false to presume that the approach is superior.

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Answer 4 is incorrect. Reinforcement learning cannot be viewed as a generalization of supervised learning to Markov Decision Processes. The reason is that reinforcement learning uses rewards to reinforce decisions, rather than labels to define the correct decision. For this reason, reinforcement learning uses a weaker form of supervision. Question 2 Answer: 1,2,3. (1) Answer 4 is incorrect. Two separate binary models {gi (X|θ )}1i=0 and (2) {gi (X|θ )}1i=0 will, in general, not produce the same output as a single, multiclass, model {gi (X|θ )}3i=0 . Consider, as a counter example, the logistic models (2) exp{−X θ1 } = g0 (X|θ2 ) = g0(1) = g0 (X|θ1 ) = 1+exp{−X T θ } and g0 1 with the multi-class model T

exp{−XT θ2 } , 1+exp{−XT θ2 }

exp{(XT θ )i } . gi (X|θ ) = softmax(exp{XT θ }) = K T k=0 exp{(X θ )k }

compared

(1.26)

If we set θ1 = θ 0 − θ 1 and θ 2 = θ 3 = 0, then the multi-class model is equivalent to Model 1. Similarly if we set θ2 = θ 2 − θ 3 and θ 0 = θ 1 = 0, then the multiclass model is equivalent to Model 2. However, we cannot simultaneously match the outputs of Model 1 and Model 2 with the multi-class model. Question 3 Answer: 1,2,3. Answer 4 is incorrect. The layers in a deep recurrent network provide more expressibility between each lagged input and the hidden state variable, but are unrelated to the amount of memory in the network. The hidden layers in any multilayered perceptron are not the hidden state variables in our time series model. It is the degree of unfolding, i.e. number of hidden state vectors which determines the amount of memory in any recurrent network. Question 4 Answer: 2.

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Chapter 2

Probabilistic Modeling

This chapter introduces probabilistic modeling and reviews foundational concepts in Bayesian econometrics such as Bayesian inference, model selection, online learning, and Bayesian model averaging. We then develop more versatile representations of complex data with probabilistic graphical models such as mixture models.

1 Introduction Not only is statistical inference from data intrinsically uncertain, but the type of data and relationships in the data that we seek to model are growing ever more complex. In this chapter, we turn to probabilistic modeling, a class of statistical models, which are broadly designed to characterize uncertainty and allow the expression of causality between variables. Probabilistic modeling is a meta-class of models, including generative modeling—a class of statistical inference models which maximizes the joint distribution, p(X, Y ), and Bayesian modeling, employing either maximum likelihood estimation or “fully Bayesian” inference. Probabilistic graphical models put the emphasis on causal modeling to simplify statistical inference of parameters from data. This chapter shall focus on the constructs of probabilistic modeling, as they relate to the application of both unsupervised and supervised machine learning in financial modeling. While it seems natural to extend the previous chapters directly to a probabilistic neural network counterpart, it turns out that this does not develop the type of intuitive explanation of complex data that is needed in finance. It also turns out that neural networks are not a natural fit for probabilistic modeling. In other words, neural networks are well suited to pointwise estimation but lead to many difficulties in a probabilistic setting. In particular, they tend to be very data intensive—offsetting one of the major advantages of Bayesian modeling.

© Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_2

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We will explore probabilistic modeling through the introduction of probabilistic graphical models—a data structure which is convenient for understanding the relationship between a multitude of different classes of models, both discriminative and generative. This representation will lead us neatly to Bayesian kernel learning, the subject of Chap. 3. We begin by introducing the reader to elementary topics in Bayesian modeling, which is a well-established approach for characterizing uncertainty in, for example, trading and risk modeling. Starting with simple probabilistic models of the data, we review some of the main constructs necessary to apply Bayesian methods in practice. The application of probabilistic models to time series modeling, using filtering and hidden variables to dynamically represent the data is presented in Chap. 7. Chapter Objectives The key learning points of this chapter are: • Apply Bayesian inference to data using simple probabilistic models; • Understand how linear regression with probabilistic weights can be viewed as a simple probabilistic graphical model; and • Develop more versatile representations of complex data with probabilistic graphical models such as mixture models and hidden Markov models.

Note that section headers ending with * are more mathematically advanced, often requiring some background in analysis and probability theory, and can be skipped by the less mathematically inclined reader.

2 Bayesian vs. Frequentist Estimation Bayesian data analysis is distinctly different from classical (or “frequentist”) analysis in its treatment of probabilities, and in its resulting treatment of model parameters when compared to classical parametric analysis.1 Bayesian analysts formulate probabilistic statements about uncertain events before collecting any additional evidence (i.e., “data”). These ex-ante probabilities (or, more generally, probability distributions plus underlying parameters) are called priors. This notion of subjective probabilities is absent in classical estimation. In the classical world, all estimation and inference is based solely on observed data. 1 Throughout

the first part of this chapter we will largely remain within the realm of parametric analysis. However, we shall later see examples of Bayesian methods for non- and semi-parametric modeling.

2 Bayesian vs. Frequentist Estimation

49

Both Bayesian and classical econometricians aim to learn more about a set of parameters, say θ . In the classical mindset, θ contains fixed but unknown elements, usually associated with an underlying population of interest (e.g., the mean and variance for credit card debt among US college students). Bayesians share with classicals the interest in θ and the definition of the population of interest. However, they assign ex ante a prior probability to θ , labeled p (θ ), which usually takes the form of a probability distribution with “known” moments. For example, Bayesians might state that the aforementioned debt amount has a normal distribution with mean $3000 and standard deviation of $1500. This prior may be based on previous research, related findings in the published literature, or it may be completely arbitrary. In any case, it is an inherently subjective construct. Both schools then develop a theoretical framework that relates θ to observed data, say a “dependent variable” y, and a matrix of explanatory variables X. This relationship is formalized via a likelihood function, say p (y | θ , X) to stay with Bayesian notation. To stress, this likelihood function takes the exact same analytical form for both schools. The classical analyst then collects a sample of observations from the underlying population of interest and, combining these data with the formulated statistical ˆ Any and all uncertainty surrounding the model, produces an estimate of θ , say θ. accuracy of this estimate is solely related to the notion that results are based on a sample, not data for the entire population. A different sample (of equal size) may produce slightly different estimates. Classicals express this uncertainty via ˆ They also have a strong focus on “standard errors” assigned to each element of θ. ˆ the behavior of θ as the sample size increases. The behavior of estimators under increasing sample size falls under the heading of “asymptotic theory.” The properties of most estimators in the classical world can only be assessed “asymptotically,” i.e. are only understood for the hypothetical case of an infinitely large sample. Also, virtually all specification tests used by frequentists hinge on asymptotic theory. This is a major limitation when the data size is finite. Bayesians, in turn, combine prior and likelihood via Bayes’ rule to derive the posterior distribution of θ as p (θ | y, X) =

p (θ , y | X) p (θ ) p (y | θ , X) = ∝ p (θ) p (y | θ , X) . p (y | X) p (y | X)

(2.1)

•

> Bayesian Modeling

Bayesian modeling is not about point estimation of a parameter value, θ , but rather updating and sharpening our subjective beliefs (our “prior”) about θ from the sample data. Thus, the sample data should contribute to “learning” about θ .

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2 Probabilistic Modeling

Bayesian Learning Simply put, the posterior distribution is just an updated version of the prior. More specifically, the posterior is proportional to the prior multiplied by the likelihood. The likelihood carries all the current information about the parameters and the data. If the data has high informational content (i.e., allows for substantial learning about θ ), the posterior will generally look very different from the prior. In most cases, it is much “tighter” (i.e., has a much smaller variance) than the prior. There is no room in Bayesian analysis for the classical notions of “sampling uncertainty,” and less a priori focus on the “asymptotic behavior” of estimators.2 Taking the Bayesian paradigm to its logical extreme, Duembgen and Rogers (2014) suggest to “estimate nothing.” They propose the replacement of the industrystandard estimation-based paradigm of calibration with an approach based on Bayesian techniques, wherein a posterior is iteratively obtained from a prior, namely stochastic filtering and MCMC. Calibration attempts to estimate, and then uses the estimates as if they were known true values—ignoring the estimation error. On the contrary, an approach based on a systematic application of the Bayesian principle is consistent: “There is never any doubt about what we should be doing to hedge or to mark-to-market a portfolio of derivatives, and whatever we do today will be consistent with what we did before, and with what we will do in the future.” Moreover, Bayesian model comparison methods enable one to easily compare models of very different types. Marginal Likelihood The term in the denominator of Eq. 2.1 is called the “marginal likelihood,” it is not a function of θ , and can usually be ignored for most components of Bayesian analysis. Thus, we usually work only with the numerator (i.e., prior times likelihood) for inference about θ . From Eq. 2.1 we know that this expression is proportional (“∝ ”) to the actual posterior. However, the marginal likelihood is crucial for model comparison, so we will learn a few methods to derive it as a by-product of or following the actual posterior analysis. For some choices of prior and likelihood there exist analytical solutions for this term. In summary, frequentists start with a “blank mind” regarding θ . They collect data ˆ They formalize the characteristics and uncertainty of θˆ for to produce an estimate θ. a finite sample context (if possible) and a hypothetical large sample (asymptotic) case. Bayesians collect data to update a prior, i.e. a pre-conceived probabilistic notion regarding θ.

2 However,

at times Bayesian analysis does rest on asymptotic results. Naturally, the general notion that a larger sample, i.e. more empirical information, is better than a small one also holds for Bayesian analysis.

3 Frequentist Inference from Data

51

3 Frequentist Inference from Data Let us begin this section with a simple example which illustrates frequentist inference. Example 2.1

Bernoulli Trials Example

Consider an experiment consisting of a single coin flip. We set the random variable Y to 0 if tails come up and 1 if heads come up. Then the probability density of Y is given by p(y | θ ) = θ y (1 − θ )1−y , where θ ∈ [0, 1] is the probability of heads showing up. You will recognize Y as a Bernoulli random variable. We view p as a function of y, but parameterized by the given parameter θ , hence the notation, p(y | θ ). More generally, suppose that we perform n such independent experiments (tosses) on the same coin. Denote these n realizations of Y as ⎛ ⎞ y1 ⎜y2 ⎟ ⎜ ⎟ y = ⎜ . ⎟ ∈ {0, 1}n , ⎝ .. ⎠ yn where, for 1 ≤ i ≤ n, yi is the result of the ith toss. What is the probability density of y? Since the coin tosses are independent, the probability density of y, i.e. the joint probability density of y1 , y2 , . . . , yn , is given by the product rule p(y | θ ) = p(y1 , y2 , . . . , yn | θ ) =

n

θ yi (1 − θ )1−yi = θ

yi

(1 − θ )n−

yi

.

i=1

Suppose that we have tossed the coin n = 50 times (performed n = 50 Bernoulli trials) and recorded the results of the trials as 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 How can we estimate θ given these data? Both the frequentists and Bayesians regard the density p(y | θ ) as a likelihood. Bayesians maintain this notation, whereas frequentists reinterpret p(y | θ ), which is

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2 Probabilistic Modeling

a function of y (given the parameters θ : in our case, there is a single parameter, so θ is univariate, but this does not have to be the case) as a function of θ (given the specific sample y), and write L(θ ) := L(θ | y) := p(y | θ ). Notice that we have merely reinterpreted this probability density, whereas its functional form remains the same, in our case: L(θ ) = θ

yi

(1 − θ )n−

yi

.

•

> Likelihood

Likelihood is one of the seminal ideas of the frequentist school. It was introduced by one of its founding fathers, Sir Ronald Aylmer Fisher: “What has now appeared is that the mathematical concept of probability is . . . inadequate to express our mental confidence or [lack of confidence] in making . . . inferences, and that the mathematical quantity which usually appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability. To distinguish it from probability, I have used the term ‘likelihood’ to designate this quantity. . . ”—R.A. Fisher, Statistical Methods for Research Workers.

It is generally more convenient to work with the log of likelihood—the loglikelihood. Since ln is a monotonically increasing function of its argument, the same values of θ maximize the log-likelihood as the ones that maximize the likelihood.

yi ln(1 − θ ). yi ln θ + n − ln L(θ ) = ln θ yi (1 − θ )n− yi = In order to find the value of θ that maximizes this expression, we differentiate with respect to θ and solve for the value of θ that sets the (partial) derivative to zero. ∂ ln L(θ ) = ∂θ

θ

yi

n − yi + . θ −1

Equating this to zero and solving for θ , we obtain the maximum likelihood estimate for θ :

4 Assessing the Quality of Our Estimator: Bias and Variance

θˆML =

53

yi . n

To confirm that this value does indeed maximize the log-likelihood, we take the second derivative with respect to θ , ∂2 n − yi yi ln L(θ ) = − 2 − < 0. ∂θ 2 θ (θ − 1)2 Since this quantity is strictly negative for all 0 < θ < 1, it is negative at θˆML , and we do indeed have a maximum. Example 2.2

Bernoulli Trials Example (continued)

Note that θˆML depends only on the sum of yi s, we can answer our question: if in a sequence of 50 coin tosses exactly twelve heads come up, then θˆML =

12 yi = = 0.24. n 50

A frequentist approach gives at a single value (a single “point”) as our estimate, 0.24—in this sense we are performing point estimation. When we apply a Bayesian approach to the same problem, we shall see that the Bayesian estimate is a probability distribution, rather than a single point. Despite some mathematical formalism, the answer is intuitively obvious. If we toss a coin fifty times, and out of those twelve times it lands with heads up, it is natural to estimate the probability of getting heads as 12 50 . It is encouraging that the result of our calculation agrees with our intuition and common sense.

4 Assessing the Quality of Our Estimator: Bias and Variance When we obtained our maximum likelihood estimate, we plugged in a specific number for yi , 12. In this sense the estimator is an ordinary function. However, we could also view it as a function of the random sample, θˆML =

Yi , n

each Yi being a random variable. A function of a random variable is itself a random variable, so we can compute its expectation and variance. In particular, an expectation of the error

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2 Probabilistic Modeling

e = θˆ − θ is known as bias, ! " ! " ˆ θ ) = E(e) = E θˆ − θ = E θˆ − E [θ ] . bias(θ, As frequentists, we view the true value of θ as a single, deterministic, fixed point, so we take it outside of the expectation: ! " ˆ θ ) = E θˆ − θ . bias(θ, In our case it is E[θˆML − θ ] = E[θˆML ] − θ = E

1 Yi E[Yi ] − θ −θ = n n =

1 · n(θ · 1 + (1 − θ ) · 0) − θ = 0, n

we see that the bias is zero, so this particular maximum likelihood estimator is unbiased (otherwise it would be biased). What about the variance of this estimator? Yi independence 1 1 1 ˆ = Var[θML ] = Var Var[Yi ] = 2 ·n·θ (1−θ ) = θ (1−θ ), 2 n n n n and we see that the variance of the estimator depends on the true value of θ . For multivariate θ, it is useful to examine the error covariance matrix given by " ! P = E[ee ] = E (θˆ − θ )(θˆ − θ ) . When estimating θ , our goal is to minimize the estimation error. This error can be expressed using loss functions. Supposing our parameter vector θ takes values on some space , a loss function L(θˆ ) is a mapping from × into R which quantifies the “loss” incurred by estimating θ with θˆ . We have already seen loss functions in earlier chapters, but we shall restate the definitions here for completeness. One frequently used loss function is the absolute error, # ˆ ˆ L1 (θ , θ ) := θ − θ 2 = (θˆ − θ ) (θˆ − θ ), where · 2 is the Euclidean norm (it coincides with the absolute value when ⊆ R). One advantage of the absolute error is that it has the same units as θ . We use the squared error perhaps even more frequently than the absolute error: L2 (θˆ , θ ) := θˆ − θ 22 = (θˆ − θ ) (θˆ − θ ).

5 The Bias–Variance Tradeoff (Dilemma) for Estimators

55

While the squared error has the disadvantage compared to the absolute error of being expressed in √quadratic units of θ , rather than the units of θ , it does not contain the cumbersome · and is therefore easier to deal with mathematically. The expected value of a loss function is known as the statistical risk of the estimator. The statistical risks corresponding to the above loss functions are, respectively, the mean absolute error, # " ! " ! ˆ θ ) :=E θˆ − θ 2 = E MAE(θˆ , θ ):=R1 (θˆ , θ ):=E L1 (θ, (θˆ − θ ) (θˆ − θ) , and, by far the most commonly used, mean squared error (MSE), " ! ! " ! " ˆ θ ) := E θˆ − θ 2 = E (θˆ − θ ) (θˆ − θ ) . MSE(θˆ , θ ) := R2 (θˆ , θ ) := E L2 (θ, 2 The square root of the mean squared error is called the root mean squared error (RMSE). The minimum mean squared error (MMSE) estimator is the estimator that minimizes the mean squared error.

5 The Bias–Variance Tradeoff (Dilemma) for Estimators It can easily be shown that the mean squared error separates into a variance and bias term: ! " MSE(θˆ , θ ) = trVar θˆ + bias(θˆ , θ ) 22 , where tr(·) is the trace operator. In the case of a scalar θ , this expression simplifies to ! " MSE(θˆ , θ ) = Var θˆ + bias(θˆ , θ )2 . In other words, the MSE is equal to the sum of the variance of the estimator and the squared bias. The bias–variance tradeoff or bias–variance dilemma consists in the need to minimize these two sources of error, the variance and bias of an estimator, in order to minimize the mean squared error. Sometimes there is a tradeoff between minimizing bias and minimizing variance to achieve the least possible MSE. The concept of a bias–variance tradeoff in machine learning will be revisited in Chap. 4, within the context of statistical learning theory.

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2 Probabilistic Modeling

6 Bayesian Inference from Data As before, let θ be the parameter of some statistical model and let y = y1 , . . . , yn be n i.i.d. observations of some random variable Y . We capture our subjective assumptions about the model parameter θ, before observing the data, in the form of a prior probability distribution p(θ). Bayes’ rule converts a prior probability into a posterior probability by incorporating the evidence provided by the observed data:

p(θ | y) =

p(y | θ ) p(θ ) p(y)

allows us to evaluate the uncertainty in θ after we have observed y. This uncertainty is characterized by the posterior probability p(θ | y). The effect of the observed data is expressed through p(y | θ)—a function of θ referred to as the likelihood function. It expresses how likely the observed dataset was generated by a model with parameter θ. Let us summarize some of the notation that will be important: • The prior is p(θ ); $ • The likelihood is p(y | θ ) = ni=1%p(yi | θ ), since the data is i.i.d.; • The marginal likelihood p(y) = p(y | θ )p(θ )dθ is the likelihood with the dependency on θ marginalized out; and • The posterior is p(θ | y).

•

> Bayesian Inference

Informally, Bayesian inference involves the following steps: 1. Formulate your statistical model as a collection of probability distributions conditional on different values for a parameter θ , about which you wish to learn; 2. Organize your beliefs about θ into a (prior) probability distribution; 3. Collect the data and insert them into the family of distributions given in Step 1; 4. Use Bayes’ rule to calculate your new beliefs about θ ; and 5. Criticize your model and revise your modeling assumptions.

6 Bayesian Inference from Data

57

The following example shall illustrate the application of Bayesian inference for the Bernoulli parameter θ . Example 2.3

Bernoulli Trials Example (continued)

θ is a probability, so it is bounded and must belong to the interval [0, 1]. We could assume that all values of θ in [0, 1] are equally likely. Thus our prior could be that θ is uniformly distributed on [0, 1], i.e. θ ∼ U(a = 0, b = 1). This assumption would constitute an application of Laplace’s principle of indifference, also known as the principle of insufficient reason: when faced with multiple possibilities, whose probabilities are unknown, assume that the probabilities of all possibilities are equal. In the context of Bayesian estimation, applying Laplace’s principle of indifference constitutes what is known as an uninformative prior. Our goal is, however, not to rely too much on the prior, but use the likelihood to proceed to a posterior based on new information. The pdf of the uniform distribution, U(a, b), is given by p(θ ) =

1 b−a

if θ ∈ [a, b] and zero elsewhere. In our case, a = 0, b = 1, and so our uninformative uniform prior is given by p(θ ) = 1, ∀θ ∈ [0, 1]. Let us derive the posterior based on this prior assumption. Bayes’ theorem tells us that posterior ∝ likelihood · prior, where ∝ stands for “proportional to,” so the left- and right-hand side are equal up to a normalizing constant which depends on the data but not on θ . The posterior is p(θ | x1:n ) ∝ p(x1:n | θ )p(θ ) = θ

xi

(1 − θ )n−

xi

· 1.

If the prior is uniform, i.e. p(θ ) = 1, then after n = 5 trials we see from the data that p(θ | x1:n ) ∝ θ (1 − θ )4 .

(2.2)

After 10 trials we have (continued)

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2 Probabilistic Modeling

Example 2.3

(continued)

p(θ | x1:n ) ∝ θ (1 − θ )4 × θ (1 − θ )4 = θ 2 (1 − θ )8 .

(2.3)

From the shape of the resulting pdf, we recognize it as the pdf of the Beta distributiona

Beta θ | xi , n − xi , and we immediately know that the missing normalizing constant factor is 1 & '= xi , n − xi B

& ' & ' xi n − xi . (n)

Let us now assume that we have tossed the coin fifty times and, out of those fifty coin tosses, we get heads on twelve. Then our posterior distribution becomes θ | x1:n ∼ Beta(θ | 12, 38). Then, from the properties of this distribution,

E[θ | x1:n ] =

xi = xi + (n − xi )

12 12 xi = = = 0.24, n 12 + 38 50

& ' & ' xi n − xi (2.4) Var[θ | x1:n ] = & ' '2 & xi + n − xi xi + n − xi + 1 & '2 xi n xi − 12 · 38 456 = = = 0.00357647058. = 2 2 n (n + 1) (12 + 38) (12 + 38 + 1) 127500 (2.5)

# 456 The standard deviation being, in units of probability, = 127500 0.05980360012. Notice that the mean of the posterior, 0.24, matches the frequentist maximum likelihood estimate of θ , θˆML , and our intuition. Again, it is not unreasonable to assume that the probability of getting heads is 0.24 if we observe heads on twelve out of fifty coin tosses. a The

function’s argument is now θ, not xi , so it is not the pdf of a Bernoulli distribution.

6 Bayesian Inference from Data first 10 trials 0 1 2 3 4 5 6 7

first 5 trials 0 1 2 3 4 5 6 7

Fig. 2.1 The posterior distribution of θ against successive numbers of trials. The x-axis shows the values of theta. The shape of the distribution tightens as the Bayesian model is observed to “learn”

59

0.0 0.2 0.4

0.0 0.2 0.4

0.6 0.8 1.0

0.6 0.8 1.0

θ

first 40 trials

first 50 trials 0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

θ

0.0 0.2 0.4

θ

0.6 0.8 1.0

0.0 0.2 0.4

0.6 0.8 1.0

θ

Note that we did not need to evaluate the marginal likelihood in the example above, only the θ dependent terms were evaluated for the purpose of the plot. Thus each plot in Fig. 2.1 is only representative of the posterior up to a scaling.

•

! The Principle of Indifference

In practice, the principle of indifference should be used with great care, as we are assuming a property of the data strictly greater than we know. Saying “the probabilities of the outcomes are equally likely” contains strictly more information than “I don’t know what the probabilities of the outcomes are.” If someone tosses a coin and then covers it with her hand, asking you, “heads or tails?” it is probably relatively sensible to assume that the two possibilities are equally likely, effectively assuming that the coin is unbiased. If an investor asks you, “Will the stock price of XYZ increase?” you should think twice before applying Laplace’s principle of indifference and replying “Well, there is a 50% chance that XYZ will grow, you can either long or short XYZ.” Clearly there are other important considerations such as the amount by which the stock could increase versus decrease, limits on portfolio exposure to market risk factors, and anticipation of other market events such as earnings announcements. In other words, the implications of going long or short will not necessarily be equal.

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6.1 A More Informative Prior: The Beta Distribution Continuing with the above example, let us question our prior. Is it somewhat too uninformative? After all, most coins in the world are probably close to being unbiased. We could use a Beta(α, β) prior instead of the Uniform prior. Picking α = β = 2, for example, will give a distribution on [0, 1] centered on 12 , incorporating a prior assumption that the coin is unbiased. The pdf of this prior is given by p(θ ) =

1 θ α−1 (1 − θ )β−1 , ∀θ ∈ [0, 1], B(α, β)

and so the posterior becomes p(θ | x1:n ) ∝ p(x1:n | θ )p(θ ) =θ

xi

(1 − θ )n−

xi

·

1 θ α−1 (1 − θ )β−1 ∝ θ (α+ xi )−1 (1 − θ )(β+n− xi )−1 , B(α, β)

which we recognize as a pdf of the distribution

xi . Beta θ | α + xi , β + n − Why did we pick this prior distribution? One reason is that its pdf is defined over the compact interval [0, 1], unlike, for example, the normal distribution, which has tails extending to −∞ and +∞. Another reason is that we are able to choose parameters which center the pdf at θ = 12 , incorporating the prior assumption that the coin is unbiased. If we initially assume a Beta(θ | α = 2, β = 2) prior, then the posterior expectation is α + xi α + xi = E [θ | x1:n ] = α + xi + β + n − xi α+β +n =

7 2 + 12 = ≈ 0.259. 2 + 2 + 50 27

Notice that both the prior and posterior belong to the same probability distribution family. In Bayesian estimation theory we refer to such prior and posterior as conjugate distributions (with respect to this particular likelihood function). Unsurprisingly, since now our prior assumption is that the coin is unbiased, 12 50 < 1 E [θ | x1:n ] < 2 . Perhaps surprisingly, we are also somewhat more certain about the posterior (its variance is smaller) than when we assumed the uniform prior.

6 Bayesian Inference from Data

61

Notice that the results of Bayesian estimation are sensitive—to varying degree in each specific case—to the choice of prior distribution: p(θ | α, β) =

(α + β − 1)! α−1 θ (1 − θ )β−1 = (α, β)θ α−1 (1 − θ )β−1 . (α − 1)!(β − 1)! (2.6)

So for the above example, this marginal likelihood would be evaluated with α = 13 and β = 39 since there are 12 observed 1s and 38 observed 0s.

6.2 Sequential Bayesian updates In the previous section we saw that, starting with the prior Beta (θ | α, β) , we arrived at the Beta-distributed posterior,

xi . Beta θ | α + xi , β + n − What would happen if, instead of observing all twelve coin tosses at once, we (i) considered each coin toss in turn; (ii) obtained our posterior; and (iii) used that posterior as a prior for an update based on the information from the next coin toss? The above two formulae give the answer to this question. We start with our initial prior, Beta (θ | α, β) , then, substituting n = 1 into the second formula, we get Beta (θ | α + x1 , β + 1 − x1 ) . Using this posterior as a prior before the second coin toss, we obtain the next posterior as Beta (θ | α + x1 + x2 , β + 2 − x1 − x2 ) . Proceeding along these lines, after all ten coin tosses, we end up with

xi , Beta θ | α + xi , β + n −

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the same result that we would have attained if we processed all ten coin tosses as a single “batch,” as we did in the previous section. This insight forms the basis for a sequential or iterative application of Bayes’ theorem—sequential Bayesian updates—the foundation for real-time Bayesian filtering. In machine learning, this mechanism for updating our beliefs in response to new data is referred to as “online learning.”

6.2.1

Online Learning

An important aspect of Bayesian learning is the capacity to update the posterior in response to the arrival of new data, y . The posterior over y now becomes the prior, and the new posterior is updated to p(θ | y , y) = %

6.2.2

p(y | θ )p(θ | y) . θ∈ p(y | θ )p(θ | y)dθ

(2.7)

Prediction

In auto-correlated data, often encountered in financial econometrics, it is common to use Bayesian models for prediction. We can write that the density of the new predicted value y given the previous data y is the expected value of the likelihood of the new data under the posterior density p(θ | y): p(y | y) = Eθ

| y [p(y

( | y, θ )] =

p(y | y, θ )p(θ | y)dθ.

(2.8)

θ∈

•

? Multiple Choice Question 1

Which of the following statements are true: 1. A frequentist performs statistical inference by finding the best fit parameters. The Bayesian finds the distribution of the parameters assuming a prior. 2. Frequentist inference can be regarded as a special case of Bayesian inference when the prior is a Dirac delta-function. 3. Bayesian inference is well suited to online learning, an experimental design under which the model is continuously updated as new data arrives. 4. Prediction, under Bayesian inference, is the conditional expectation of the predicted variable under the posterior distribution of the parameter.

7 Model Selection

63

6.3 Practical Implications of Choosing a Classical or Bayesian Estimation Framework If the sample size is large and the likelihood function “well-behaved” (which usually means a simple function with a clear maximum, plus a small dimension for θ), classical and Bayesian analysis are essentially on the same footing and will produce virtually identical results. This is because the likelihood function and empirical data will dominate any prior assumptions in the Bayesian approach. If the sample size is large but the dimensionality of θ is high and the likelihood function is less tractable (which usually means highly non-linear, with local maxima, flat spots, etc.), a Bayesian approach may be preferable purely from a computational standpoint. It can be very difficult to attain reliable estimates via maximum likelihood estimation (MLE) techniques, but it is usually straightforward to derive a posterior distribution for the parameters of interest using Bayesian estimation approaches, which often operate via sequential draws from known distributions. If the sample size is small, Bayesian analysis can have substantial advantages over a classical approach. First, Bayesian results do not depend on asymptotic theory to hold for their interpretability. Second, the Bayesian approach combines the sparse data with subjective priors. Well-informed priors can increase the accuracy and efficiency of the model. Conversely, of course, poorly chosen priors3 can produce misleading posterior inference in this case. Thus, under small sample conditions, the choice between Bayesian and classical estimation often distills to a choice between trusting the asymptotic properties of estimators and trusting one’s priors.

7 Model Selection Beyond the inference challenges described above, there are a number of problems with the classical approach to model selection which Bayesian statistics solves. For example, it has been shown by Breiman (2001) that the following three linear regression models have a residual sum of squares (RSS) which are all within 1%: Model 1

Yˆ = 2.1 + 3.8X3 − 0.6X8 + 83.2X13 − 2.1X17 + 3.2X27 ,

Model 2

Yˆ = − 8.9 + 4.6X5 + 0.01X6 + 12.0X15 + 17.5X21 + 0.2X22 , (2.10)

Model 3

Yˆ = − 76.7 + 9.3X2 + 22.0X7 − 13.2X8 + 3.4X11 + 7.2X28 .

(2.9)

(2.11) 3 For

example, priors that place substantial probability mass on practically infeasible ranges of θ—this often happens inadvertently when parameter transformations are involved in the analysis.

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You could, for example, think of each model being used to find the fair price of an asset Y , where each Xi are the contemporaneous (i.e., measured at the same time) firm characteristics. • Which model is better? • How would your interpretation of which variables are the most important change between models? • Would you arrive at different conclusions about the market signals if you picked, say, Model 1 versus Model 2? • How would you eliminate some of the ambiguity resulting from this outcome of statistical inference? Of course one direction is to simply analyze the F-scores of each independent variable and select the model which has the most statistically significant fitted coefficients. But this is unlikely to reliably discern the models when the fitted coefficients are comparable in statistical significance. It is well known that the goodness-of-fit measures, such as RSS’s and F-scores, do not scale well to more complex datasets where there are several independent variables. This leads to modelers drawing different conclusions about the same data, and is famously known as the “Rashomon effect.” Yet many studies and models in finance are still built this way and make use of information criterion and regularization techniques such as Akaike’s information criteria (AIC). A limitation for more robust frequentist model comparison is the requirement that the models being compared are “nested.” That is, one model should be a subset of the other model being compared, e.g. Model 1

Yˆ = β0 + β1 X1 + β2 X2

(2.12)

Model 2

Yˆ =

(2.13)

β0 + β1 X1 + β2 X2 + β11 X12 .

Model 1 is nested in Model 2 and we refer to the model selection as a “nested model selection.” In contrast to classical model selection, Bayesian model selection need not be restricted to nested models.

7.1 Bayesian Inference We now consider the more general setting—selection and updating of several candidate models in response to a dataset y. The “model” can be any data model, not just a regression, and the notation used here reflects that. In Bayesian inference, a model is a family of probability distributions, each of which can explain the observed data. More precisely, a model M is the set of likelihoods p(x n | θ ) over all possible parameter values .

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65

For example, consider the case of flipping a coin n times with an unknown bias θ ∈ ≡ [0, 1]. The data x n = {xi }ni=1 is now i.i.d. Bernoulli and if we observe the number of heads X = x, the model is the family of binomial distributions M := {P [X = x | n, θ ] =

) * n x θ (1 − θ )n−x }θ∈ . x

(2.14)

Each one of these distributions is a potential explanation of the observed head count x. In the Bayesian method, we maintain a belief over which elements in the model are considered plausible by reasoning about p(θ | x n ). See Example 1.1 for further details of this experiment. We start by re-writing the Bayesian inference formula with explicit inclusion of model indexes. You will see that we have dropped X since the exact composition of explanatory data is implicitly covered by model index Mi : p (θ i | x n , Mi ) =

p (θ i | Mi ) p (x n | θ i , Mi ) p (x n | Mi )

i = 1, 2.

(2.15)

This expression shows that differences across models can occur due to differing priors for θ and/or differences in the likelihood function. The marginal likelihood in the denominator will usually also differ across models.

7.2 Model Selection So far, we just considered parameter inference when the model has already been selected. The Bayesian setting offers a very flexible framework for the comparison of competing models—this is formally referred to as “model selection.” The models do not have to be nested—all that is required is that the competing specifications share the same x n . Suppose there are two models, denoted M1 and M2 , each associated with a respective set of parameters θ 1 and θ 2 . We seek the most “probable” model given the observed data x n . We first apply Bayes’ rule to derive an expression for the posterior model probability p (Mi ) p (x n | Mi ) ' & ' p (Mi | x n ) = & j p x n | Mj p Mj

i = 1, 2.

(2.16)

Here p (Mi ) is a prior distribution over models that we have selected; a common practice is to set this to a uniform distribution over the models. The value p (x n | Mi ) is a marginal likelihood function—a likelihood function over the space of models in which the parameters have been marginalized out:

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( p(x n | Mi ) =

θ i ∈i

p(x n | θ i , Mi )p(θ i | Mi )dθ i .

(2.17)

From a sampling perspective, this marginal likelihood can be interpreted as the probability that the model could have generated the observed data, under the chosen prior belief over its parameters. More precisely, the marginal likelihood can be viewed as the probability of generating x n from a model Mi whose parameters θ i ∈ i are sampled at random from the prior p(θ i | Mi ). For this reason, it is often referred to here as the model evidence and plays an important role in model selection that we will see later. We can now construct the posterior odds ratio for the two models as p (M1 ) p (x n | M1 ) p (M1 | x n ) = , p (M2 | x n ) p (M2 ) p (x n | M2 )

(2.18)

which is simply the prior odds multiplied by the ratio of the evidence for each model. Under equal model priors (i.e., p (M1 ) = p (M2 )) this reduces to the Bayes’ factor for Model 1 vs. 2, i.e. B1,2 =

p (x n | M1 ) , p (x n | M2 )

(2.19)

which is simply the ratio of marginal likelihoods for the two models. Since Bayes’ factors can become quite large, we usually prefer to work with its logged version logB1,2 = logp (x n | M1 ) − logp (x n | M2 ) .

(2.20)

The derivation of BFs and thus model comparison is straightforward if expressions for marginal likelihoods are analytically known or can be easily derived. However, often this can be quite tricky, and we will learn a few techniques to compute marginal likelihoods in this book.

7.3 Model Selection When There Are Many Models Suppose now that a set of models {Mi } may be used to explain the data x n . θ i represents the parameters of model Mi . Which model is “best”? We answer this question by estimating the posterior distribution over models: % p(Mi | x n ) =

θ i ∈i

p(x n | θ i , Mi )p(θ i | Mi )dθ i p(Mi ) . j p(x n | Mj )p(Mj )

(2.21)

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Table 2.1 Jeffreys’ scale is used to assess the comparative strength of evidence in favor of one model over another

As before we can compare any two models via the posterior odds, or if we assume equal priors, by the BFs. Model selection is always relative rather than absolute. We must always pick a reference model M2 and decide whether model M1 has more strength. We use Jeffreys’ scale to assess the strength of evidence as shown in Table 2.1. Example 2.4

Model Selection

You compare two models for explaining the behavior of a coin. The first model, M1 , assumes that the probability of a head is fixed to 0.5. Notice that this model does not have any parameters. The second model, M2 , assumes the probability of a head is set to an unknown θ ∈ = (0, 1) with a uniform prior on θ : p(θ | M2 ) = 1. For simplicity, we additionally choose a uniform model prior p(M1 ) = p(M2 ). Suppose we flip the coin n = 200 times and observe X = 115 heads. Which model should we prefer in light of this data? We compute the model evidence for each model. The model evidence for M1 ) * n 1 p(x | M1 ) = ≈ 0.005956. (2.22) x 2200 The model evidence of M2 requires integrating over θ : ( p(x | M2 ) = 0

( =

=

1

p(x | θ, M2 )p(θ | M2 )dθ

1 )n*

x

θ 115 (1 − θ )200−115 dθ

1 ≈ 0.004975. 201

(2.23) (2.24) (2.25) (continued)

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2 Probabilistic Modeling

Example 2.4

(continued)

Note that we have used the definition of the Beta density function p(θ | α, β) =

(α + β − 1)! α−1 θ (1 − θ )β−1 (α − 1)!(β − 1)!

(2.26)

to evaluate the integral in the marginal density function above. The Bayes’ factor in favor of M1 is 1.2 and thus |lnB| = 0.182 and there is no evidence in favor of M1 .

•

! Frequentist Approach

An interesting aside here is that a frequentist hypothesis test would reject the null hypothesis θ = 0.5 at the α = 0.05 level. The probability of generating at least 115 heads under model M1 is approximately 0.02. The probability of generating at least 115 tails is also 0.02. So a two-sided test would give a p-value of approximately 4%.

•

! Hyperparameters

We note in passing that the prior distribution in the example above does not involve any parameterization. If the prior is a parameterized distribution, then the parameters of the prior are referred to as hyperparameters. The distributions of the hyperparameters are known as hyperpriors. “Bayesian hierarchical modeling” is a statistical model written in multiple levels (hierarchical form) that estimates the parameters of the posterior distribution using the Bayesian method.

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69

7.4 Occam’s Razor The model evidence performs a vital role in the prevention of model over-fitting. Models that are too simple are unlikely to generate the dataset. On the other hand, models that are too complex can generate many possible data sets, but they are unlikely to generate any particular dataset at random. Bayesian inference therefore automates the determination of model complexity using the training data x n alone and does not need special “fixes” (a.k.a regularization and information criteria) to prevent over-fitting. The underlying philosophical principle of selecting the simplest model, if given a choice, is known as “Occam’s razor” (Fig. 2.2). We maintain a belief over which parameters in the model we consider plausible by reasoning with the posterior p(θ i | x n , Mi ) =

p(x n | θ i , Mi )p(θ i | Mi ) , p(x n | Mi )

(2.27)

and we may choose the parameter value which maximizes the posterior distribution (MAP).

7.5 Model Averaging Marginal likelihoods can also be used to derive model weights in Bayesian model averaging (BMA). Informally, the intuition behind BMA is that we are never fully convinced that a single model is the correct one for our analysis at hand. There are usually several (and often millions of) competing specifications. To explicitly incorporate this notion of “model uncertainty,” one can estimate every model separately, compute relative probability weights for each model, and then generate model-averaged posterior distributions for the parameters (and predictions) Fig. 2.2 The model evidence p(D | m) performs a vital role in the prevention of model over-fitting. Models that are too simple are unlikely to generate the dataset. Models that are too complex can generate many possible data sets, but they are unlikely to generate any particular dataset at random. Source: Rasmussen and Ghahramani (2001)

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of interest. We often choose BMA when there is not strong enough evidence for any particular model. Prediction of a new point y∗ under BMA is given over m models as the weighted average p(y∗ |y) =

m

p(y∗ |y, Mi )p(Mi |y).

(2.28)

i=1

Note that model-averaged prediction would be cumbersome to accomplish in a classical framework, and thus constitutes another advantage of employing a Bayesian estimation approach.

•

? Multiple Choice Question 2

Which of the following statements are true: 1. Bayesian inference is ideally suited to model selection because the model evidence effectively penalizes over-parameterized models. 2. The principle of Occam’s razor is to simply choose the model with the least bias. 3. Bayesian model averaging uses the uncertainty from the model to weight the output from each model. 4. Bayesian model averaging is a method of consensus voting between models—the best candidate model is selected for each new observation. 5. Hierarchical Bayesian modeling involves nesting Bayesian models through parameterizations of prior distributions and their distributions.

8 Probabilistic Graphical Models Graphical models (a.k.a. Bayesian networks) are a method for representing relationships between random variables in a probabilistic model. They provide a useful tool for big data, providing graphical representations of complex datasets. To see how graphical models arise, we can revisit the familiar perceptron model from the previous chapter in a probabilistic framework, i.e. the network weights are now assumed to be probabilistic. As a starting point, consider a logistic regression classifier with probabilistic output: P [G | X] = σ (U ) =

1 , U = wX + b, G ∈ {0, 1}, X ∈ Rp . 1 + e−U

(2.29)

8 Probabilistic Graphical Models

71

By Bayes’ law, we know that the posterior probabilities must be given by the likelihood, prior and evidence: P [G | X] =

P [X | G] P [G] = P [X]

1

1+e

, | G]) P[G] − log PP[X [X | Gc ] +log P[Gc ]

(2.30)

where Gc is the complement of G. So the outputs are only posterior probabilities when the weights and biases are, respectively, likelihood and log-odds ratios: P Xj | G , ∀j ∈ {1, . . . , p}, wj = P Xj | Gc

* P [G] . b = log P [Gc ] )

(2.31)

In particular, the Xj s must be conditionally independent over G; otherwise, the outputs from the logistic regression are not the true posterior probabilities. Put differently, the posteriors of the input given the class can only be found when the input is mutually independent given the class G. In this case, the logistic regression is a naive Bayes’ classifier—a type of generative model which the joint dis$p models tribution as the products of marginals, P [X, G] = P [G] j =1 P Xj | G . Hence, under this data assumption, logistic regression is the discriminative counterpart to naive Bayes. Figure 2.3b shows an example of an equivalent logistic regression models and naive Bayes’ binary classifier for the case when the inputsare binary. Furthermore, if the conditional density functions of the inputs, P Xj | G , are Gaussian (but not necessarily identical), then we can establish equivalence between logistic regression and a Gaussian naive Bayes’ classifier. See Exercise 2.7 for establishing the equivalence when the inputs are binary. The graphical model captures the causal process (Pearl, 1988) by which the observed data was generated. For this reason, such models are often called generative models. Fig. 2.3 Logistic regression fw (X) = σ (w T X) and an equivalent naive Bayes’ classifier. (a) Logistic regression weights and resulting predictions. (b) A naive Bayes’ classifier with the same probabilistic output

X 1 X 2 F w (X )

−1.16 w = 2.23 −0.20

1 1 0 0

1 0 1 0

0.70 0.74 0.20 0.24

(a) G X1

X2

P 1 (c )

G

0.5

1 0

(b)

1 [X 1

| G]

0.8 0.3

G 1 0

1 [X 2

| G]

0.45 0.5

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2 Probabilistic Modeling

Naive Bayes’ classification is the simplest form of a probabilistic graphical model (PGM) with a directed graph G = (X, E), where the edges, E, represent the conditional dependence between the random variables X = (X, Y ). For example, Fig. 2.3b shows the dependence of the response G on X in the naive Bayes’ classifier. Such graphs, provided they are directed, are often referred to as “Bayesian networks.” Such graphical model captures the causal process by which the observed data was generated. If the graph is undirected—an undirected graphical model (UGM), as in restricted Boltzmann machines (RBMs), then the network is referred to as a “Markov network” or Markov random field. RBMs have the specific restriction that there are no observed–observed and hidden–hidden node connections. RBMs are an example of continuous latent variable models which find the probability of an observed variable by marginalizing over the continuous latent variable, Z, ( p(x) = p(x | z)p(z)dz. (2.32) This type of graphical model corresponds to that of factor analysis. Other related types of graphical models include mixture models.

8.1 Mixture Models A standard mixture probability density of a continuous and independently (but not identically) distributed random variable X, whose value is denoted by x, is defined as p(x; υ) =

K

πk p(x; θk ).

(2.33)

k=1

The mixture density has K components (or states) and is defined by the parameter set υ = {θ, π }, where π = {π1 , · · · , πK } is the set of weights given to each component and θ = {θ1 , · · · , θK } is the set of parameters describing each component distribution. A well-known mixture model is the Gaussian mixture model (GMM): p(x) =

K

πk N x; μk , σk2 ,

(2.34)

k=1

where each component parameter vector θk is the mean and variance parameters, μk and σk2 .

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73

When X is discrete, the graphical model is referred to as a “discrete mixture model.” Examples of GMMs are common in finance. Risk managers, for example, speak in terms of “correlation breakdowns,” “market contagion,” and “volatility shocks.” Finger (1997) presents a two-component GMM for modeling risk under normal and stressed market scenarios which has become standard methodology for stressed Value-at-Risk and Economic Capital modeling in the investment banking sector. Mixture models can also be used to cluster data and have a non-probabilistic analog called the K-means algorithm which is a well-known unsupervised learning method used in finance and other fields. Before such a model can be fitted, it is necessary to introduce an additional variable which represents the current state of the data, i.e. which of the mixture component distributions is the current observation drawn from.

8.1.1

Hidden Indicator Variable Representation of Mixture Models

Let us first suppose that the independent random variable, X, has been observed over N data points, xN = {x1 , · · · , xN }. The set is assumed to be generated by a K-component mixture model. To indicate the mixture component from which a sample was drawn, we introduce an independent hidden (a.k.a. latent) discrete random variable, S ∈ {1, . . . , K}. For each observation xi , the value of S is denoted as si , and is encoded as a binary vector of length K. We set the vector’s k-th component, (si )k = 1 to indicate that the k-th mixture component is selected, while all other states are set to 0. As a consequence, 1=

K (si )k .

(2.35)

k=1

We can now specify the joint probability distribution of X and S in terms of a marginal density p(si ; π ) and a conditional density p(xi | si ; θ ) as p(xn , sn ; υ) =

N

p(xi |si ; θ )p(si ; π ),

(2.36)

i

where the marginal densities p(si ; π ) are drawn from a multinomial distribution that is parameterized by the mixing weights π = {π1 , · · · , πK }: p(si ; π ) =

K k=1

or, more simply,

(si )k

πk

,

(2.37)

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2 Probabilistic Modeling

P [(si )k = 1] = πk .

(2.38)

Naturally the mixing weights, πk ∈ [0, 1], must satisfy 1=

K

πk .

(2.39)

k=1

8.1.2

Maximum Likelihood Estimation

The maximum likelihood method of estimating mixture models is known as the expectation–maximization (EM) algorithm. The goal of the EM is to maximize the likelihood of the data given the model, i.e. maximize L(υ) = log

+ p(xn , sn ; υ) =

s

K N (si )k log {πk p(xi ; θk )} .

(2.40)

i=1 k=1

If the sequence of states sn were known, then the estimation of the model parameters π, θ would be straightforward; conditioned on the state variables and the observations, Eq. 2.40 could be maximized with respect to the model parameters. However, the value of the state variable is unknown. This suggests an alternative two-stage iterated optimization algorithm: If we know the expected value of S, one could use this expectation in the first step to perform a weighted maximum likelihood estimation of Eq. 2.40 with respect to the model parameters. These estimates will be incorrect since the expectation S is inaccurate. So, in the second step, one could update the expected value of all S pretending the model parameters υ := (π, θ ) are known and held fixed at their values from the past iteration. This is precisely the strategy of the expectation–maximization (EM) algorithm—a statistically selfconsistent, iterative, algorithm for maximum likelihood estimation. With the context of mixture models, the EM algorithm is outlined as follows: • E-step: In this step, the parameters υ are held fixed at the old values, υ old , obtained from the previous iteration (or at their initial settings during the algorithm’s initialization). Conditioned on the observations, the E-step then computes the probability density of the state variables Si , ∀i given the current model parameters and observation data, i.e. p(si | xi , υ old ) ∝ p(xi | si ; θ )p(si ; π old ). In particular, we compute

(2.41)

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75

p(xi | (si )k = 1; θk )πk P((si )k = 1 | xi , υ old ) = . p(xi | (si ) = 1; θ )π

(2.42)

The likelihood terms p(xi | (si )k = 1; θk ) are evaluated using the observation densities defined for each of the states. • M-step: In this step, the hidden state probabilities are considered given and maximization is performed with respect to the parameters: υ new = arg max L(υ). υ

(2.43)

This results in the following update equations for the parameters in the probability distributions: 1 N i=1 (γi )k xi μk = N N i=1 (γi )k N 1 i=1 (γi )k (xi − μk )2 , ∀k ∈ {1, . . . , K}, σk2 = N N i=1 (γi )k

(2.44)

(2.45)

where (γi )k := E[(si )k | xi ] are the responsibilities—conditional expectations which measure how strongly a data point, xi , “belongs” to each component, k, of the mixture model. The number of components needed to model the data depends on the data and can be determined by a Kolmogorov–Smirnoff test or based on entropy criteria. Heavy tailed data required at least two light tailed components to compensate. More components require larger sample sizes to ensure adequate fitting. In the extreme case there may be insufficient data available to calibrate a given mixture model with a certain degree of accuracy. In summary, while GMMs are flexible they may not be the most appropriate model. If more is known about the data distribution, such as its behavior in the tails, incorporation of this knowledge can only help improve the model.

•

? Multiple Choice Question 3

Which of the following statements are true: 1. Mixture models assume that the data is multi-modal and drawn from a linear combination of uni-modal distributions. 2. The expectation–maximization (EM) algorithm is a type of iterative unsupervised learning algorithm which alternates between updating the probability density of the state variables, based on model parameters (E-step) and updating the parameters by maximum likelihood estimation (M-step).

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3. The EM algorithm automatically determines the modality of the distribution and hence the number of components. 4. A mixture model is only appropriate for use in finance if the modeler specifies which component is the most relevant for each observation.

9 Summary Probabilistic modeling is an important class of models in financial data, which is often noisy and incomplete. Additionally much of finance rests on being able to make financial decisions under uncertainty, a task perfectly suited to probabilistic modeling. In this chapter we have identified and demonstrated how probabilistic modeling is used for financial modeling. In particular we have: – Applied Bayesian inference to data using simple probabilistic models; – Show how linear regression with probabilistic weights can be viewed as a simple probabilistic graphical model; and – Developed more versatile representations of complex data with probabilistic graphical models such as mixture models and hidden Markov models.

10 Exercises Exercise 2.1: Applied Bayes’ Theorem An accountant is 95 percent effective in detecting fraudulent accounting when it is, in fact, present. However, the audit also yields a “false positive” result for one percent of the non-fraudulent companies audited. If 0.1 percent of the companies are actually fraudulent, what is the probability that a company is fraudulent given that the audit revealed fraudulent accounting? Exercise 2.2*: FX and Equity A currency strategist has estimated that JPY will strengthen against USD with probability 60% if S&P 500 continues to rise. JPY will strengthen against USD with probability 95% if S&P 500 falls or stays flat. We are in an upward trending market at the moment, and we believe that the probability that S&P 500 will rise is 70%. We then learn that JPY has actually strengthened against USD. Taking this new information into account, what is the probability that S&P 500 will rise? Hint: | A) Recall Bayes’ rule: P (A | B) = P (B P (B) P (A). Exercise 2.3**: Bayesian Inference in Trading Suppose there are n conditionally independent, but not identical, Bernoulli trials G1 , . . . , Gn generated from the map P (Gi = 1 | X = xi ) = g1 (xi | θ ) with θ ∈ [0, 1]. Show that the likelihood of G | X is given by

10 Exercises

77

p(G | X, θ ) =

n

(g1 (xi | θ ))Gi · (g0 (xi | θ ))1−Gi

(2.46)

i=1

and the log-likelihood of G | X is given by ln p(G | X, θ ) =

n

Gi ln(g1 (xi | θ )) + (1 − Gi )ln(g0 (xi | θ )).

(2.47)

i=1

Using Bayes’ rule, write the condition probability density function of θ (the “posterior”) given the data (X, G) in terms of the above likelihood function. From the previous example, suppose that G = 1 corresponds to JPY strengthening against the dollar and X are the S&P 500 daily returns and now g1 (x | θ ) = θ 1x>0 + (θ + 35)1x≤0 .

(2.48)

Starting with a neutral view on the parameter θ (i.e., θ ∈ [0, 1]), learn the distribution of the parameter θ given that JPY strengthens against the dollar for two of the three days and S&P 500 is observed to rise for 3 consecutive days. Hint: You can use the Beta density function with a scaling constant (α, β) (α + β − 1)! α−1 θ (1 − θ )β−1 = (α, β)θ α−1 (1 − θ )β−1 (α − 1)!(β − 1)! (2.49) to evaluate the integral in the marginal density function. If θ represents the currency analyst’s opinion of JPY strengthening against the dollar, what is the probability that the model overestimates the analyst’s estimate? p(θ |α, β) =

Exercise 2.4*: Bayesian Inference in Trading Suppose that you observe the following daily sequence of directional changes in the JPY/USD exchange rate (U (up), D(down or stays flat)): U, D, U, U, D and the corresponding daily sequence of S&P 500 returns is -0.05, 0.01, -0.01, -0.02, 0.03 You propose the following probability model to explain the behavior of JPY against USD given the directional changes in S&P 500 returns: Let G denote a Bernoulli R.V., where G = 1 corresponds to JPY strengthening against the dollar and r are the S&P 500 daily returns. All observations of G are conditionally independent (but *not* identical) so that the likelihood is p(G | r, θ ) =

n i=1

p(G = Gi | r = ri , θ )

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2 Probabilistic Modeling

where p(Gi = 1 | r = ri , θ ) =

θu ,

ri > 0

θd , ri ≤ 0.

Compute the full expression for the likelihood that the data was generated by this model. Exercise 2.5: Model Comparison Suppose you observe the following daily sequence of direction changes in the stock market (U (up), D(down)): U, D, U, U, D, D, D, D, U, U, U, U, U, U, U, D, U, D, U, D, U, D, D, D, D, U, U, D, U, D, U, U, U, D, U, D, D, D, U, U, D, D, D, U, D, U, D, U, D, D

You compare two models for explaining its behavior. The first model, M1 , assumes that the probability of an upward movement is fixed to 0.5 and the data is i.i.d. The second model, M2 , also assumes the data is i.i.d. but that the probability of an upward movement is set to an unknown θ ∈ = (0, 1) with a uniform prior on θ : p(θ |M2 ) = 1. For simplicity, we additionally choose a uniform model prior p(M1 ) = p(M2 ). Compute the model evidence for each model. Compute the Bayes’ factor and indicate which model should we prefer in light of this data? Exercise 2.6: Bayesian Prediction and Updating Using Bayesian prediction, predict the probability of an upward movement given the best model and data in Exercise 2.5. Suppose now that you observe the following new daily sequence of direction changes in the stock market (U (up), D(down)): D, U, D, D, D, D, U, D, U, D, U, D, D, D, U, U, D, U, D, D, D, U, U, D, D, D, U, D, U, D, U, D, D, D, U, D, U, D, U, D, D, D, D, U, U, D, U, D, U, U

Using the best model from Exercise 2.5, compute the new posterior distribution function based on the new data and the data in the previous question and predict the probability of an upward price movement given all data. State all modeling assumptions clearly. Exercise 2.7: Logistic Regression Is Naive Bayes Suppose that G and X ∈ {0, 1}p are Bernoulli random $ variables and the Xi s are p mutually independent given G—that is, P [X | G] = i=1 P [Xi | G]. Given a naive Bayes’ classifier P [G | X], show that the following logistic regression model produces equivalent output if the weights are

10 Exercises

79

P [G] P [Xi = 0 ∈ G] + log P [Gc ] P [Xi = 0 ∈ Gc ] p

w0 = log

i=1

P [Xi = 1 ∈ G] P [Xi = 0 ∈ Gc ] , · wi = log P [Xi = 1 ∈ Gc ] P [Xi = 0 ∈ G]

i = 1, . . . , p.

Exercise 2.8**: Restricted Boltzmann Machines Consider a probabilistic model with two types of binary variables: visible binary stochastic units v ∈ {0, 1}D and hidden binary stochastic units h ∈ {0, 1}F , where D and F are the number of visible and hidden units, respectively. The joint probability density to observe their values is given by the exponential distribution 1 exp (−E(v, h)) exp (−E(v, h)) , Z = Z

p(v, h) =

v,h

and where the energy E(v, h) of the state {v, h} is E(v, h) = −v W h − b v − a h = − T

T

T

F D i=1 j =1

Wij vi hj −

D

bi vi −

F

aj hj ,

j =1

i=1

with model parameters a, b, W . This probabilistic model is called the restricted Boltzmann machine. Show that conditional probabilities for visible and hidden nodes are given by the sigmoid function σ (x) = 1/(1 + e−x ): ⎛ P [vi = 1 | h] = σ ⎝

⎛ ⎞ Wij hj + bi ⎠ , P [hi = 1 | v] = σ ⎝ Wij vj + ai ⎠ . ⎞

j

j

Appendix Answers to Multiple Choice Questions Question 1 Answer: 1,3,4. Question 2 Answer: 1,3,5. Question 3 Answer: 1,2. Mixture models assume that the data is multi-modal—the data is drawn from a linear combination of uni-modal distributions. The expectation– maximization (EM) algorithm is a type of iterative, self-consistent, unsupervised

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learning algorithm which alternates between updating the probability density of the state variables, based on model parameters (E-step) and updating the parameters by maximum likelihood estimation (M-step). The EM algorithm does not automatically determine the modality of the data distribution, although there are statistical tests to determine this. A mixture model assigns a probabilistic weight for every component that each observation might belong to. The component with the highest weight is chosen.

References Breiman, L. (2001). Statistical modeling: The two cultures (with comments and a rejoinder by the author). Statistical Science 16(3), 199–231. Duembgen, M., & Rogers, L. C. G. (2014). Estimate nothing. https://arxiv.org/abs/1401.5666. Finger, C. (1997). A methodology to stress correlations, fourth Quarter. RiskMetrics Monitor. Rasmussen, C. E., & Ghahramani, Z. (2001). Occam’s razor. In In Advances in Neural Information Processing Systems 13, (pp. 294–300). MIT Press.

Chapter 3

Bayesian Regression and Gaussian Processes

This chapter introduces Bayesian regression and shows how it extends many of the concepts in the previous chapter. We develop kernel based machine learning methods—specifically Gaussian process regression, an important class of Bayesian machine learning methods—and demonstrate their application to “surrogate” models of derivative prices. This chapter also provides a natural starting point from which to develop intuition for the role and functional form of regularization in a frequentist setting—the subject of subsequent chapters.

1 Introduction In general, it is difficult to develop intuition about how the distribution of weights in a parametric regression model represents the data. Rather than induce distributions over variables, as we have seen in the previous chapter, we could instead induce distributions over functions. Specifically, we can express those intuitions using a “covariance kernel .” We start by exploring Bayesian regression in a more general setup that enables us to easily move from a toy regression model to a more complex non-parametric Bayesian regression model, such as Gaussian process regression. By introducing Bayesian regression in more depth, we show how it extends many of the concepts in the previous chapter. We develop kernel based machine learning methods (specifically Gaussian process regression), and demonstrate their application to “surrogate” models of derivative prices.1

1 Surrogate

models learn the output of an existing mathematical or statistical model as a function of input data. © Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_3

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Chapter Objectives The key learning points of this chapter are: – Formulate a Bayesian linear regression model; – Derive the posterior distribution and the predictive distribution; – Describe the role of the prior as an equivalent form of regularization in maximum likelihood estimation; and – Formulate and implement Gaussian Processes for kernel based probabilistic modeling, with programming examples involving derivative modeling.

2 Bayesian Inference with Linear Regression Consider the following linear regression model which is affine in x ∈ R: y = f (x) = θ0 + θ1 x, θ0 , θ1 ∼ N(0, 1), x ∈ R,

(3.1)

and suppose that we observe the value of the function over the inputs x := [x1 , . . . , xn ]. The random parameter vector θ := [θ0 , θ1 ] is unknown. This setup is referred to as “noise-free,” since we assume that y is strictly given by the function f (x) without noise. The graphical model representation of this model is given in Fig. 3.1 and clearly specifies that the ith model output only depends on xi . Note that the graphical model also holds in the case when there is noise. In the noise-free setting, the expectation of the function under known data is Eθ [f (xi )|xi ] = Eθ [θ0 ] + Eθ [θ1 ]xi = 0, ∀i,

(3.2)

where the expectation operator is w.r.t. the prior density of θ ( Eθ [·] =

Fig. 3.1 This graphical model represents Bayesian linear regression. The features x := {xi }ni=1 and responses y := {yi }ni=1 are known and the random parameter vector θ is unknown. The ith model output only depends on xi

(·)p(θ )dθ .

(3.3)

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Then the covariance of the function values between any two points, xi and xj is Eθ [f (xi )f (xj )|xi , xj ] = Eθ [θ02 + θ0 θ1 (xi + xj ) + θ12 xi xj ]

(3.4)

= Eθ [θ02 ] + Eθ [θ02 ]xi xj + Eθ [θ0 θ1 ](xi + xj ), (3.5) = 1 + xi xj ,

(3.6)

where the last term is zero because of the independence of θ0 and θ1 . Then any collection of function values [f (x1 ), . . . , f (xn )] with given data has a joint Gaussian distribution with covariance matrix Kij := Eθ [f (xi )f (xj )|xi , xj ] = 1 + xi xj . Such a probabilistic model is the simplest example of a more general, non-linear, Bayesian kernel learning method referred to as “Gaussian Process Regression” or simply “Gaussian Processes” (GPs) and is the subject of the later material in this chapter. Noisy Data The above example is in a noise-free setting where the function values [f (x1 ), . . . , f (xn )] are observed. In practice, we do not observe these function values, but rather some target values y = [y1 , . . . , yn ] which depend on x by the function, f (x), and some zero-mean Gaussian i.i.d. additive noise with known variance σn2 : yi = f (xi ) + i , i ∼ N(0, σn2 ).

(3.7)

Hence the observed i.i.d. data is D := (x, y). Following Rasmussen and Williams (2006), under this noise assumption and the linear model we can write down the likelihood function of the data: p(y|x, θ ) =

n

p(yi |xi , θ )

i=1

= √

1 2π σn

exp{−(yi − xi θ1 − θ0 )2 /(2σn2 )}

and hence y|x, θ ∼ N(θ0 + θ1 x, σn2 I ). Bayesian inference of the parameters in this linear regression model is based on the posterior distribution over the weights: p(θi |y, x) =

p(y|x, θi )p(θi ) , i ∈ {0, 1}, p(y|x)

(3.8)

where the marginal likelihood in the denominator is given by integrating over the parameters as

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( p(y|x) =

p(y|x, θ )p(θ)dθ .

(3.9)

If we define the matrix X, where [X]i := [1, xi ], and under more general conjugate priors, we have θ ∼ N(μ, ), y|X, θ ∼ N(θ T X, σn2 I ), and the product of Gaussian densities is also Gaussian, we can simply use standard results of moments of affine transformations to give E[y|X] = E[θ T X + ] = E[θ T ]X = μT X.

(3.10)

The conditional covariance is Cov(y|X) = Cov(θ T X)+σn2 I = XCov(θ )XT +σn2 I = XXT +σn2 I.

(3.11)

To derive the posterior of θ , it is convenient to transform the prior density function from a moment parameterization to a natural parameterization by completing the square. This is useful for multiplying normal density functions such as normalized likelihoods and conjugate priors. The quadratic form for the prior transforms to 1 p(θ ) ∝ exp{− (θ − μ)T −1 (θ − μ)}, 2 1 ∝ exp{μT −1 θ − θ T −1 θ}, 2

(3.12) (3.13)

where the 12 μT −1 μ term is absorbed in the normalizing term as it is independent of θ . Using this transformation, the posterior p(θ|D) is proportional to: p(y|X, θ )p(θ ) ∝ exp{−

1 1 (y − θ T X)T (y − θ T X)} exp{μT −1 − θ T −1 θ} 2 2σn2 (3.14)

∝ exp{−

1 1 (−2yθ T X + θ T XXT θ )} exp{μT −1 − θ T −1 θ } 2 2σn2 (3.15)

= exp{( −1 μ +

1 T T T 1 1 y X) θ − θ T ( −1 + 2 XXT )θ } 2 σn2 σn (3.16)

1 = exp{a T θ − θ T Aθ }. 2

(3.17)

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The posterior follows the distribution θ | D ∼ N(μ , ),

(3.18)

where the moments of the posterior are μ = a = ( −1 +

1 1 XXT )−1 ( −1 μ + 2 yT X) 2 σn σn

(3.19)

= A−1 = ( −1 +

1 XXT )−1 σn2

(3.20)

and we use the inverse of transformation above, from natural back to moment parameterization to write 1 p(θ |D) ∝ exp{− (θ − μ )T ( )−1 (θ − μ )}. 2

(3.21)

−1 , the inverse of a covariance matrix, is referred to as the precision matrix. The mean of this distribution is the maximum a posteriori (MAP) estimate of the weights—it is the mode of the posterior distribution. We will show shortly that it corresponds to the penalized maximum likelihood estimate of the weights, with a L2 (ridge) penalty term given by the log prior. Figure 3.2 demonstrates Bayesian learning of the posterior distribution of the weights. A bi-variate Gaussian prior is initially chosen for the prior distribution and there are an infinite number of possible lines that could be drawn in the data space [−1, 1] × [−1, 1]. The data is generated under the model f (x) = 0.3 + 0.5x with a small amount of additive i.i.d. Gaussian noise. As the number of points that the likelihood function is evaluated over increases, the posterior distribution sharpens and eventually contracts to a point. See the Bayesian Linear regression Python notebook for details of the implementation.

•

? Multiple Choice Question 1

Which of the following statements are true: 1. Bayesian regression treats the regression weights as random variables. 2. In Bayesian regression the data function f (x) is assumed to always be observed. 3. The posterior distribution of the parameters is always Gaussian if the prior is Gaussian. 4. The posterior distribution of the regression weights will typically contract with increasing data. 5. The mean of the posterior distribution depends on both the mean and covariance of the prior if it is Gaussian.

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Fig. 3.2 This figure demonstrates Bayesian inference for the linear model. The data has been generated from the function f (x) = 0.3 + 0.5x with a small amount of additive white noise. Source: Bishop (2006)

2.1 Maximum Likelihood Estimation Let us briefly revisit parameter estimation in a frequentist setting to solidify our understanding of Bayesian inference. Assuming that σn2 is a known parameter, we can easily derive the maximum likelihood estimate of the parameter vector, θˆ . The gradient of the negative log-likelihood function (a.k.a. loss function) w.r.t. θ is d d L(θ ) := − dθ dθ

,

n

log p(yi |xi , θ ) ,

i=1

=

1 d T 2 ||y − θ X|| + c 2 2σn2 dθ

=

1 (−yT X + θ T XT X), σn2

where the constant c := − n2 (log(2π ) + log(σn2 )). Setting this gradient to zero gives the orthogonal projection of y on to the subspace spanned by X:

2 Bayesian Inference with Linear Regression

θˆ = (XT X)−1 XT y,

87

(3.22)

where θˆ is the vector in the subspace spanned by X which is closest to y. This result states that the maximum likelihood estimate of an unpenalized loss function (i.e., without including the prior) is the OLS estimate when the noise variance is known. If the noise variance is unknown then the loss function is L(θ , σn2 ) =

n 1 ||y − θ T X||22 + c, log(σn2 ) + 2 2σn2

(3.23)

where now c = n2 log(2π ). Taking the partial derivative n 1 ∂L(θ, σn2 ) = − ||y − θ T X||22 , 2 2 ∂σn 2σn 2σn4

(3.24)

and setting it to zero gives2 σˆ n2 = n1 ||y − θ T X||22 . Maximum likelihood estimation is prone to overfitting and therefore should be avoided. We instead maximize the posterior distribution to arrive at the MAP estimate, θˆ MAP . Returning to the above computation under known noise: , n d d L(θ ) := − log p(yi |xi , θ ) + log p(θ ) , dθ dθ i=1 ) * d 1 1 T 2 T −1 = ||y − θ X||2 + (θ − μ) (θ − μ) + c dθ 2σn2 2 =

1 (−yT X + θ T XXT ) + (θ − μ)T −1 . σn2

Setting this derivative to zero gives 1 T (y X − θ T XXT ) = (θ − μ)T −1 , σn2

(3.25)

and after some rearrangement we obtain θˆ MAP = (XXT + σn2 −1 )−1 (σn2 Σ −1 μ + XT y) = A−1 (Σ −1 μ + σn−2 XT y), (3.26) which is equal to the mean of the posterior derived in Eq. 3.19. Of course, this is to be expected since the mean of a Gaussian distribution is also its mode. The difference between θˆ MAP and θˆ are the σn2 −1 terms. This term has the effect of

2 Note

that the factor of 2 in the denominator of the second term does not cancel out because the derivative is w.r.t. σn2 and not σn .

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reducing the condition number of XT X. Forgetting the mean of the prior, the linear system (XT X)θ = XT y becomes the regularized linear system: Aθ = σn−2 XT y. 1 Note that choosing the isotropic Gaussian prior p(θ ) = N(0, 2λ I ) gives the 2 ridge penalty term in the loss function: λ||θ||2 , i.e. the negative log Gaussian prior matches the ridge penalty term up to a constant. In the limit, λ → 0 recovers maximum likelihood estimation—this corresponds to using the uninformative prior. Of course, in Bayesian inference, we do not perform point-estimation of the parameters, however it was a useful exercise to confirm that the mean of the posterior in Eq. 3.19 did indeed match the MAP estimate. Furthermore, we have made explicit the interpretation of the prior as a regularization term used in ridge regression.

2.2 Bayesian Prediction Recall from Chap. 2 that Bayesian prediction requires evaluating the density of f∗ := f (x∗ ) w.r.t. a new data point x∗ and the training data D. In general, we predict the model output at a new point, f∗ , by averaging the model output over all possible weights, with the weight density function given by the posterior. That is we seek to find the marginal density p(f∗ |x∗ , D) = Eθ |D [p(f∗ |x∗ , θ )], where the dependency on θ has been integrated out. This conditional density is Gaussian f∗ |x∗ , D ∼ N(μ∗ , ∗ ),

(3.27)

with moments μ∗ = Eθ|D [f∗ |x∗ , D] = x∗T Eθ|D [θ|x∗ , D] = x∗T Eθ|D [θ|D] = x∗T μ ∗ = Eθ|D [(f∗ − μ∗ )(f∗ − μ∗ )T |x∗ , D] = x∗T Eθ|D [(θ − μ )(θ − μ )|x∗ , D]x∗ = x∗T Eθ|D [(θ − μ )(θ − μ )|D]x∗ = x∗T x∗ , where we have avoided taking the expectation of the entire density function p(f∗ |x∗ , θ ), but rather just the moments because we know that f∗ is Gaussian.

•

? Multiple Choice Question 2

Which of the following statements are true: 1. Prediction under a Bayesian linear model requires first estimating the posterior distribution of the parameters; 2. The predictive distribution is Gaussian only if the posterior and likelihood distributions are Gaussian;

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3. The predictive distribution depends on the weights in the models; 4. The variance of the predictive distribution typically contracts with increasing training data.

2.3 Schur Identity There is another approach to deriving the predictive distribution from the conditional distribution of the model output which relies on properties of inverse matrices. We can write the joint density between Gaussian random variables X and Y in terms of the partitioned covariance matrix: * ) xx xy X μx , , =N μy Σyx Σyy Y where xx = V(X), xy = Cov(XY ) and yy = V(Y ), how can we find the conditional density p(y|x)? In order to express the moments in terms of the partitioned covariance matrix we shall use the following Schur identity:

AB CD

−1

−MBD −1 . = −D −1 CM D −1 + D −1 CMBD −1

M

where the Schur complement w.r.t. the submatrix D is M := (A − BD −1 C)−1 . Applying the Schur identity to the partitioned precision matrix A gives −1 yy yx Ayy Ayx , = xy xx Axy Axx

(3.28)

where −1 Ayy = (yy − yx xx xy )−1

(3.29)

−1 −1 Ayx = −(yy − yx xx xy )−1 yx xx ,

(3.30)

and thus the moments of the Gaussian distribution p(y|x) are −1 (x − μx ), μy|x = μy + yx xx

(3.31)

−1 y|x = yy − yx xx xy .

(3.32)

Hence the density of the condition distribution Y |X can alternatively be derived by using the Schur identity. In the special case when the joint density p(x, y) is bi-Gaussian, the expression for the moments simplify to

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μy|x = μy + y|x = σy −

σyx (x − μx ), σx2 2 σyx

σx2

,

(3.33) (3.34)

where σxy is the covariance between X and Y . Now returning to the predictive distribution, the joint density between y and f∗ is ) * μy yy yf∗ y =N , . (3.35) μf∗ f∗ y f∗ f∗ f∗ We can immediately write down the moments of the condition distribution −1 μf∗ |X,y,x∗ = μf∗ + f∗ y yy (y − μy ),

(3.36)

−1 yf∗ . f∗ |X,y,x∗ = f∗ f∗ − f∗ y yy

(3.37)

Since we know the form of the function f (x), we can simplify this expression by writing that yy = KX,X + σn2 I,

(3.38)

where KX,X is the covariance of f (X), which for linear regression takes the form KX,X = E[θ12 (X − μx )2 ], f∗ f∗ = Kx∗ ,x∗ and yf∗ = KX,x∗ . Now we can write the moments of the predictive distribution as −1 (y − μy ), μf∗ |X,y,x∗ = μf∗ + Kx∗ ,X KX,X

(3.39)

−1 KX,x∗ . Kf∗ |X,y,x∗ = Kx∗ ,x∗ − Kx∗ ,X KX,X

(3.40)

Discussion Note that we have assumed that the functional form of the map, f (x) is known and parameterized. Here we assumed that the map is linear in the parameters and affine in the features. Hence our approximation of the map is in the data space and, for prediction, we can subsequently forget about the map and work with its moments. The moments of the prior on the weights also no longer need to be specified.

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If we do not know the form of the map but want to specify structure on the covariance of the map (i.e., the kernel), then we are said to be approximating in the kernel space rather than in the data space. If the kernels are given by continuous functions of X, then such an approximation corresponds to learning a posterior distribution over an infinite dimensional function space rather than a finite dimensional vector space. Put differently, we perform non-parametric regression rather than parametric regression. This is the remaining topic of this chapter and is precisely how Gaussian process regression models data.

3 Gaussian Process Regression Whereas, statistical inference involves learning a latent function Y = f (X) of the training data, (X, Y ) := {(xi , yi ) | i = 1, . . . , n}, the idea of GPs is to, without parameterizing3 f (X), place a probabilistic prior directly on the space of functions (MacKay 1998). Restated, the GP is hence a Bayesian non-parametric model that generalizes the Gaussian distributions from finite dimensional vector spaces to infinite dimensional function spaces. GPs do not provide some parameterized map, Yˆ = fθ (X), but rather the posterior distribution of the latent function given the training data. The basic theory of prediction with GPs dates back to at least as far as the time series work of Kolmogorov or Wiener in the 1940s (see (Whittle and Sargent 1983)). GPs are an example of a more general class of supervised machine learning techniques referred to as “kernel learning,” which model the covariance matrix from a set of parameterized kernels over the input. GPs extend and put in a Bayesian framework spline or kernel interpolators, and Tikhonov regularization (see (Rasmussen and Williams 2006) and (Alvarez et al. 2012)). On the other hand, (Neal 1996) observed that certain neural networks with one hidden layer converge to a Gaussian process in the limit of an infinite number of hidden units. We refer to the reader to (Rasmussen and Williams 2006) for an excellent introduction to GPs. In addition to a number of favorable statistical and mathematical properties, such as universality (Micchelli et al. 2006), the implementation support infrastructure is mature—provided by GpyTorch, scikit-learn, Edward, STAN, and other open-source machine learning packages. In this section we restrict ourselves to the simpler case of single-output GPs where f is real-valued. Multi-output GPs are considered in the next section.

3 This

is in contrast to non-linear regressions commonly used in finance, which attempt to parameterize a non-linear function with a set of weights.

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3.1 Gaussian Processes in Finance The adoption of GPs in financial derivative modeling is more recent and sometimes under the name of “kriging” (see, e.g., (Cousin et al. 2016) or (Ludkovski 2018)). Examples of applying GPs to financial time series prediction are presented in (Roberts et al. 2013). These authors helpfully note that AR(p) processes are discrete-time equivalents of GP models with a certain class of covariance functions, known as Matérn covariance functions. Hence, GPs can be viewed as a Bayesian non-parametric generalization of well-known econometrics techniques. da Barrosa et al. (2016) present a GP method for optimizing financial asset portfolios. Other examples of GPs include metamodeling for expected shortfall computations (Liu and Staum 2010), where GPs are used to infer portfolio values in a scenario based on inner-level simulation of nearby scenarios, and Crépey and Dixon (2020), where multiple GPs infer derivative prices in a portfolio for market and credit risk modeling. The approach of Liu and Staum (2010) significantly reduces the required computational effort by avoiding inner-level simulation in every scenario and naturally takes account of the variance that arises from inner-level simulation. The caveat is that the portfolio remains fixed. The approach of Crépey and Dixon (2020), on the other hand, allows for the composition of the portfolio to be changed, which is especially useful for portfolio sensitivity analysis, risk attribution and stress testing. Derivative Pricing, Greeking, and Hedging In the general context of derivative pricing, Spiegeleer et al. (2018) noted that many of the calculations required for pricing a wide array of complex instruments, are often similar. The market conditions affecting OTC derivatives may often only slightly vary between observations by a few variables, such as interest rates. Accordingly, for fast derivative pricing, greeking, and hedging, Spiegeleer et al. (2018) propose offline learning the pricing function, through Gaussian Process regression. Specifically, the authors configure the training set over a grid and then use the GP to interpolate at the test points. We emphasize that such GP estimates depend on option pricing models, rather than just market data - somewhat counter the motivation for adopting machine learning, but also the case in other computational finance applications such as Hernandez (2017), Weinan et al. (2017), or Hans Bühler et al. (2018). Spiegeleer et al. (2018) demonstrate the speed up of GPs relative to MonteCarlo methods and tolerable accuracy loss applied to pricing and Greek estimation with a Heston model, in addition to approximating the implied volatility surface. The increased expressibility of GPs compared to cubic spline interpolation, a popular numerical approximation techniques useful for fast point estimation, is also demonstrated. However, the applications shown in (Spiegeleer et al. 2018) are limited to single instrument pricing and do not consider risk modeling aspects. In particular, their study is limited to single-output GPs, without consideration

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of multi-output GPs (respectively referred to as single- vs. multi-GPs for brevity hereafter). By contrast, multi-GPs directly model the uncertainty in the prediction of a vector of derivative prices (responses) with spatial covariance matrices specified by kernel functions. Thus the amount of error in a portfolio value prediction, at any point in space and time, can only be adequately modeled using multi-GPs (which, however, do not provide any methodology improvement in estimation of the mean with respect to single-GPs). See Crépey and Dixon (2020) for further details of how multi-GPs can be applied to estimate market and credit risk. The need for uncertainty quantification in the prediction is certainly a practical motivation for using GPs, as opposed to frequentist machine learning techniques such as neural networks, etc., which only provide point estimates. A high uncertainty in a prediction might result in a GP model estimate being rejected in favor of either retraining the model or even using full derivative model repricing. Another motivation for using GPs, as we will see, is the availability of a scalable training method for the model hyperparameters.

3.2 Gaussian Processes Regression and Prediction We say that a random function f : Rp → R is drawn from a GP with a mean function μ and a covariance function, called kernel, k, i.e. f ∼ GP(μ, k), if for any input points x1 , x2 , . . . , xn in Rp , the corresponding vector of function values is Gaussian: [f (x1 ), f (x2 ), . . . , f (xn )] ∼ N(μ, KX,X ), for some mean vector μ, such that μi = μ(xi ), and covariance matrix KX,X that satisfies (KX,X )ij = k(xi , xj ). We follow the convention4 in the literature of assuming μ = 0. Kernels k can be any symmetric positive semidefinite function, which is the infinite dimensional analogue of the notion of a symmetric positive semidefinite (i.e., covariance) matrix, i.e. such that n

k(xi , xj )ξi ξj ≥ 0, for any points xk ∈ Rp and reals ξk .

i,j =1

Radial basis functions (RBF) are kernels that only depend on ||x − x ||, such as the squared exponential (SE) kernel

4 This

choice is not a real limitation in practice (since it is for the prior) and does not prevent the mean of the predictor from being nonzero.

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k(x, x ) = exp{−

1 ||x − x ||2 }, 22

(3.41)

where the length-scale parameter can be interpreted as “how far you need to move in input space for the function values to become uncorrelated,” or the Matern (MA) kernel ) *ν ) * √ ||x − x || 21−ν √ ||x − x || (3.42) k(x, x ) = σ 2 2ν Kν 2ν (ν) (which converges to (3.41) in the limit where ν goes to infinity), where is the gamma function, Kν is the modified Bessel function of the second kind, and and ν are non-negative parameters. GPs can be seen as distributions over the reproducing kernel Hilbert space (RKHS) of functions which is uniquely defined by the kernel function, k (Scholkopf and Smola 2001). GPs with RBF kernels are known to be universal approximators with prior support to within an arbitrarily small epsilon band of any continuous function (Micchelli et al. 2006). Assuming additive Gaussian noise, y | x ∼ N(f (x), σn2 ), and a GP prior on f (x), given training inputs x ∈ X and training targets y ∈ Y , the predictive distribution of the GP evaluated at an arbitrary test point x∗ ∈ X∗ is: f∗ | X, Y, x∗ ∼ N(E[f∗ |X, Y, x∗ ], var[f∗ |X, Y, x∗ ]),

(3.43)

where the moments of the posterior over X∗ are E[f∗ |X, Y, X∗ ] = μX∗ + KX∗ ,X [KX,X + σn2 I ]−1 Y, var[f∗ |X, Y, X∗ ] = KX∗ ,X∗ − KX∗ ,X [KX,X + σn2 I ]−1 KX,X∗ .

(3.44)

Here, KX∗ ,X , KX,X∗ , KX,X , and KX∗ ,X∗ are matrices that consist of the kernel, k : Rp × Rp → R, evaluated at the corresponding points, X and X∗ , and μX∗ is the mean function evaluated on the test inputs X∗ . One key advantage of GPs over interpolation methods is their expressibility. In particular, one can combine the kernels, using convolution, to generalize the base kernels (c.f. “multi-kernel” GPs (Melkumyan and Ramos 2011)).

3.3 Hyperparameter Tuning GPs are fit to the data by optimizing the evidence-the marginal probability of the data given the model with respect to the learned kernel hyperparameters. The evidence has the form (see, e.g., (Murphy 2012, Section 15.2.4, p. 523)):

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! " n log p(Y | X, λ) = − Y (KX,X + σn2 I )−1 Y + log det(KX,X + σn2 I ) − log 2π, 2 (3.45) where KX,X implicitly depends on the kernel hyperparameters λ (e.g., [, σ ], assuming an SE kernel as per (3.41) or an MA kernel for some exogenously fixed value of ν in (3.42) ). The first and second term in the [· · · ] in (3.45) can be interpreted as a model fit and a complexity penalty term (see (Rasmussen and Williams 2006, Section 5.4.1)). Maximizing the evidence with respect to the kernel hyperparameters, i.e. computing λ∗ = argmaxλ log p(y | x, λ), results in an automatic Occam’s razor (see (Alvarez et al. 2012, Section 2.3) and (Rasmussen and Ghahramani 2001)), through which we effectively learn the structure of the space of functional relationships between the inputs and the targets. In practice, the negative evidence is minimized by stochastic gradient descent (SGD). The gradient of the evidence is given analytically by

∂λ log p(y | x, λ) = tr (αα T − (K + σn2 I )−1 )∂λ (K + σn2 I )−1 ,

(3.46)

where α := (K + σn2 I )−1 y and ∂ (K + σn2 I )−1 = −(K + σn2 I )−2 ∂ K, ∂σ (K

+ σn2 I )−1

= −2σ (K

+ σn2 I )−2 ,

(3.47) (3.48)

with ∂ k(x, x ) = −3 ||x − x ||2 k(x, x ).

(3.49)

•

? Multiple Choice Question 3

Which of the following statements are true: 1. Gaussian Processes are a Bayesian modeling approach which assumes that the data is Gaussian distributed. 2. Gaussian Processes place a probabilistic prior directly on the space of functions. 3. Gaussian Processes model the posterior of the predictor using a parameterized kernel representation of the covariance matrix. 4. Gaussian Processes can be fitted to data by maximizing the evidence for the kernel parameters. 5. During evidence maximization, different kernels are evaluated, and the optimal kernel is chosen.

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3.4 Computational Properties If uniform grids $pare used (as opposed to a mesh-free GP as described in Sect. 5.2), we have n = k=1 nk , where nk are the number of grid points per variable. However, although each kernel matrix KX,X is n × n, we only store the n-vector α in (3.46), which brings reduced memory requirements. Training time, required for maximizing (3.45) numerically, scales poorly with the number of observations n. This complexity stems from the need to solve linear systems and compute log determinants involving an n × n symmetric positive definite covariance matrix K. This task is commonly performed by computing the Cholesky decomposition of K incurring O(n3 ) complexity. Prediction, however, is faster and can be performed in O(n2 ) with a matrix–vector multiplication for each test point, and hence the primary motivation for using GPs is real-time risk estimation performance. Online Learning If the option pricing model is recalibrated intra-day, then the corresponding GP model should be retrained. Online learning techniques permit performing this incrementally (Pillonetto et al. 2010). To enable online learning, the training data should be augmented with the constant model parameters. Each time the parameters are updated, a new observation (x , y ) is generated from the option model prices under the new parameterization. The posterior at test point x∗ is then updated with the new training point following p(f∗ |X, Y, x , y , x∗ ) = %

p(x , y |f∗ )p(f∗ |X, Y, x∗ ) , f∗ p(x , y |z)p(z|X, Y, x∗ )dz

(3.50)

where the previous posterior p(f∗ |X, Y, x∗ ) becomes the prior in the update. Hence the GP learns over time as model parameters (which are an input to the GP) are updated through pricing model recalibration.

4 Massively Scalable Gaussian Processes Massively scalable Gaussian processes (MSGP) are a significant extension of the basic kernel interpolation framework described above. The core idea of the framework, which is detailed in (Gardner et al. 2018), is to improve scalability by combining GPs with “inducing point methods.” The basic setup is as follows; Using structured kernel interpolation (SKI), a small set of m inducing points are carefully selected from the original training points. The covariance matrix has a Kronecker and Toeplitz structure, which is exploited by the Fast Fourier Transform (FFT). Finally, output over the original input points is interpolated from the output at the

4 Massively Scalable Gaussian Processes

97

inducing points. The interpolation complexity scales linearly with dimensionality p of the input data by expressing the kernel interpolation as a product of 1D kernels. Overall, SKI gives O(pn + pmlogm) training complexity and O(1) prediction time per test point.

4.1 Structured Kernel Interpolation (SKI) Given a set of m inducing points, the n × m cross-covariance matrix, KX,U , between the training inputs, X, and the inducing points, U, can be approximated as K˜ X,U = WX KU,U using a (potentially sparse) n × m matrix of interpolation weights, WX . This allows to approximate KX,Z for an arbitrary set of inputs Z as KX,Z ≈ K˜ X,U WZ . For any given kernel function, K, and a set of inducing points, U, structured kernel interpolation (SKI) procedure (Gardner et al. 2018) gives rise to the following approximate kernel: KSKI (x, z) = WX KU,U Wz ,

(3.51)

which allows to approximate KX,X ≈ WX KU,U WX . Gardner et al. (2018) note that standard inducing point approaches, such as subset of regression (SoR) or fully independent training conditional (FITC), can be reinterpreted from the SKI perspective. Importantly, the efficiency of SKI-based MSGP methods comes from, first, a clever choice of a set of inducing points which exploit the algebraic structure of KU,U , and second, from using very sparse local interpolation matrices. In practice, local cubic interpolation is used.

4.2 Kernel Approximations If inducing points, U , form a regularly spaced P -dimensional grid, and we use a stationary product kernel (e.g., the RBF kernel), then KU,U decomposes as a Kronecker product of Toeplitz matrices: KU,U = T1 ⊗ T2 ⊗ · · · ⊗ TP .

(3.52)

The Kronecker structure allows one to compute the eigendecomposition of KU,U by separately decomposing T1 , . . . , TP , each of which is much smaller than KU,U . Further, a Toeplitz matrix can be approximated by a circulant matrix5 which eigendecomposes by simply applying a discrete Fourier transform (DFT) to its

5 Gardner et al. (2018) explored 5 different approximation methods known in the numerical analysis

literature.

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first column. Therefore, an approximate eigendecomposition of each T1 , . . . , TP is computed via the FFT in only O(m log m) time.

4.2.1

Structure Exploiting Inference

To perform inference, we need to solve (KSKI + σn2 I )−1 y; kernel learning requires evaluating log det(KSKI + σn2 I ). The first task can be accomplished by using an iterative scheme—linear conjugate gradients—which depends only on matrix vector multiplications with (KSKI + σn2 I ). The second is performed by exploiting the Kronecker and Toeplitz structure of KU,U for computing an approximate eigendecomposition, as described above. In this chapter, we primarily use the basic interpolation approach for simplicity. However for completeness, Sect. 5.3 shows the scaling of the time taken to train and predict with MSGPs.

5 Example: Pricing and Greeking with Single-GPs In the following example, the portfolio holds a long position in both a European call and a put option struck on the same underlying, with K = 100. We assume that the underlying follows Heston dynamics: . dSt = μdt + Vt dWt1 , St . dVt = κ(θ − Vt )dt + σ Vt dWt2 , dW 1 , W 2 t = ρdt,

(3.53) (3.54) (3.55)

where the notation and fixed parameter values used for experiments are given in Table 3.1 under μ = r0 . We use a Fourier Cosine method (Fang and Oosterlee 2008) to generate the European Heston option price training and testing data for the GP. We also use this method to compare the GP Greeks, obtained by differentiating the kernel function. Table 3.1 lists the values of the parameters for the Heston dynamics and terms of the European Call and Put option contract used in our numerical experiments. Table 3.2 shows the values for the Euler time stepper used for simulating Heston dynamics and the credit risk model. For each pricing time ti , we simultaneously√fit a multi-GP to both gridded call and put prices over stock price S and volatility V , keeping time to maturity fixed. Figure 3.3 shows the gridded call (top) and put (bottom) price surfaces at various time to maturities, together with the GP estimate. Within each column in the figure, the same GP model has been simultaneously fitted to both the call and put price

5 Example: Pricing and Greeking with Single-GPs

99

Table 3.1 This table shows the values of the parameters for the Heston dynamics and terms of the European Call and Put option contracts

Parameter description Mean reversion rate Mean reversion level Vol. of Vol. Risk-free rate Strike Maturity Correlation

Table 3.2 This table shows the values for the Euler time stepper used for market risk factor simulation

Parameter description Number of simulation Number of time steps Initial stock price Initial variance

Symbol κ θ σ r0 K T ρ Symbol M ns S0 V0

Value 0.1 0.15 0.1 0.002 100 2.0 −0.9 Value 1000 100 100 0.1

(a) Call: T − t = 1.0

(b) Call: T − t = 0.5

(c) Call: T − t = 0.1

(a) Put: T − t = 1.0

(b) Put: T − t = 0.5

(c) Put: T − t = 0.1

Fig. 3.3 This figure compares the gridded Heston model call (top) and put (bottom) price surfaces at various time to maturities, with the GP estimate. The GP estimate is observed to be practically identical (slightly below in the first five panels and slightly above in the last one). Within each column in the figure, the same GP model has been simultaneously fitted to both the Heston model call and put price surfaces over a 30 × 30 grid of prices and volatilities, fixing the time to maturity. Across each column, corresponding to different time to maturities, a different GP model has been fitted. The GP is then evaluated out-of-sample over a 40 × 40 grid, so that many of the test samples are new to the model. This is repeated over various time to maturities

surfaces over a 30 × 30 grid h ⊂ := [0, 1] × [0, 1] of prices and volatilities,6 fixing the time to maturity. The scaling to the unit domain is not essential. However, we observed superior numerical stability when scaling.

6 Note

that the plot uses the original coordinates and not the re-scaled coordinates.

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Across each column, corresponding to different time to maturities, a different GP model has been fitted. The GP is then evaluated out-of-sample over a 40 × 40 grid h ⊂ , so that many of the test samples are new to the model. This is repeated over various time to maturities.7 Extrapolation One instance where kernel combination is useful in derivative modeling is for extrapolation—the appropriate mixture or combination of kernels can be chosen so that the GP is able to predict outside the domain of the training set. Noting that the payoff is linear when a call or put option is respectively deeply in and out-of-the money, we can configure a GP as a combination of a linear kernel and, say, a SE kernel. The linear kernel is included to ensure that prediction outside the domain preserves the linear property, whereas the SE kernel captures non-linearity. Figure 3.4 shows the results of using this combination of kernels to extrapolate the prices of a call struck at 110 and a put struck at 90. The linear property of the payoff function is preserved by the GP prediction and the uncertainty increases as the test point is further from the training set.

Fig. 3.4 This figure assesses the GP option price prediction in the setup of a Black–Scholes model. The GP with a mixture of a linear and SE kernel is trained on n = 50 X, Y pairs, where X ∈ h ⊂ (0, 300] is the gridded underlying of the option prices and Y is a vector of call or put prices. These training points are shown by the black “+” symbols. The exact result using the Black–Scholes pricing formula is given by the black line. The predicted mean (blue solid line) and variance of the posterior are estimated from Eq. 3.44 over m = 100 gridded test points, X∗ ∈ h∗ ⊂ [300, 400], for the (left) call option struck at 110 and (center) put option struck at 90. The shaded envelope represents the 95% confidence interval about the mean of the posterior. This confidence interval is observed to increase the further the test point is from the training set. The time to maturity of the options are fixed to two years. (a) Call price. (b) Put price

7 Such

maturities might correspond to exposure evaluation times in CVA simulation as in Crépey and Dixon (2020). The option model versus GP model are observed to produce very similar values.

5 Example: Pricing and Greeking with Single-GPs

101

5.1 Greeking The GP provides analytic derivatives with respect to the input variables ∂X∗ E[f∗ |X, Y, X∗ ] = ∂X∗ μX∗ + ∂X∗ KX∗ ,X α,

(3.56)

where ∂X∗ KX∗ ,X = 12 (X − X∗ )KX∗ ,X and we recall from Sect. (3.46) that α = [KX,X + σn2 I ]−1 y (and in the numerical experiments we set μ = 0). Second-order sensitivities are obtained by differentiating once more with respect to X∗ . Note that α is already calculated at training time (for pricing) by Cholesky matrix factorization of [KX,X + σn2 I ] with O(n3 ) complexity, so there is no significant computational overhead from Greeking. Once the GP has learned the derivative prices, Eq. 3.56 is used to evaluate the first order MtM Greeks with respect to the input variables over the test set. Example source code illustrating the implementation of this calculation is presented in the notebook Example-2-GP-BS-Derivatives.ipynb. Figure 3.5 shows (left) the GP estimate of a call option’s delta := ∂C ∂S and (right) vega ν := ∂C ∂σ , having trained on the underlying, respectively implied volatility, and on the BS option model prices. For avoidance of doubt, the model is not trained on the BS Greeks. For comparison in the figure, the BS delta and vega are also shown. In each case, the two graphs are practically indistinguishable, with one graph superimposed over the other.

5.2 Mesh-Free GPs The above numerical examples have trained and tested GPs on uniform grids. This approach suffers from the curse of dimensionality, as the number of training points grows exponentially with the dimensionality of the data. This is why, in order to estimate the MtM cube, we advocate divide-and-conquer, i.e. the use of numerous

Fig. 3.5 This figure shows (left) the GP estimate of the call option’s delta := ∂C ∂S and (right) vega ν := ∂C ∂σ , having trained on the underlying, respectively implied volatility, and on the BS option model prices

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low input dimensional space, p, GPs run in parallel on specific asset classes. However, use of fixed grids is by no means necessary. We show here how GPs can show favorable approximation properties with a relatively few number of simulated reference points (cf. also (Gramacy and Apley 2015)). Figure 3.6 shows the predicted Heston call prices using (left) 50 and (right) 100 simulated training points, indicated by “+”s, drawn from a uniform random distribution. The Heston call option is struck at K = 100 with a maturity of T = 2 years. Figure 3.7 (left) shows the convergence of the GP MSE of the prediction, based on the number of Heston simulated training points. Fixing the number of simulated points to 100, but increasing the input space dimensionality, p, of each observation point (to include varying Heston parameters, Fig. 3.7 (right) shows the wall-clock time for training a GP with SKI (see Sect. 3.4), Note that the number of SGD iterations has been fixed to 1000. 120

GP Prediction Analytical Model

120

100

100

80

80

V 60

V 60

40

40

20

20

60

80

100

120

S

140

160

180

200

GP Prediction Analytical Model

60

80

100

120

S

140

160

180

200

Fig. 3.6 Predicted Heston Call prices using (left) 50 and (right) 100 simulated training points, indicated by “+”s, drawn from a uniform random distribution

Fig. 3.7 (Left) The convergence of the GP MSE of the prediction is shown based on the number of simulated Heston training points. (Right) Fixing the number of simulated points to 100, but increasing the dimensionality p of each observation point (to include varying Heston parameters), the figure shows the wall-clock time for training a GP with SKI

6 Multi-response Gaussian Processes

103

Fig. 3.8 (Left) The elapsed wall-clock time is shown for training against the number of training points generated by a Black–Scholes model. (Right) The elapsed wall-clock time for prediction of a single point is shown against the number of testing points. The reason that the prediction time increases (whereas the theory reviewed in Sect. 3.4 says it should be constant) is due to memory latency in our implementation—each point prediction involves loading a new test point into memory

5.3 Massively Scalable GPs Figure 3.8 shows the increase of MSGP training time and prediction time against the number of training points n from a Black Scholes model. Fixing the number of inducing points to m = 30 (see Sect. 3.4), we increase the number of observations, n, in the p = 1 dimensional training set. Setting the number of SGD iterations to 1000, we observe an approximate 1.4x increase in training time for a 10x increase in the training sample. We observe an approximate 2x increase in prediction time for a 10x increase in the training sample. The reason that the prediction time does not scale independently of n is due to memory latency in our implementation—each point prediction involves loading a new test point into memory. Fast caching approaches can be used to reduce this memory latency, but are beyond the scope of this section. Note that training and testing times could be improved with CUDA on a GPU, but are not evaluated here.

6 Multi-response Gaussian Processes A multi-output Gaussian process is a collection of random vectors, any finite number of which have a matrix-variate Gaussian distribution. We borrow from Chen et al. (2017) the following formulation of a separable multi-output kernel specification as per (Alvarez et al. 2012, Eq. (9)): Definition (MGP) f is a d variate Gaussian process on Rp with vector-valued mean function μ : Rp → Rd , kernel k : Rp × Rp → R, and positive semi-definite

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parameter covariance matrix ∈ Rd×d , if the vectorization of any finite collection of vectors f(x1 ), . . . , f(xn ) have a joint multivariate Gaussian distribution, vec([f(x1 ), . . . , f(xn )]) ∼ N(vec(M), ⊗ ), where f(xi ) ∈ Rd is a column vector whose components are the functions fl (xi )}dl=1 , M is a matrix in Rd×n with Mli = μl (xi ), is a matrix in Rn×n with ij = k(xi , xj ), and ⊗ is the Kronecker product ⎞ 11 · · · 1n ⎜ .. .. ⎟. .. ⎝ . . . ⎠ m1 · · · mn ⎛

Sometimes is called the column covariance matrix while is the row (or task) covariance matrix. We denote f ∼ MGP(mμ, k, ). As explained after Eq. (10) in (Alvarez et al. 2012), the matrices and encode dependencies among the inputs, respectively outputs.

6.1 Multi-Output Gaussian Process Regression and Prediction Given n pairs of observations {(xi , yi )}ni=1 , xi ∈ Rp , yi ∈ Rd , we assume the model yi = f(xi ), i ∈ {1, . . . , n}, where f ∼ MGP(μ, k , ) with k = k(xi , xj ) + δij σn2 , in which σn2 is the variance of the additive Gaussian noise. That is, the vectorization of the collection of functions [f(x1 ), . . . , f(xn )] follows a multivariate Gaussian distribution vec([f(x1 ), . . . , f(xn )]) ∼ N(0, K ⊗ ), where K is the n × n covariance matrix of which the (i, j )-th element [K ]ij = k (xi , xj ). To predict a new variable f∗ = [f∗1 , . . . , f∗m ] at the test locations X∗ = [xn+1 , . . . , xn+m ], the joint distribution of the training observations Y = [y1 , . . . , yn ] and the predictive targets f∗ are given by ) * Y K (X, X) K (X∗ , X)T ∼ MN 0, , , K (X∗ , X) K (X∗ , X∗ ) f∗

(3.57)

where K (X, X) is an n × n matrix of which the (i, j )-th element [K (X, X)]ij = k (xi , xj ), K (X∗ , X) is an m × n matrix of which the (i, j )-th element

7 Summary

105

[K (X∗ , X)]ij = k (xn+i , xj ), and K (X∗ , X∗ ) is an m × m matrix with the (i, j )-th element [K (X∗ , X∗ )]ij = k (xn+i , xn+j ). Thus, taking advantage of conditional distribution of multivariate Gaussian process, the predictive distribution is: ˆ ˆ ⊗ ), ˆ p(vec(f∗ )|X, Y, X∗ ) = N(vec(M),

(3.58)

where Mˆ = K (X∗ , X)T K (X, X)−1 Y,

(3.59)

ˆ = K (X∗ , X∗ ) − K (X∗ , X) K (X, X)

ˆ = .

T

−1

K (X∗ , X),

(3.60) (3.61)

The hyperparameters and elements of the covariance matrix are found by minimizing the negative log marginal likelihood of observations: nd d 1 n ln(2π ) + ln |K | + ln || + tr((K )−1 Y −1 Y T ). 2 2 2 2 (3.62) Further details of the multi-GP are given in (Bonilla et al. 2007; Alvarez et al. 2012; Chen et al. 2017). The computational remarks made in Sect. 3.4 also apply here, with the additional comment that the training and prediction time also scale linearly (proportionally) with the number of dimensions d. Note that the task covariance matrix is estimated via a d-vector factor b by = bbT + σΩ2 I (where the σ2 component corresponds to a standard white noise term). An alternative computational approach, which exploits separability of the kernel, is the one described in Section 6.1 of (Alvarez et al. 2012), with complexity O(d 3 + n3 ). L(Y |X, λ, ) =

7 Summary In this chapter we have introduced Bayesian regression and shown how it extends many of the concepts in the previous chapter. We develop kernel based machine learning methods, known as Gaussian processes, and demonstrate their application to surrogate models of derivative prices. The key learning points of this chapter are: – Introduced Bayesian linear regression; – Derived the posterior distribution and the predictive distribution; – Described the role of the prior as an equivalent form of regularization in maximum likelihood estimation; and – Developed Gaussian Processes for kernel based probabilistic modeling, with programming examples in derivative modeling.

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8 Exercises Exercise 3.1: Posterior Distribution of Bayesian Linear Regression Consider the Bayesian linear regression model yi = θ T X + , ∼ N(0, σn2 ), θ ∼ N(μ, ). Show that the posterior over data D is given by the distribution θ|D ∼ N(μ , ), with moments: μ = a = ( −1 +

1 1 XXT )−1 ( −1 μ + 2 yT X) 2 σn σn

= A−1 = ( −1 +

1 XXT )−1 . σn2

Exercise 3.2: Normal Conjugate Distributions Suppose that the prior is p(θ ) = φ(θ ; μ0 , σ02 ) and the likelihood is given by p(x1:n | θ ) =

n

φ(xi ; θ, σ 2 ),

i=1

where σ 2 is assumed to be known. Show that the posterior is also normal, 2 ), where p(θ | x1:n ) = φ(θ ; μpost , σpost μpost =

σ02 σ2 n

+ σ02

2 σpost =

where x¯ :=

1 n

x¯ +

σ2 σ2 n

1 1 σ02

+

n σ2

+ σ02

μ0 ,

,

n

i=1 xi .

Exercise 3.3: Prediction with GPs Show that the predictive distribution for a Gaussian Process, with model output over a test point, f∗ , and assumed Gaussian noise with variance σn2 , is given by f∗ | D, x∗ ∼ N(E[f∗ |D, x∗ ], var[f∗ |D, x∗ ]), where the moments of the posterior over X∗ are

Appendix

107

E[f∗ |D, X∗ ] = μX∗ + KX∗ ,X [KX,X + σn2 I ]−1 Y, var[f∗ |D, X∗ ] = KX∗ ,X∗ − KX∗ ,X [KX,X + σn2 I ]−1 KX,X∗ .

8.1 Programming Related Questions* Exercise 3.4: Derivative Modeling with GPs Using the notebook Example-1-GP-BS-Pricing.ipynb, investigate the effectiveness of a Gaussian process with RBF kernels for learning the shape of a European derivative (call) pricing function Vt = ft (St ) where St is the underlying stock’s spot price. The risk free rate is r = 0.001, the strike of the call is KC = 130, the volatility of the underlying is σ = 0.1 and the time to maturity τ = 1.0. Your answer should plot the variance of the predictive distribution against the stock price, St = s, over a dataset consisting of n ∈ {10, 50, 100, 200} gridded values of the stock price s ∈ h := {is | i ∈ {0, . . . , n−1}, s = 200/(n−1)} ⊆ [0, 200] and the corresponding gridded derivative prices V (s). Each observation of the dataset, (si , vi = ft (si )) is a gridded (stock, call price) pair at time t.

Appendix Answers to Multiple Choice Questions Question 1 Answer: 1,4,5. Parametric Bayesian regression always treats the regression weights as random variables. In Bayesian regression the data function f (x) is only observed if the data is assumed to be noise-free. Otherwise, the function is not directly observed. The posterior distribution of the parameters will only be Gaussian if both the prior and the likelihood function are Gaussian. The distribution of the likelihood function depends on the assumed error distribution. The posterior distribution of the regression weights will typically contract with increasing data. The precision matrix grows with decreasing variance and hence the variance of the posterior shrinks with increasing data. There are exceptions if, for example, there are outliers in the data. The mean of the posterior distribution depends on both the mean and covariance of the prior if it is Gaussian. We can see this from Eq. 3.19. Question 2 Answer: 1, 2, 4. Prediction under a Bayesian linear model requires first estimating the moments of the posterior distribution of the parameters. This is because the prediction is the expected likelihood of the new data under the posterior distribution.

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The predictive distribution is Gaussian only if the posterior and likelihood distributions are Gaussian. The product of Gaussian density functions is also Gaussian. The predictive distribution does not depend on the weights in the models it is marginalized out under the expectation w.r.t. the posterior distribution. The variance of the predictive distribution typically contracts with increasing training data because the variance of the posterior and the likelihood typically decreases with increasing training data. Question 3 Answer: 2, 3, 4. Gaussian Process regression is a Bayesian modeling approach but they do not assume that the data is Gaussian distributed, neither do they make such an assumption about the error. Gaussian Processes place a probabilistic prior directly on the space of functions and model the posterior of the predictor using a parameterized kernel representation of the covariance matrix. Gaussian Processes are fitted to data by maximizing the evidence for the kernel parameters. However, it is not necessarily the case that the choice of kernel is effectively a hyperparameter that can be optimized. While this could be achieved in an ad hoc way, there are other considerations which dictate the choice of kernel concerning smoothness and ability to extrapolate.

Python Notebooks A number of notebooks are provided in the accompanying source code repository, beyond the two described in this chapter. These notebooks demonstrate the use of Multi-GPs and application to CVA modeling (see Crépey and Dixon (2020) for details of these models). Further details of the notebooks are included in the README.md file.

References Alvarez, M., Rosasco, L., & Lawrence, N. (2012). Kernels for vector-valued functions: A review. Foundations and Trends in Machine Learning, 4(3), 195–266. Bishop, C. M. (2006). Pattern recognition and machine learning (information science and statistics). Berlin, Heidelberg: Springer-Verlag. Bonilla, E. V., Chai, K. M. A., & Williams, C. K. I. (2007). Multi-task Gaussian process prediction. In Proceedings of the 20th International Conference on Neural Information Processing Systems, NIPS’07, USA (pp. 153–160). Curran Associates Inc. Chen, Z., Wang, B., & Gorban, A. N. (2017, March). Multivariate Gaussian and student−t process regression for multi-output prediction. ArXiv e-prints. Cousin, A., Maatouk, H., & Rullière, D. (2016). Kriging of financial term structures. European Journal of Operational Research, 255, 631–648.

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Crépey, S., & M. Dixon (2020). Gaussian process regression for derivative portfolio modeling and application to CVA computations. Computational Finance. da Barrosa, M. R., Salles, A. V., & de Oliveira Ribeiro, C. (2016). Portfolio optimization through kriging methods. Applied Economics, 48(50), 4894–4905. Fang, F., & Oosterlee, C. W. (2008). A novel pricing method for European options based on Fourier-cosine series expansions. SIAM J. SCI. COMPUT. Gardner, J., Pleiss, G., Wu, R., Weinberger, K., & Wilson, A. (2018). Product kernel interpolation for scalable Gaussian processes. In International Conference on Artificial Intelligence and Statistics (pp. 1407–1416). Gramacy, R., & D. Apley (2015). Local Gaussian process approximation for large computer experiments. Journal of Computational and Graphical Statistics, 24(2), 561–578. Hans Bühler, H., Gonon, L., Teichmann, J., & Wood, B. (2018). Deep hedging. Quantitative Finance. Forthcoming (preprint version available as arXiv:1802.03042). Hernandez, A. (2017). Model calibration with neural networks. Risk Magazine (June 1–5). Preprint version available at SSRN.2812140, code available at https://github.com/Andres-Hernandez/ CalibrationNN. Liu, M., & Staum, J. (2010). Stochastic kriging for efficient nested simulation of expected shortfall. Journal of Risk, 12(3), 3–27. Ludkovski, M. (2018). Kriging metamodels and experimental design for Bermudan option pricing. Journal of Computational Finance, 22(1), 37–77. MacKay, D. J. (1998). Introduction to Gaussian processes. In C. M. Bishop (Ed.), Neural networks and machine learning. Springer-Verlag. Melkumyan, A., & Ramos, F. (2011). Multi-kernel Gaussian processes. In Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence - Volume Two, IJCAI’11 (pp. 1408–1413). AAAI Press. Micchelli, C. A., Xu, Y., & Zhang, H. (2006, December). Universal kernels. J. Mach. Learn. Res., 7, 2651–2667. Murphy, K. (2012). Machine learning: a probabilistic perspective. The MIT Press. Neal, R. M. (1996). Bayesian learning for neural networks, Volume 118 of Lecture Notes in Statistics. Springer. Pillonetto, G., Dinuzzo, F., & Nicolao, G. D. (2010, Feb). Bayesian online multitask learning of Gaussian processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(2), 193–205. Rasmussen, C. E., & Ghahramani, Z. (2001). Occam’s razor. In In Advances in Neural Information Processing Systems 13 (pp. 294–300). MIT Press. Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian processes for machine learning. MIT Press. Roberts, S., Osborne, M., Ebden, M., Reece, S., Gibson, N., & Aigrain, S. (2013). Gaussian processes for time-series modelling. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371(1984). Scholkopf, B., & Smola, A. J. (2001). Learning with kernels: support vector machines, regularization, optimization, and beyond. Cambridge, MA, USA: MIT Press. Spiegeleer, J. D., Madan, D. B., Reyners, S., & Schoutens, W. (2018). Machine learning for quantitative finance: fast derivative pricing, hedging and fitting. Quantitative Finance, 0(0), 1–9. Weinan, E, Han, J., & Jentzen, A. (2017). Deep learning-based numerical methods for highdimensional parabolic partial differential equations and backward stochastic differential equations. arXiv:1706.04702. Whittle, P., & Sargent, T. J. (1983). Prediction and regulation by linear least-square methods (NED - New edition ed.). University of Minnesota Press.

Chapter 4

Feedforward Neural Networks

This chapter provides a more in-depth description of supervised learning, deep learning, and neural networks—presenting the foundational mathematical and statistical learning concepts and explaining how they relate to real-world examples in trading, risk management, and investment management. These applications present challenges for forecasting and model design and are presented as a reoccurring theme throughout the book. This chapter moves towards a more engineering style exposition of neural networks, applying concepts in the previous chapters to elucidate various model design choices.

1 Introduction Artificial neural networks have a long history in financial and economic statistics. Building on the seminal work of (Gallant and White 1988; Andrews 1989; Hornik et al. 1989; Swanson and White 1995; Kuan and White 1994; Lo 1994; Hutchinson, Lo, and Poggio Hutchinson et al.; Baillie and Kapetanios 2007; Racine 2001) develop various studies in the finance, economics, and business literature. Most recently, the literature has been extended to include deep neural networks (Sirignano et al. 2016; Dixon et al. 2016; Feng et al. 2018; Heaton et al. 2017). In this chapter we shall introduce some of the theory of function approximation and out-of-sample estimation with neural networks when the observation points are independent and typically also identically distributed. Such a case is not suitable for times series data and shall be the subject of later chapters. We shall restrict our attention to feedforward neural networks in order to explore some of the theoretical arguments which help us reason scientifically about architecture design and approximation error. Understanding these networks from a statistical, mathematical, and information-theoretic perspective is key to being able to successfully apply them in practice. While this chapter does present some simple financial examples to

© Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_4

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highlight problematic conceptual issues, we defer the realistic financial applications to later chapters. Also, note that the emphasis of this chapter is how to build statistical models suitable for financial modeling, thus our emphasis is less on engineering considerations and more on how theory can guide the design of useful machine learning methods. Chapter Objectives By the end of this chapter, the reader should expect to accomplish the following: – Develop mathematical reasoning skills to guide the design of neural networks; – Gain familiarity with the main theory supporting statistical inference with neural networks; – Relate feedforward neural networks with other types of machine learning methods; – Perform model selection with ridge and LASSO neural network regression; – Learn how to train and test a neural network; and – Gain familiarity with Bayesian neural networks.

Note that section headers ending with * are more mathematically advanced, often requiring some background in analysis and probability theory, and can be skipped by the less mathematically advanced reader.

2 Feedforward Architectures 2.1 Preliminaries Feedforward neural networks are a form of supervised machine learning that use hierarchical layers of abstraction to represent high-dimensional non-linear predictors. The paradigm that deep learning provides for data analysis is very different from the traditional statistical modeling and testing framework. Traditional fit metrics, such as R 2 , t-values, p-values, and the notion of statistical significance has been replaced in the machine learning literature by out-of-sample forecasting and understanding the bias–variance tradeoff ; that is the tradeoff between a more complex model versus over-fitting. Deep learning is data-driven and focuses on finding structure in large datasets. The main tools for variable or predictor selection are regularization and dropout. There are a number of issues in any architecture design. How many layers? How many neurons Nl in each hidden layer? How to perform “variable selection?” Many of these problems can be solved by a stochastic search technique, called dropout

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Srivastava et al. (2014), which we discuss in Sect. 5.2.2. Recall from Chap. 1 that a feedforward neural network model takes the general form of a parameterized map Y = FW,b (X) + ,

(4.1)

where FW,b is a deep neural network with L layers (Fig. 4.1) and is i.i.d. error. The deep neural network takes the form of a composition of simpler functions: (L) (1) Yˆ (X) := FW,b (X) = fW (L) ,b(L) ◦ · · · ◦ fW (1) ,b(1) (X),

(4.2)

where W = (W (1) , . . . , W (L) ) and b = (b(1) , . . . , b(L) ) are weight matrices and bias vectors. Any weight matrix W () ∈ Rm×n can be expressed as n column m () () () vectors W () = [w() . ,1 , . . . , w,n ]. We denote each weight as wij := W ij More formally and under additional restrictions, we can form this parameterized map in the class of compositions of semi-affine functions.

•

> Semi-Affine Functions

Let σ : R → B ⊂ R denote a continuous, monotonically increasing function () whose codomain is a bounded subset of the real line. A function fW () ,b() : (continued)

Input layer x1 x2 x3

Hidden layer 1

Hidden layer 2 Output layer yˆ1

x4 yˆ2 x5 x6 Fig. 4.1 An illustrative example of a feedforward neural network with two hidden layers, six features, and two outputs. Deep learning network classifiers typically have many more layers, use a large number of features and several outputs or classes. The goal of learning is to find the weight on every edge and the bias for every neuron (not illustrated) that minimizes the out-of-sample error

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Rn → Rm , given by f (v) = W () σ (−1) (v) + b() , W () ∈ Rm×n and b() ∈ Rm , is a semi-affine function in v, e.g. f (v) = wtanh(v) + b. σ (·) are the activation functions of the output from the previous layer.

If all the activation functions are linear, FW,b is just linear regression, regardless of the number of layers L and the hidden layers are redundant. For any such network we can always find an equivalent network without hidden units. This follows from the fact that the composition of successive linear transformations is itself a linear transformation.1 For example if there is one hidden layer and σ (1) is the identify function, then ˜ Yˆ (X) = W (2) (W (1) X + b(1) ) + b(2) = W (2) W (1) X + W (2) b(1) + b(2) = W˜ X + b. (4.3) Informally, the main effect of activation is to introduce non-linearity into the model, and in particular, interaction terms between the input. The geometric interpretation of the activation units will be discussed in the next section. We can view the special case when the network has one hidden layer and will see that the activation function introduces interaction terms Xi Xj . Consider the partial derivative (2) (1) w,i σ (Ii(1) )wij , (4.4) ∂Xj Yˆ = i (2)

where w,i is the ith column vector of W (2) , I () (X) := W () X + b() , and differentiate again with respect to Xk , k = i to give ∂X2 j ,Xk Yˆ = −2

(1) (1) (1) (1) w(2) ,i σ (Ii )σ (Ii )wij wik ,

(4.5)

i

which is not in general zero unless σ is the identity map.

2.2 Geometric Interpretation of Feedforward Networks We begin by considering a simple feedforward binary classifier with only two features, as illustrated in Fig. 4.2. The simplest configuration we shall consider has just two inputs and one output unit—this is a multivariate regression model. More precisely, because we shall fit the model to binary responses, this network 1 Note

that there is a potential degeneracy in this case; There may exist “flat directions”—hypersurfaces in the parameter space that have exactly the same loss function.

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Fig. 4.2 Simple two variable feedforward networks with and without hidden layers. The yellow nodes denote input variables, the green nodes denote hidden units, and the red nodes are outputs. A feedforward network without hidden layers is a linear regressor. A feedforward network with one hidden layer is a shallow learner and a feedforward network with two or more hidden layers is a deep learner

is a logistic regression. Recall that only one output unit is required to represent the probability of a positive label, i.e. P [G = 1 | X]. The next configuration we shall consider has one hidden layer—the number of hidden units shall be equal to the number of input neurons. This choice serves as a useful reference point as many hidden units are often needed for sufficient expressibility. The final configuration has substantially more hidden units. Note that the second layer has been introduced purely to visualize the output from the hidden layer. This set of simple configurations (a.k.a. architectures) is ample to illustrate how a neural network method works. In Fig. 4.3 the data has been arranged so that no separating linear plane can perfectly separate the points in [−1, 1] × [−1, 1]. The activation function is chosen to be ReLU (x). The weight and biases of the network have been trained on this data. For each network, we can observe how the input space is transformed by the layers by viewing the top row of the figure. We can also view the linear regression in the original, input, space in the bottom row of the figure. The number of units in the first hidden layers is observed to significantly affect the classifier performance.2 Determining the weight and bias matrices, together with how many hidden units are needed for generalizable performance is the goal of parameter estimation and model selection. However, we emphasize that some conceptual understanding of neural networks is needed to derive interpretability, the topic of Chap. 5. Partitioning The partitioning of the input space is a distinguishing feature of neural networks compared to other machine learning methods. Each hidden unit defines a manifold 2 There

is some redundancy in the construction of the network and around 50 units are needed.

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No hidden units

Two hidden units

Many hidden units

Fig. 4.3 This figure compares various feedforward neural network classifiers applied to a toy, non-linearly separable, binary classification dataset. Its purpose is to illustrate that increasing the number of hidden units in the first hidden layer provides substantial expressibility, even when the number of input variables is small. (Top) Each neural network classifier attempts to separate the labels with a hyperplane in the space of the output from the last hidden layer, Z (L−1) . If the network has no hidden layers, then Z (L−1) = Z (0) = X. The features are shown in the space of Z (L−1) . (Bottom) The separating hyperplane in the space of Z (L−1) is projected to the input space in order to visualize how the layers partition the input space. (Left) A feedforward classifier with no hidden layers is a logistic regression model—it partitions the input space with a plane. (Center) One hidden layer transforms the features by rotation, dilatation, and truncation. (Right) Two hidden layers with many hidden units perform an affine projection into high-dimensional space where points are more separable. See the Deep Classifiers notebook for an implementation of the classifiers and additional diagnostic tests (not shown here)

which divides the input space into convex regions. In other words, each unit in the hidden layer implements a half-space predictor. In the case of a ReLU activation function f (x) = max(x, 0), each manifold is simply a hyperplane and the neuron gets activated when the observation is on the “best” side of this hyperplane, the activation amount is equal to how far from the boundary the given point is. The set of hyperplanes defines a hyperplane arrangement (Montúfar & '2014). In general, pet al. an arrangement of n ≥ p hyperplanes in Rp has at most j =0 nj convex regions. For example, in a two-dimensional input space, threeneurons & ' with ReLU activation functions will divide the space into no more than 2j =0 j3 = 7 regions, as shown in Fig. 4.4. Multiple Hidden Layers We can easily extend this geometrical interpretation to three-layered perceptrons (L = 3). Clearly, the neurons in the first (hidden) layer partition the network input space by corresponding hyperplanes into various half-spaces. Hence, the number of these half-spaces equals the number of neurons in the first layer. Then, the neurons in the second layer can classify the intersections of some of these half-spaces, i.e.

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Fig. 4.4 Hyperplanes defined by three neurons in the hidden layer, each with ReLU activation functions, form a hyperplane arrangement. An arrangement of 3 hyperplanes in R2 has at & ' most 2j =0 j3 = 7 convex regions

they represent convex regions in the input space. This means that a neuron from the second layer is active if and only if the network input corresponds to a point in the input space that is located simultaneously in all half-spaces, which are classified by selected neurons from the first layer. The maximal number of linear regions of the functions computed by a neural network with p input units and L − 1 hidden layers, with equal n() = n ≥ p )! width "(L−2)p * rectifiers at the th layer, can compute functions that have pn np linear regions (Montúfar et al. 2014). We see that the number of linear regions of deep models grows exponentially in L and polynomially in n. See Montúfar et al. (2014) for a more detailed exposition of how the additional layers partition the input space. While this form of reasoning guides our intuition towards designing neural network architectures it falls short at explaining why projection into a higher dimensional space is complementary to how the networks partition the input space. To address this, we turn to some informal probabilistic reasoning to aid our understanding.

2.3 Probabilistic Reasoning Data Dimensionality First consider any two independent standard Gaussian random p-vectors X, Y ∼ N(0, I ) and define their distance in Euclidean space by the 2-norm

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d(X, Y )2 := ||X − Y ||22 =

p

(Xi − Yi )2 .

(4.6)

E[Xi2 ] + E[Yi2 ] = 2p.

(4.7)

i=1

Taking expectations gives E[d(X, Y )2 ] =

p i=1

Under these i.i.d. assumptions, the mean of the pairwise distance squared between any random points in Rp is increasingly linear with the√dimensionality of the space. By Jensen’s inequality for concave functions, such as x . . . E[d(X, Y )] = E[ d(X, Y )2 ] ≤ E[d(X, Y )2 ] = 2p,

(4.8)

and hence the expected distance is bounded above by a function which grows to the power of p1/2 . This simple observation supports the characterization of random points as being less concentrated as the dimensionality of the input space increases. In particular, this property suggests machine learning techniques which rely on concentration of points in the input space, such as linear kernel methods, may not scale well with dimensionality. More importantly, this notion of loss of concentration with dimensionality of the input space does not conflict with how the input space is partitioned—the model defines a convex polytope with a less stringent requirement for locality of data for approximation accuracy. Size of Hidden Layer A similar simple probabilistic reasoning can be applied to the output from a onelayer network to understand how concentration varies with the number of units in the hidden layer. Consider, as before two i.i.d. random vectors X and Y in Rp . Suppose now that these vectors are projected by a bounded semi-affine function g : Rp → Rq . Assume that the output vectors g(X), g(Y ) ∈ Rq are i.i.d. with zero mean and variance σ 2 I . Defining the distance between the output vectors as the 2−norm dg2

:=

||g(X) − g(Y )||22

q = (gi (X) − gi (Y ))2 .

(4.9)

i=1

Under expectations E[dg2 ] =

p i=1

E[g(X)2i ] + E[g(Y )2i ] = 2qσ 2 ≤ q(g¯ − g)

(4.10)

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119

and again by Jensen’s inequality, E[d] ≤

√ √ . 2 qσ ≤ q(g¯ − g),

(4.11)

we observe that the distance between the two output vectors, corresponding to the output of a hidden layer g under different inputs X and Y , can be less concentrated as the dimensionality of the output space increases. In other words, points in the codomain of g are on average more separate as q increases.

2.4 Function Approximation with Deep Learning* While the above informal geometric and probabilistic reasoning provides some intuition for the need for multiple units in the hidden layer of a two-layer MLP, it does not address why deep networks are needed. The most fundamental mathematical concept in neural networks is the universal representation theorem. Simply put, this is a statement about the ability of a neural network to approximate any continuous, and unknown, function between input and output pairs with a simple, and known, functional representation. Hornik et al. (1989) show that a feedforward network with a single hidden layer can approximate any continuous function, regardless of the choice of activation function or data. Formally, let C p := {F : Rp → R | F (x) ∈ C(R)} be the set of continuous functions from Rp to R. Denote p (g) as the class of functions {F : Rp → R : F (x) = W (2) σ (W (1) x + b(1) ) + b(2) }. Consider = (0, 1] and let C0 be the collection of all open intervals in (0, 1]. Then σ (C0 ), the σ -algebra generated by C0 , is called the Borel σ -algebra. It is denoted by B((0, 1]). An element of B((0, 1]) is called a Borel set. A map f : X → Y between two topological spaces X, Y is called Borel measurable if f −1 (A) is a Borel set for any open set A. Let M p := {F : Rp → R | F (x) ∈ B(R)} be the set of all Borel measurable functions from Rp to R. We denote the Borel σ -algebra of Rp as Bp .

•

> Universal Representation Theorem

(Hornik et al. (1989)) For every monotonically increasing activation function σ , every input dimension size p, and every probability measure μ on (Rp , Bp ), p (g) is uniformly dense on compacta in C p and ρμ -dense in M p .

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This theorem shows that standard feedforward networks with only a single hidden layer can approximate any continuous function uniformly on any compact set and any measurable function arbitrarily well in the ρμ metric, regardless of the activation function (provided it is measurable), regardless of the dimension of the input space, p, and regardless of the input space. In other words, by taking the number of hidden units, k, large enough, every continuous function over Rp can be approximated arbitrarily closely, uniformly over any bounded set by functions realized by neural networks with one hidden layer. The universal approximation theorem is important because it characterizes feedforward networks with a single hidden layer as a class of approximate solutions. However, the theorem is not constructive—it does not specify how to configure a multilayer perceptron with the required approximation properties. The theorem has some important limitations. It says nothing about the effect of adding more layers, other than to suggest they are redundant. It assumes that the optimal network weight vectors are reachable by gradient descent from the initial weight values, but this may not be possible in finite computations. Hence there are additional limitations introduced by the learning algorithm which are not apparent from a functional approximation perspective. The theorem cannot characterize the prediction error in any way, the result is purely based on approximation theory. An important concern is over-fitting and performance generalization on out-of-sample datasets, both of which it does not address. Moreover, it does not inform how MLPs can recover other approximation techniques, as a special case, such as polynomial spline interpolation. As such we shall turn to alternative theory in this section to assess the learnability of a neural network and to further understand it, beginning with a perceptron binary classifier. The reason why multiple hidden layers are needed is still an open problem, but various clues are provided in the next section and later in Sect. 2.7.

2.5 VC Dimension In addition to expressive power, which determines the approximation error of the model, there is the notion of learnability, which determines the level of estimation error. The former measures the error introduced by an approximating function and the latter error measures the performance lost as a result of using a finite training sample. One classical measure of the learnability of neural network classifiers is the Vapnik–Chervonenkis (VC) dimension. The VC dimension of a binary model g = FW,b (X) is the maximum number of points that can be arranged so that FW,b (X) shatters them, i.e. for all possible assignments of labels to those points, there exists a W, b such that FW,b makes no errors when classifying that set of data points. In the simplest case, a perceptron with n inputs units and a linear threshold activation σ (x) := sgn(x) has a VC dimension of n+1. For example, if n = 1, then

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Fig. 4.5 For the points {−0.5, 0.5}, there are weights and biases that activate only one of them (W = 1, b = 0 or W = −1, b = 0), none of them (W = 1, b = −0.75), and both of them (W = 1, b = 0.75)

only two distinct points can always be correctly classified under all possible binary label assignments. As shown in Fig. 4.5, for the points {−0.5, 0.5}, there are weights and biases that activate both of them (W = 1, b = 0.75), only one of them (W = 1, b = 0 or W = −1, b = 0), and none of them (W = 1, b = −0.75). Every distinct pair of points is separable with the linear threshold perceptron. So every dataset of size 2 is shattered by the perceptron. However, this linear threshold perceptron is incapable of shattering triplets, for example, X ∈ {−0.5, 0, 0.5} and Y ∈ {0, 1, 0}. In general, the VC dimension of the class of half-spaces in Rk is k + 1. For example, a 2d plane shatters any three points, but cannot shatter four points. The VC dimension determines both the necessary and sufficient conditions for the consistency and rate of convergence of learning processes (i.e., the process of choosing an appropriate function from a given set of functions). If a class of functions has a finite VC dimension, then it is learnable. This measure of capacity is more robust than arbitrary measures such as the number of parameters. It is possible, for example, to find a simple set of functions that depends on only one parameter and that has infinite VC dimension.

•

? VC Dimension of an Indicator Function

Determine the VC dimension of the indicator function over = [0, 1] F (x) = {f : → {0, 1}, f (x) = 1x∈[t1 ,t2 ) , or f (x) = 1−1x∈[t1 ,t2 ) , t1 < t2 ∈ }. (4.12) Suppose there are three points x1 , x2 , and x3 and assume x1 < x2 < x3 without loss of generality. All possible labeling of the points is reachable; therefore, we assert that V C(F ) ≥ 3. With four points x1 , x2 , x3 , and x4 (assumed increasing as always), you cannot label x1 and x3 with the value 1 and x2 and x4 with the value 0, for example. Hence V C(F ) = 3.

Recently (Bartlett et al. 2017a) prove upper and lower bounds on the VC dimension of deep feedforward neural network classifiers with the piecewise linear activation function, such as ReLU activation functions. These bounds are tight for almost the entire range of parameters. Letting |W | be the number of weights and L be the number of layers, they proved that the VC dimension is O(|W |Llog(|W |)).

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They further showed the effect of network depth on VC dimension with different non-linearities: there is no dependence for piecewise constant, linear dependence for piecewise-linear, and no more than quadratic dependence for general piecewisepolynomials. Vapnik (1998) formulated a method of VC dimension based inductive inference. This approach, known as structural empirical risk minimization, achieved the smallest bound on the test error by using the training errors and choosing the machine (i.e., the set or functions) with the smallest VC dimension. The minimization problem expresses the bias–variance tradeoff . On the one hand, to minimize the bias, one needs to choose a function from a wide set of functions, not necessarily with a low VC dimension. On the other hand, the difference between the training error and the test error (i.e., variance) increases with VC dimension (a.k.a. expressibility). The expected risk is an out-of-sample measure of performance of the learned model and is based on the joint probability density function (pdf) p(x, y): R[Fˆ ] = E[L(Fˆ (X), Y )] =

(

L(Fˆ (x), y)dp(x, y).

(4.13)

If one could choose Fˆ to minimize the expected risk, then one would have a definite measure of optimal learning. Unfortunately, the expected risk cannot be measured directly since this underlying pdf is unknown. Instead, we typically use the risk over the training set of N observations, also known as the empirical risk measure (ERM): Remp (Fˆ ) :=

N 1 L(Fˆ (xi ), yi ). N

(4.14)

i=1

Under i.i.d. data assumptions, the law of large numbers ensures that the empirical risk will asymptotically converge to the expected risk. However, for small samples, one cannot guarantee that ERM will also minimize the expected risk. A famous result from statistical learning theory (Vapnik 1998) is that the VC dimension provides bounds on the expected risk as a function of the ERM and the number of training observations N , which holds with probability (1 − η):

R[Fˆ ] ≤ Remp (Fˆ ) +

/

0 0 h ln 2N + 1 − ln & η ' 1 h 4 N

,

(4.15)

where h is the VC dimension of Fˆ (X) and N > h. Figure 4.6 shows the tradeoff between VC dimension and the tightness of the bound. As the ratio N/ h gets larger, i.e. for a fixed N, we decrease h, the VC confidence becomes smaller, and the actual risk becomes closer to the empirical risk. On the other hand, choosing a model with a higher VC dimension reduces the ERM at the expense of increasing the VC confidence.

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Fig. 4.6 This figure shows the tradeoff between VC dimension and the tightness of the bound. As the ratio N/ h gets larger, i.e. for a fixed N, we decrease h, the VC confidence becomes smaller, and the actual risk becomes closer to the empirical risk. On the other hand, choosing a model with a higher VC dimension reduces the ERM at the expense of increasing the VC confidence

The VC dimension plays a more dominant role in small-scale learning problems, where i.i.d. training data is limited and optimization error, that is the error introduced by the optimizer, is negligible. Beyond a certain sample size, computing power and the optimization algorithm become more dominant and the VC dimension is limited as a measure of learnability. Several studies demonstrate that VC dimension based error bounds are too weak and its usage, while providing some intuitive notion of model complexity, have faded in favor of alternative theories. Perhaps most importantly for finance, the bound in Eq. 4.15 only holds for i.i.d. data and little is known in the case when the data is auto-correlated.

•

? Multiple Choice Question 1

Which of the following statements are true: 1. The hidden units of a shallow feedforward networkpartition, with n hidden units, p & ' partition the input space in Rp into no more than j =0 nj convex regions. 2. The VC dimension of a Heaviside activated shallow feedforward network, with one hidden unit, and p features, is p + 1. 3. The bias–variance tradeoff is equivalently expressed through the VC confidence and the empirical risk measure. 4. The upper bound on the out-of-sample error of a feedforward network depends on its VC dimension and the number of training samples. 5. The VC dimension always grows linearly with the number of layers in a deep network.

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2.6 When Is a Neural Network a Spline?* Under certain choices of the activation function, we can construct MLPs which are a certain type of piecewise polynomial interpolants referred to as “splines.” Let f (x) be any function whose domain is and the function values fk := f (xk ) are known only at grid points h := {xk | xk = kh, k ∈ {1, . . . , K}} ⊂ ⊂ R which are spaced by h. Note that the requirement that the data is gridded is for ease of exposition and is not necessary. We construct an orthogonal basis over to give the interpolant fˆ(x) =

K

φk (x)fk , x ∈ ,

(4.16)

i=1

where the {φk }K k=1 are orthogonal basis functions. Under additional restrictions of the function space of f , we can derive error bounds which are a function of h. We can easily demonstrate how a MLP with hidden units activated by Heaviside functions (unit step functions) is a piecewise constant functional approximation. Let f (x) be any function whose domain is = [0, 1]. Suppose that the function is Lipschitz continuous, that is, ∀x, x ∈ [0, 1), |f (x ) − f (x)| ≤ L|x − x|, for some constant L ≥ 0. Using Heaviside functions to activate the hidden units H (x) =

1,

x ≥ 0,

0,

x < 0,

(4.17)

L we construct a neural network with K = 2 + 1 units in a single hidden layer that approximates f (x) within > 0. That is, ∀x ∈ [0, 1), |f (x) − fˆ(x)| ≤ , where fˆ(x) is the output of the neural network given input x. Let = L . We shall show that the neural network is a linear combination of indicator functions, φk , with compact support over [xk − , xk + ) and centered about xk :

φk (x) =

1 [xk − , xk + ), 0 otherwise.

(4.18)

The {φk }K k=1 are piecewise constant basis functions, φi (xj ) = δij , and the first few are illustrated in Fig. 4.7 below. The basis functions satisfy the partition of unity property K k=1 φk (x) = 1, ∀x ∈ .

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We shall construct such basis functions as the difference of Heaviside functions φk (x) = H (x − (xk − )) − H (x − (xk + )), xk = (2k − 1) , by choosing the bias bk(1) = −2(k − 1) and W (1) = 1 so that the neural network, fˆ(X) = W (2) H (W (1) X + b(1) ) has values based on ⎤ H (x) ⎥ ⎢ H (x − 2 ) ⎥ ⎢ ⎥ ⎢ . . . ⎥ ⎢ H (W (1) x + b(1) ) = ⎢ ⎥. ⎢ H (x − 2(k − 1) ) ⎥ ⎥ ⎢ ⎦ ⎣ ... H (x − (2K − 1) ) ⎡

(4.19)

Then W (2) is set equal to exact function values and their differences: W (2) = [f (x1 ), f (x2 ) − f (x1 ), . . . , f (xK ) − f (xK−1 )], so that

(4.20)

4 Feedforward Neural Networks

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⎧ ⎪ f (x1 ), ⎪ ⎪ ⎪ ⎪ ⎪ f (x2 ), ⎪ ⎪ ⎪ ⎨. . . fˆ = ⎪ f (xk ), ⎪ ⎪ ⎪ ⎪ ⎪ ... ⎪ ⎪ ⎪ ⎩ f (xK−1 ),

x ≤ 2 , 2 < x ≤ 4 , ... 2(k) < x ≤ 2(k + 1) ,

(4.21)

... 2(K − 1) < x ≤ 2K .

Figure 4.8 illustrates the function approximation for the case when f (x) = cos(2π x). Since xk = (2k −1) ,we have that Yˆ = f (xk ) over the interval [xk − , xk + ], which is the support of φk (x). By the Lipschitz continuity of f (x), it follows that the worst-case error appears at the mid-point of any interval [xk , xk+1 ) |f (xk + ) − fˆ(x + )| = |f (x + ) − f (xk )| ≤ |f (xk )| + L − |f (xk )| = . (4.22) This example is a special case of a more general representation permitted by MLPs. If we relax that the points need to be gridded, but instead just assume there are K data points in Rp , then the region boundaries created by the K hidden units define a Voronoi diagram. Informally, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. The set of points are referred to as “seeds.” For each seed there is a corresponding region consisting of all points closer to that seed than to any other. The discussion of Voronoi diagrams is beyond the scope of this chapter, but suffice to say that the representation of MLPs as splines extends to higher dimensional input spaces and higher degree splines.

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Hence, under a special configuration of the weights and biases, with the hidden units defining Voronoi cells for each observation, we can show that a neural network is a univariate spline. This result generalizes to higher dimensional and higher order splines. Such a result enables us to view splines as a special case of a neural network which is consistent with our reasoning of neural networks as generalized approximation and regression techniques. The formulation of neural networks as splines allows approximation theory to guide the design of the network. Unfortunately, equating neural networks with splines says little about why and when multiples layers are needed.

2.7 Why Deep Networks? The extension to deep neural networks is in fact well motivated on statistical and information-theoretical grounds (Tishby and Zaslavsky 2015; Poggio 2016; Mhaskar et al. 2016; Martin and Mahoney 2018; Bartlett et al. 2017a). Poggio (2016) shows that deep networks can achieve superior performance versus linear additive models, such as linear regression, while avoiding the curse of dimensionality. There are additionally many recent theoretical developments which characterize the approximation behavior as a function of network depth, width, and sparsity level (Polson and Rockova 2018). Recently (Bartlett et al. 2017b) prove upper and lower bounds on the expressibility of deep feedforward neural network classifiers with the piecewise linear activation function, such as ReLU activation functions. These bounds are tight for almost the entire range of parameters. Letting n denote the total number of weights, they prove that the VC dimension is O(nLlog(n)).

•

> VC Dimension Theorem

Theorem (Bartlett et al. (2017b)) There exists a universal constant C such that the following holds. Given any W, L with W > CL > C 2 , there exists a ReLU network with ≤ L layers and ≤ W parameters with VC dimension ≥ W Llog(W/L)/C.

They further showed the effect of network depth on VC dimension with different non-linearities: there is no dependence for piecewise constant, linear dependence for piecewise-linear, and no more than quadratic dependence for general piecewisepolynomial. Thus the relationship between expressibility and depth is determined by the degree of the activation function. There is further ample theoretical evidence to

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suggest that shallow networks cannot approximate the class of non-linear functions represented by deep ReLU networks without blow-up. Telgarsky (2016) shows that there is a ReLU network with L layers such that any network approximating it with 1/3 only O(L1/3 ) layers must have (2L ) units. Mhaskar et al. (2016) discuss the differences between composition versus additive models and show that it is possible to approximate higher polynomials much more efficiently with several hidden layers than a single hidden layer. Martin and Mahoney (2018) shows that deep networks are implicitly selfregularizing behaving like Tikhonov regularization. Tishby and Zaslavsky (2015) characterizes the layers as “statistically decoupling” the input variables.

2.7.1

Approximation with Compositions of Functions

To gain some intuition as to why function composition can lead to successively more accurate function representation with each layer, consider the following example of a binary expansion of a decimal x. Example 4.1

Binary Expansion of a Decimal

For each integer n ≥ 1 and x ∈ [0, 1], define fn (x)= xn , where xn is the xn nth binary digit of x. The binary expansion of x = ∞ n=1 2n , where xn is 1 1 or 0 depends on whether Xn−1 ≥ 2n or otherwise, respectively, and Xn := x − ni=1 x2ii . For example, we can find the first binary digit, x1 as either 1 or 0 depending on whether x0 = x ≥ 12 . Now consider X1 = x − x1 /2 and set x2 = 1 if X1 ≥ 212 or x2 = 0 otherwise.

Example 4.2

Neural Network for Binary Expansion of a Decimal

A deep feedforward network for such a binary expansion of a decimal uses two neurons in each layer with different activations—Heaviside and identity functions. The input weight matrix, W (1) , is the identity matrix, the other weight matrices, {W () }>1 are W () ()

()

1 − 2−1 1 = , 1 1 − 2−1

and σ1 (x) = H (x, 21 ) and σ2 (x) = id(x) = x. There are no bias terms. The output after hidden layers is the error, X ≤ 21 .

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While the example of converting a decimal in binary format using a binary expansion is simple, the approach can be readily generalized to the binary expansion of polynomials. Theorem 4.2 (Liang and Srikant (2016)) For the pth order polynomial f (x) = p p i , x ∈ [0, 1] and a x |a there exists a multilayer neural network i=0 i i=1 &i | ≤ 1, & & ' p' p' ˆ f (x) with O p + log ε layers, O log ε Heaviside units, and O p log pε rectifier linear units such that |f (x) − fˆ(x)| ≤ ε, ∀x ∈ [0, 1]. Proof The sketch of the proof is as follows. Liang and Srikant (2016) use the deep structure shown in Fig. 4.9 to find the n-step binary expansion ni=0 ai x i of x. Then they construct a multilayer network to approximate polynomials gi (x) = x i , i = 1, . . . , p. Finally, they analyze the approximation error which is |f (x) − fˆ(x)| ≤

p . 2n−1

See Appendix “Proof of Theorem 4.2” for the proof.

2.7.2

Composition with ReLU Activation

An intuitive way to understand the importance of multiple network layers is to consider the effect of composing piecewise affine functions instead of adding them. It is easy to see that combinations of ReLU activated neurons give piecewise affine

X3 x3

H (x − x 1 /2 − x 2 / 4, 1/8)

id (x − x 1 /2 − x 2 / 4) x2

x2

H (x − x 1 /2, 1/4)

id (x − x 1 /2) x1

x1

H (x, 1/2)

id (x )

x Fig. 4.9 An illustrative example of a deep feedforward neural network for binary expansion of a decimal. Each layer has two neurons with different activations—Heaviside and identity functions

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4 Feedforward Neural Networks

approximations. For example, consider the shallow ReLU network with 2 and 4 perceptrons in Fig. 4.10: FW,b = W (2) σ (W (1) x + b(1) ), σ := max(x, 0). Let us start by defining σ : R → R to be t-sawtooth if it is piecewise affine with t pieces, meaning R is partitioned into t consecutive intervals, and σ is affine within each interval. Consequently, ReLU (x) is 2-sawtooth, but this class also includes many other functions, for instance, the decision stumps used in boosting are 2-sawtooth, and decision trees with t − 1 nodes correspond to t-sawtooths. The following lemma serves to build intuition about the effect of adding versus composing sawtooth functions which is illustrated in Fig. 4.11. Lemma 4.1 Let f : R → R and g : R → R be, respectively, k- and l-sawtooth. Then f + g is (k + l)-sawtooth, and f ◦ g is kl-sawtooth.

Fig. 4.10 A Shallow ReLU network with (left) two perceptrons and (right) four perceptrons

1

1

1/2

1

2σ (x) − 4σ (x − Two units W (2) = [2, −4] b(1) = [0, − 12 ]T

1/2

1

4σ (x) − 8σ (x − 14 )+ 4σ (x − 12 ) − 8σ (x − 34 ). Four units W (2) = [4, −8, 4, −8] b(1) = [0, − 14 , − 12 , − 34 ]T

0.6

0.6

f(x) + g(x)

1 2)

f(x) y 0.4

y 0.4

f(x)

g(x) f(g(x))

0.0

0.0

0.2

0.2

g(x)

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

x

(a)

0.6

0.8

1.0

x

(b)

Fig. 4.11 Adding versus composing 2-sawtooth functions. (a) Adding 2-sawtooths. (b) Composing 2-sawtooths

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131

Let us now build on this result by considering the mirror map fm : R → R, which is shown in Fig. 4.12, and defined as

fm (x) :=

⎧ ⎪ ⎪ ⎨2x

when 0 ≤ x ≤ 1/2,

⎪ ⎪ ⎩0

otherwise.

2(1 − x)

when 1/2 < x ≤ 1,

Note that fm can be represented by a two-layer ReLU activated network with two neurons; For instance, fm (x) = (2σ (x)−4σ (x−1/2)). Hence fmk is the composition of k (identical) ReLU sub-networks. A key observation is that fewer hidden units are needed to shatter a set of points when the network is deep versus shallow. Consider, for example, the sequence of n = 2k points with alternating labels, referred to as the n-ap, and illustrated in Fig. 4.13 for the case when k = 3. As the x values pass from left to right, the labels change as often as possible and provide the most challenging arrangement for shattering n points. There are many ways to measure the representation power of a network, but we will consider the classification error here. Suppose that we have a σ activated network with m units per layer and l layers. Given a function f : Rp → R let f˜ : Rp → {0, 1} denote the corresponding classifier f˜(x) := 1f (x)≥1/2 , and additionally given a sequence of points ((xi , yi ))ni=1 with xi ∈ Rp and yi ∈ {0, 1}, define the classification error as E(f ) := n1 i 1f˜(xi )=yi . Given a sawtooth function, its classification error on the n-ap may be lower bounded as follows.

1

1

1

1/2

1

1/2

fm fm2

1

1/2

1

fm3 .

Fig. 4.12 The mirror map composed with itself 1

1/2

1

Fig. 4.13 The n-ap consists of n uniformly spaced points with alternating labels over the interval [0, 1 − 2−n ]. That is, the points ((xi , yi ))ni=1 with xi = i2−n and yi = 0 when i is even, and otherwise yi = 1

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Lemma 4.2 Let ((xi , yi ))ni=1 be given according to the n-ap. Then every t-sawtooth function f : R → R satisfies E(f ) ≥ (n − 4t)/(3n). The proof in the appendix relies on a simple counting argument for the number of crossings of 1/2. If there are m t-saw-tooth functions, then by Lemma 4.1, the resultant is a piecewise affine function over mt intervals. The main theorem now directly follows from Lemma 4.2. Theorem 4.3 Let positive integer k, number of layers l, and number of nodes per layer m be given. Given a t-sawtooth σ : R → R and n := 2k points as specified by the n-ap, then min E(f ) ≥ W,b

n − 4(tm)l . 3n

From this result one can say, for example, that on the n-ap one needs m = 2k−3 many units when classifying with a ReLU activated shallow network versus only m = 2(1/ l(k−2)−1) units per layer for a l ≥ 2 deep network. Research on deep learning is very active and there are still many questions that need to be addressed before deep learning is fully understood. However, the purpose of these examples is to build intuition and motivate the need for many hidden layers in addition to the effect of increasing the number of neurons in each hidden layer. In the remaining part of this chapter we turn towards the practical application of neural networks and consider some of the primary challenges in the context of financial modeling. We shall begin by considering how to preserve the shape of functions being approximated and, indeed, how to train and evaluate a network.

3 Convexity and Inequality Constraints It may be necessary to restrict the range of fˆ(x) or impose certain properties which are known about the shape of the function f (x) being approximated. For example, V = f (S) might be an option price and S the value of the underlying asset and convexity and non-negativity of fˆ(S) are necessary. Consider the following feedforward network architecture FW,b (X) : Rp → Rd : (L) (1) Yˆ = FW,b (X) = fW (L) ,b(L) ◦ · · · ◦ fW (1) ,b(1) (X),

(4.23)

where ()

fW () ,b() (x) = σ (W () x + b() ), ∀ ∈ {1, . . . , L}.

(4.24)

3 Convexity and Inequality Constraints

133

Convexity For convexity of Yˆ w.r.t. x, the activation function, σ (x), must be a convex function of x. For avoidance of doubt, this convexity constraint should not be confused with convexity of the loss function w.r.t. the weights as in, for example, Bengio et al. (2006). Examples3 include ReLU(x) := max(x, 0) and softplus(x; t) := 1t ln(1 + exp{tx}). For this class of activation functions, the semi-affine function () fW () ,b() (x) = σ (W () x + b() ) must also be convex in x since a convex function of a linear combination of x is also convex in x. The composition, (+1) () (+1) fW (+1) ,b(+1) ◦ fW () ,b() (x), is convex if and only if fW (+1) ,b(+1) (x) is non()

decreasing convex and fW () ,b() (x) is convex. The proof is left to the reader as a straightforward exercise. Hence, for convexity of fˆ(x) = FW,b (x) w.r.t. x we require that the weights in all but the first layer be positive: ()

wij ≥ 0, ∀i, j, ∀ ∈ {2, . . . , L}.

(4.25)

The constraints on the weights needed to enforce convexity guarantee non-negative (L) (L) output if the bias bi ≥ 0, ∀i ∈ {1, . . . , d} and σ (x) ≥ 0, ∀x. Since wij ≥ 0, ∀i, j it follows that (L)

(L−1)

wij σ (Ii

) ≥ 0,

(4.26)

and with non-negative bias terms, fˆi is non-negative. We now separately consider bounding the network output independently of imposing convexity on fˆi (x) w.r.t. x. If we choose a bounded activation function σ ∈ [σ , σ¯ ], then we can easily impose linear inequality constraints to ensure that fˆi ∈ [ci , di ] ci ≤ fˆi (x) ≤ di , di > ci , i ∈ {1, . . . , d},

(4.27)

by setting bi(L)

= ci −

(L−1) n

(L) (L) min(sij |σ |, sij |σ¯ |)|wij |, sij := sign(wij ).

(4.28)

j =1

= 1t ln(1+exp{tx}), with a model parameter t >> 1, converges to the ReLU function in the limit t → ∞. 3 The parameterized softplus function σ (x; t)

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4 Feedforward Neural Networks

Note that the expression inside the min function can be simplified further to min(sij |σ |, sij |σ¯ |)|wij | = min(wij |σ |, wij |σ¯ |). Training of the weights and biases is a constrained optimization problem with the linear constraints (L−1) n

(L) max(sij |σ |, sij |σ¯ |)|wij | ≤ di − bi(L) ,

(4.29)

j =1

which can be solved with the method of Lagrange multipliers or otherwise. If we require that fˆi should be convex and bounded in the interval [ci , di ], then the (L) additional constraint, wij ≥ 0, ∀i, j , is needed of course and the above simplifies to (L) bi

= ci − σ

(L−1) n

(L)

wij

(4.30)

j =1 (L)

and solving the underdetermined system for wij , ∀j : (L−1) n

(L) σ¯ wij ≤ di − bi(L)

(4.31)

j =1 (L−1) n

j =1

(L) wij ≤

di − ci . (σ¯ − σ )

(4.32)

The following toy examples provide simplified versions of constrained learning problems that arise in derivative modeling and calibration. The examples are intended only to illustrate the methodology introduced here. The first example is motivated by the need to learn an arbitrage free option price as a function of the underlying asset price. In particular, there are three scenarios where neural networks, and more broadly, supervised machine learning is useful for pricing. First, it provides a “model-free” framework, where no data generation process is assumed for the underlying dynamics. Second, machine learning can price complex derivatives where no analytic solution is known. Finally, machine learning does not suffer from the curse of dimensionality w.r.t. to the input space and can thus scale to basket options, options on many underlying assets. Each of these aspects merits further exploration and our example illustrates some of the challenges with learning pricing functions. Perhaps the single largest defect of conventional derivative pricing models, however, is their calibration to data. Machine learning, too, provides an answer here—it provides a method for learning the relationship between market and contract variables and the model parameters.

3 Convexity and Inequality Constraints

Example 4.3

135

Approximating Option Prices

The payoff of a European call option at expiry time T is VT = max(ST − K, 0) and is convex with respect to S. Under the risk-neutral measure the option price at time t is the conditional expectation Vt = Et [exp{−r(T − t)}VT ]. Since the conditional expectation is a linear operator, it preserves the convexity of the payoff function, so that the option price is always convex w.r.t. the underlying price. Thus, the second derivative, γ is always non-negative. Furthermore, the option price must always be non-negative. Let us approximate the surface of a European call option with strike K over ¯ The input variable X ∈ R+ are underlying all underlying values St ∈ (0, S). asset prices and the outputs are call prices, so that the data is {Si , Vi }. We use a neural network to learn the relation V = f (S) and enforce the property that f is non-negative and convex w.r.t. S. In the following example, we train the MLP over a uniform grid of 100 training points Si ∈ h ⊂ [0.001, 300], and Vi = f (Si ) generated by the Black–Scholes (BS) pricing formula. The risk-free rate r = 0.01, the strike is 130, the volatility is σ , and time to maturity is T = 2.0. The test data of 100 observations are on a different uniform gridded over a wider domain [0.001, 600]. The network uses one hidden layer (L = 2) with 100 units, a (L) (L) softplus activation function, and wij , bi ≥ 0, ∀i, j . Figure 4.14 compares the prediction with the BS model over the test set. Yˆ is observed to be convex (2) w.r.t. S because wij is non-negative. Additionally, because bi(2) ≥ 0 and σ = 0, Yˆ ≥ 0. The figure also compares the Black–Scholes formula for the delta of the call option, (X), the derivative of the price w.r.t. to S with the gradient of Yˆ :

ˆ (X) = ∂X Yˆ = (W (2) )T DW (1) , Dii =

1 (1) 1 + exp{−w(1) i, X − bi }

.

(4.33)

Under the BS model, the delta of a call option is in the interval [0, 1]. Note that the delta, although observed positive here, could be negative since there are no restrictions on W (1) . Similarly, the delta approximation is observed to exceed unity. Thus, additional constraints are needed to bound the delta. For (1) this architecture, imposing wij ≥ 0 preserves the non-negativity of the delta (1) (2) (1) and nj wij wj, ≤ 1, ∀i bounds the delta at unity.

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4 Feedforward Neural Networks

(a) Estimated call prices

(b) Derived delta

Fig. 4.14 (a) The out-of-sample call prices are estimated using a single-layer neural network with constraints to ensure non-negativity and convexity of the price approximation w.r.t. the underlying price S. (b) The analytic derivative of Yˆ is taken as the approximation of delta and compared over the test set with the Black–Scholes delta. We observe that additional constraints on the weights are needed to ensure that ∂X Yˆ ∈ [0, 1]

Example 4.4

Calibrating Options

The goal is to learn the inverse of the Black–Scholes formula, as a function of moneyness, M = S/K. For simplicity, it considers the calibration of a chain of European in-the-money put or in-the-money equity call options with fixed time to maturity only. The input is moneyness for each option in the chain. The output of the neural network is the BS implied volatility—this is the implied volatility needed to calibrate the BS model to option price data corresponding to each moneyness. The neural network preserves the positivity of the volatility and, in this example, imposes a convexity constraint on the surface w.r.t. to moneyness. The latter ensures consistency with liquid option markets, the implied volatility for both puts and calls typically monotonically increases as the strike price moves away from the current stock price—the so-called implied volatility smile. In markets, such as the equity markets, an implied volatility skew occurs because money managers usually prefer to write calls over puts. The input variable X ∈ R+ is moneyness and the output is volatility so that the training data is {Mi , σi }. We use a neural network to learn the relation σ = f (M) and enforce the property that f is non-negative and convex w.r.t. M. Note, in this example, that we do not directly learn the relationship between option prices and implied volatilities. Instead we learn how a BS root finder approximates the implied volatility as a function of the moneyness. (continued)

3 Convexity and Inequality Constraints

Example 4.4

137

(continued)

In the following example, we train the MLP over a uniform grid of n = 100 training points Mi ∈ h ⊂ [0.5, 1 × 104 ], and σi = f (Mi ) is generated by using a root finder for V (σ ; S, Ki , τ, r) − Vˆi = 0, ∀i = 1, . . . , n and τ = 0.2 years using the option price with strike Ki and time to maturity τ . The risk-free rate r = 0.01. The test data of 100 observations are on a different uniform gridded over a wider domain [0.4166, 1 × 104 ]. The network uses one hidden layer (L = 2) with 100 units, a softplus activation function, and (L) (L) wij , bi ≤ 0, ∀i, j . Figure 4.15 compares the out-of-sample model output with the root finder for the BS model over the test set. Yˆ is observed to be (2) (2) convex w.r.t. M because wij is non-negative. Additionally, because bi ≥ 0 and σ = 0, Yˆ ≥ 0.

No-Arbitrage Pricing The previous examples are simple enough to illustrate the application of constraints in neural networks. However, one would typically need to enforce more complex constraints for no-arbitrage pricing and calibration. Pricing approximations should be monotonically increasing w.r.t. to maturity and convex w.r.t. strike. Such constraints require that the neural network is fitted with more input variables K and T . Accelerating Calibrations One promising direction, which does not require neural network derivative pricing, is to simply learn a stochastic volatility based pricing model, such as the Heston model, as a function of underlying price, strike, and maturity, and then use the neural network pricing function to calibrate the pricing model. Such a calibration Fig. 4.15 The out-of-sample MLP estimation of implied volatility as a function of moneyness is compared with the true values

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4 Feedforward Neural Networks

avoids fitting a few parameters to the chain of observed option prices or implied volatilities. Replacement of expensive pricing functions, which may require FFTs or Monte Carlo methods, with trained neural networks reduces calibration time considerably. See Horvath et al. (2019) for further details. Dupire Local Volatility Another challenge is how to price exotic options consistently with the market prices of their European counterpart. The former are typically traded over-thecounter, whereas the latter are often exchange traded and therefore “fully” observable. To fix ideas, let C(K, T ) denote an observed call price, for some fixed strike, K, maturity, T , and underlying price St . Modulo a short rate and dividend term, the unique “effective” volatility, σ02 , is given by the Dupire formula: σ02 =

2∂T C(K, T ) . 2 C(K, T ) K 2 ∂K

(4.34)

The challenge arises when calibrating the local volatility model, extracting effective volatility from market option prices is an ill-posed inverse problem. Such a challenge has recently been addressed by Chataigner et al. (2020) in their paper on deep local volatility.

•

? Multiple Choice Question 2

Which of the following statements are true: 1. A feedforward architecture is always convex w.r.t. each input variables if every activation function is convex and the weights are constrained to be either all positive or all negative. 2. A feedforward architecture with positive weights is a monotonically increasing function of the input for any choice of monotonically increasing activation function. 3. The weights of a feedforward architecture must be constrained for the output of a feedforward network to be bounded. 4. The bias terms in a network simply shift the output and have no effect on the derivatives of the output w.r.t. to the input.

3.1 Similarity of MLPs with Other Supervised Learners Under special circumstances, MLPs are functionally equivalent to a number of other machine learning techniques. As previously mentioned, when the network has no hidden layer, it is either a regression or logistic regression. Neural networks with

3 Convexity and Inequality Constraints

139

one hidden layer is essentially a projection pursuit regression (PPR), both project the input vector onto a hyperplane, apply a non-linear transformation into feature space, followed by an affine transformation. The mapping of input vectors to feature space by the hidden layer is conceptually similar to kernel methods, such as support vector machines (SVMs), which map to a kernel space, where classification and regression are subsequently performed. Boosted decision stumps, one level boosted decision trees, can even be expressed as a single-layer MLP. Caution must be exercised in over-stretching these conceptual similarities. Data generation assumptions aside, there are differences in the classes of non-linear functions and learning algorithms used. For example, the non-linear function being fitted in PPR can be different for each combination of input variables and is sequentially estimated before updating the weights. In contrast, neural networks fix these functions and estimate all the weights belonging to a single layer simultaneously. A summary of other machine learning approaches is given in Table 4.1 and we refer the reader to numerous excellent textbooks (Bishop 2006; Hastie et al. 2009) covering such methods. Table 4.1 This table compares supervised machine learning algorithms (reproduced from Mullainathan and Spiess (2017)) Function class F (and its parameterization) Global/parametric predictors Linear β x (and generalizations)

Regularizer R(f ) Subset selection β 0 = kj =1 1βj =0 k LASSO β 1 = j =1 |βj | Ridge β 22 = kj =1 βj2 Elastic net α β 1 + (1 − α) β 22

Local/non-parametric predictors Decision/regression trees Random forest (linear combination of trees) Nearest neighbors Kernel regression Mixed predictors Deep learning, neural nets, convolutional neural networks Splines Combined predictors Bagging: unweighted average of predictors from bootstrap draws Boosting: linear combination of predictions of residual Ensemble: weighted combination of different predictors

Depth, number of nodes/leaves, minimal leaf size, information gain at splits Number of trees, number of variables used in each tree, size of bootstrap sample, complexity of trees (see above) Number of neighbors Kernel bandwidth Number of levels, number of neurons per level, connectivity between neurons Number of knots, order Number of draws, size of bootstrap samples (and individual regularization parameters) Learning rate, number of iterations (and individual regularization parameters) Ensemble weights (and individual regularization parameters)

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4 Feedforward Neural Networks

4 Training, Validation, and Testing Deep learning is a data-driven approach which focuses on finding structure in large datasets. The main tools for variable or predictor selection are regularization and dropout. Out-of-sample predictive performance helps assess the optimal amount of regularization, the problem of finding the optimal hyperparameter selection. There is still a very Bayesian flavor to the modeling procedure and the modeler follows two key steps: 1. Training phase: pair the input with expected output, until a sufficiently close match has been found. Gauss’ original least squares procedure is a common example. 2. Validation and test phase: assess how well the deep learner has been trained for out-of-sample prediction. This depends on the size of your data, the value you would like to predict, the input, etc., and various model properties including the mean-error for numeric predictors and classification errors for classifiers. Often, the validation phase is split into two parts. 2.a First, estimate the out-of-sample accuracy of all approaches (a.k.a. validation). 2.b Second, compare the models and select the best performing approach based on the validation data (a.k.a. verification). Step 2.b. can be skipped if there is no need to select an appropriate model from several rivaling approaches. The researcher then only needs to partition the dataset into a training and test set. To construct and evaluate a learning machine, we start with training data of input– output pairs D = {Y (i) , X(i) }N i=1 . The goal is to find the machine learner of Y = F (X), where we have a loss function L(Y, Yˆ ) for a predictor, Yˆ , of the output signal, Y . In many cases, there is an underlying probability model, p(Y | Yˆ ), then the loss function is the negative log probability L(Y, Yˆ ) = − log p(Y | Yˆ ). For example, under a Gaussian model L(Y, Yˆ ) = ||Y − Yˆ ||2 is a L2 norm, for binary classification, L(Y, Yˆ ) = −Y log Yˆ is the negative cross-entropy. In its simplest form, we then solve an optimization problem minimizef (W, b) + λφ(W, b) W,b

f (W, b) =

N 1 L(Y (i) , Yˆ (X(i) )) N i=1

with a regularization penalty, φ(W, b). The loss function is non-convex, possessing many local minima and is generally difficult to find a global minimum. An important assumption, which is often not explicitly stated, is that the errors are assumed to be “homoscedastic.” Homoscedasticity is the assumption that the error has an identical distribution over each observation. This assumption can be relaxed by

4 Training, Validation, and Testing

141

weighting the observations differently. However, we shall regard such extensions as straightforward and compatible with the algorithms for solving the unweighted optimization problem. Here λ is a global regularization parameter which we tune using the out-of-sample predictive mean squared error (MSE) of the model. The regularization penalty, φ(W, b), introduces a bias–variance tradeoff . ∇L is given in closed form by a chain rule and, through back-propagation, each layer’s weights Wˆ () are fitted with stochastic gradient descent. Recall from Chap. 1 that a 1-of-K encoding is used for a categorical response, so that G is a K-binary vector G ∈ [0, 1]K and the value k is presented as Gk = ˆ k := 1 and Gj = 0, ∀j = k, where ||Gk ||1 = 1. The predictor is given by G ˆ k ||1 = 1 and the loss function is the negative cross-entropy for gk (X|(W, b)), ||G discrete random variables ˆ ˆ L(G, G(X)) = −GT lnG.

(4.35)

For example, if there are K = 3 classes, then G = [0, 0, 1], G = [0, 1, 0], or G = [1, 0, 0] to represent the three classes. When K > 2, the output layer has K neurons and the loss function is the negative cross-entropy ˆ L(G, G(X)) =−

K

ˆ k. Gk lnG

(4.36)

k=1

For the case when K = 2, i.e. binary classification, there is only one neuron in the output layer and the loss function is ˆ ˆ − (1 − G)ln(1 − G), ˆ L(G, G(X)) = −GlnG

(4.37)

ˆ = g1 (X|(W, b)) = σ (I (L−1) ) and σ is a sigmoid function. where G We observe that when there are no hidden layers, I (1) = W (1) X + b(1) and g1 (X|(W, b)) is a logistic regression. The softmax function, σs generalizes binary classifiers to multi-classifiers. σs : RK → [0, 1]K is a continuous K-vector function given by σs (x)k =

exp(xk ) , k ∈ {1, . . . , K}, || exp(x)||1

(4.38)

where ||σs (x)||1 =1. The softmax function is used to represent a probability distribution over K possible states: ˆ k = P (G = k | X) = σs (W X + b) = exp((W X + b)k ) . G || exp(W X + b)||1

(4.39)

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4 Feedforward Neural Networks

•

> Derivative of the Softmax Function

Using the quotient rule f (x) = can be written as:

g (x)h(x)−h (x)g(x) , [h(x)]2

the derivative σ := σs (x)

∂σi exp(xi )|| exp(x)||1 − exp(xi ) exp(xi ) = ∂xi || exp(x)||21 =

exp(xi ) || exp(x)||1 − exp(xi ) · || exp(x)||1 || exp(x)||1

= σi (1 − σi )

(4.40) (4.41) (4.42)

For the case i = j , the derivative of the sum is 0 − exp(xi ) exp(xj ) ∂σi = ∂xj || exp(x)||21 =−

exp(xj ) exp(xi ) · || exp(x)||1 || exp(x)||1

= −σj σi This can be written compactly as Kronecker delta function.

(4.43) (4.44) (4.45)

∂σi ∂xj

= σi (δij − σj ), where δij is the

5 Stochastic Gradient Descent (SGD) Stochastic gradient descent (SGD) method or its variation is typically used to find the deep learning model weights by minimizing the penalized loss function, f (W, b). The method minimizes the function by taking a negative step along an estimate g k of the gradient ∇f (W k , bk ) at iteration k. The approximate gradient is calculated by gk =

1 ∇LW,b (Y (i) , Yˆ k (X(i) )), bk i∈Ek

5 Stochastic Gradient Descent (SGD)

143

where Ek ⊂ {1, . . . , N } and bk = |Ek | is the number of elements in Ek (a.k.a. batch size). When bk > 1 the algorithm is called batch SGD and simply SGD otherwise. A usual strategy to choose subset E is to go cyclically and pick consecutive elements of {1, . . . , N } and Ek+1 = [Ek mod N ]+1, where modular arithmetic is applied to the set. The approximated direction g k is calculated using a chain rule (a.k.a. backpropagation) for deep learning. It is an unbiased estimator of ∇f (W k , bk ), and we have E(g k ) =

N

1 ∇LW,b Y (i) , Yˆ k (X(i) ) = ∇f (W k , bk ). N i=1

At each iteration, we update the solution (W, b)k+1 = (W, b)k − tk g k . Deep learning applications use a step size tk (a.k.a. learning rate) as constant or a reduction strategy of the form, tk = a exp{−kt}. Appropriate learning rates or the hyperparameters of reduction schedule are usually found empirically from numerical experiments and observations of the loss function progression. In order to update the weights across the layers, back-propagation is needed and will now be explained.

•

? Multiple Choice Question 3

Which of the following statements are true: 1. The training of a neural network involves minimizing a loss function w.r.t. the weights and biases over the training data. 2. L1 regularization is used during model selection to penalize models with too many parameters. 3. In deep learning, regularization can be applied to each layer of the network. 4. Back-propagation uses the chain rule to update the weights of the network but is not guaranteed to convergence to a unique minimum. 5. Stochastic gradient descent and back-propagation are two different optimization algorithms for minimizing the loss function and the user must choose the best one.

5.1 Back-Propagation Staying with a multi-classifier, we can begin by informally motivating the need for a recursive approach to updating the weights and biases. Let us express Yˆ ∈ [0, 1]K as a function of the final weight matrix W ∈ RK×M and output bias RK so that Yˆ (W, b) = σ ◦ I (W, b),

(4.46)

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4 Feedforward Neural Networks

where the input function I : RK×M ×RK → RK is of the form I (W, b) := W X +b and σ : RK → RK is the softmax function. Applying the multivariate chain rule gives the Jacobian of Yˆ (W, b): ∇ Yˆ (W, b) = ∇(σ ◦ I )(W, b) = ∇σ (I (W, b)) · ∇I (W, b).

5.1.1

(4.47) (4.48)

Updating the Weight Matrices

Recall that the loss function for a multi-classifier is the cross-entropy L(Y, Yˆ (X)) = −

K

Yk lnYˆk .

(4.49)

k=1

Since Y is a constant vector we can express the cross-entropy as a function of (W, b) L(W, b) = L ◦ σ (I (W, b)).

(4.50)

Applying the multivariate chain rule gives ∇L(W, b) = ∇(L ◦ σ )(I (W, b)) = ∇L(σ (I (W, b))) · ∇σ (I (W, b)) · ∇I (W, b).

(4.51) (4.52)

Stochastic gradient descent is used to find the minimum ˆ = arg minW,b (Wˆ , b)

N 1 L(yi , Yˆ W,b (xi )). N

(4.53)

i=1

Because of the compositional form of the model, the gradient must be derived using the chain rule for differentiation. This can be computed by a forward and then a backward sweep (“back-propagation”) over the network, keeping track only of quantities local to each neuron. Forward Pass Set Z (0) = X and for ∈ {1, . . . , L} set ()

Z () = fW () ,b() (Z (−1) ) = σ () (W () Z (−1) + b() ).

(4.54)

On completion of the forward pass, the error Yˆ − Y is evaluated using Yˆ := Z (L) .

5 Stochastic Gradient Descent (SGD)

145

Back-Propagation Define the back-propagation error δ () := ∇b() L, where given δ (L) = Yˆ − Y , and for = L − 1, . . . , 1 the following recursion relation gives the updated backpropagation error and weight update for layer : δ () = (∇I () σ () )W (+1)T δ (+1) , ∇W () L = δ () ⊗ Z (−1) ,

(4.55) (4.56)

and ⊗ is the outer product of two vectors. See Appendix “Back-Propagation” for a derivation of Eqs. 4.55 and 4.56. The weights and biases are updated for all ∈ {1, . . . , L} according to the expression W () = −γ ∇W () L = −γ δ () ⊗ Z (−1) , b() = −γ δ () , where γ is a user defined learning rate parameter. Note the negative sign: this indicates that weight changes are in the direction of decrease in error. Mini-batch or off-line updates involve using many observations of X at the same time. The batch size refers to the number of observations of X used in each pass. An epoch refers to a round-trip (i.e., forward+backward pass) over all training samples. Example 4.5

Back-Propagation with a Three-Layer Network

Suppose that a feedforward network classifier has two sigmoid activated hidden layers and a softmax activated output layer. After a forward pass, the values of {Z () }3=1 are stored and the error Yˆ − Y , where Yˆ = Z (3) , is calculated for one observation of X. The back-propagation and weight updates in the final layer are evaluated: δ (3) = Yˆ − Y ∇W (3) L = δ (3) ⊗ Z (2) . Now using Eqs. 4.55 and 4.56, we update the back-propagation error and weight updates for hidden layer 2 δ (2) = Z (2) (1 − Z (2) )(W (3) )T δ (3) , ∇W (2) L = δ (2) ⊗ Z (1) . (continued)

146

Example 4.4

4 Feedforward Neural Networks

(continued)

Repeating for hidden layer 1 δ (1) = Z (1) (1 − Z (1) )(W (2) )T δ (2) , ∇W (1) L = δ (1) ⊗ X. We update the weights and biases using Eqs. 4.57 and 4.57, so that b(3) → b(3) − γ δ (3) , W (3) → W (3) − γ δ (3) ⊗ Z (2) and repeat for the other weight-bias pairs, {(W () , b() )}2=1 . See the back-propagation notebook for further details of a worked example in Python and then complete Exercise 4.12.

5.2 Momentum One disadvantage of SGD is that the descent in f is not guaranteed or can be very slow at every iteration. Furthermore, the variance of the gradient estimate g k is near zero as the iterates converge to a solution. To address those problems a coordinate descent (CD) and momentum-based modifications of SGD are used. Each CD step evaluates a single component Ek of the gradient ∇f at the current point and then updates the Ek th component of the variable vector in the negative gradient direction. The momentum-based versions of SGD or the so-called accelerated algorithms were originally proposed by Nesterov (2013). The use of momentum in the choice of step in the search direction combines new gradient information with the previous search direction. These methods are also related to other classical techniques such as the heavy-ball method and conjugate gradient methods. Empirically momentum-based methods show a far better convergence for deep learning networks. The key idea is that the gradient only influences changes in the “velocity” of the update v k+1 =μv k − tk g k , (W, b)k+1 =(W, b)k + v k . The parameter μ controls the dumping effect on the rate of update of the variables. The physical analogy is the reduction in kinetic energy that allows “slow down” in the movements at the minima. This parameter is also chosen empirically using cross-validation. Nesterov’s momentum method (a.k.a. Nesterov acceleration) instead calculate gradient at the point predicted by the momentum. We can think of it as a look-ahead strategy. The resulting update equations are

5 Stochastic Gradient Descent (SGD)

147

v k+1 =μv k − tk g((W, b)k + v k ), (W, b)k+1 =(W, b)k + v k . Another popular modification to the SGD method is the AdaGrad method, which adaptively scales each of the learning parameters at each iteration ck+1 =ck + g((W, b)k )2 ,

. (W, b)k+1 =(W, b)k − tk g(W, b)k )/( ck+1 − a), where a is usually a small number, e.g. a = 10−6 that prevents dividing by zero. PRMSprop takes the AdaGrad idea further and places more weight on recent values of the gradient squared to scale the update direction, i.e. we have ck+1 = dck + (1 − d)g((W, b)k )2 . The Adam method combines both PRMSprop and momentum methods and leads to the following update equations: v k+1 =μv k − (1 − μ)tk g((W, b)k + v k ), ck+1 =dck + (1 − d)g((W, b)k )2 , . (W, b)k+1 =(W, b)k − tk v k+1 /( ck+1 − a). Second-order methods solve the optimization problem by solving a system of nonlinear equations ∇f (W, b) = 0 with Newton’s method (W, b)+ = (W, b) − {∇ 2 f (W, b)}−1 ∇f (W, b). SGD simply approximates ∇ 2 f (W, b) by 1/t. The advantages of a second-order method include much faster convergence rates and insensitivity to the conditioning of the problem. In practice, second-order methods are rarely used for deep learning applications (Dean et al. 2012). The major disadvantage is the inability to train the model using batches of data as SGD does. Since typical deep learning models rely on large-scale datasets, second-order methods become memory and computationally prohibitive at even modest-sized training datasets.

5.2.1

Computational Considerations

Batching alone is not sufficient to scale SGD methods to large-scale problems on modern high-performance computers. Back-propagation through a chain rule creates an inherit sequential dependency in the weight updates which limits the

148

4 Feedforward Neural Networks

dataset dimensions for the deep learner. Polson et al. (2015) consider a proximal Newton method, a Bayesian optimization technique which provides an efficient solution for estimation and optimization of such models and for calculating a regularization path. The authors present a splitting approach, alternating direction method of multipliers (ADMM), which overcomes the inherent bottlenecks in backpropagation by providing a simultaneous block update of parameters at all layers. ADMM facilitates the use of large-scale computing. A significant factor in the widespread adoption of deep learning has been the creation of TensorFlow (Abadi et al. 2016), an interface for easily expressing machine learning algorithms and mapping compute intensive operations onto a wide variety of different hardware platforms and in particular GPU cards. Recently, TensorFlow has been augmented by Edward (Tran et al. 2017) to combine concepts in Bayesian statistics and probabilistic programming with deep learning.

5.2.2

Model Averaging via Dropout

We close this section by briefly mentioning one final technique which has proved indispensable in preventing neural networks from over-fitting. Dropout is a computationally efficient technique to reduce model variance by considering many model configurations and then averaging the predictions. The layer input space Z = (Z1 , . . . , Zn ), where n is large, needs dimension reduction techniques which are designed to avoid over-fitting in the training process. Dropout works by removing layer inputs randomly with a given probability θ . The probability, θ , can be viewed as a further hyperparameter (like λ) which can be tuned via cross-validation. Heuristically, if there are 1000 variables, then a choice of θ = 0.1 will result in a search for models with 100 variables. The dropout architecture with stochastic search for the predictors can be used ()

di

∼ Ber(θ ),

Z˜ () = d () ◦ Z () , 1 ≤ < L, Z () = σ () (W () Z˜ (−1) + b() ). Effectively, this replaces the layer input Z by d ◦ Z, where ◦ denotes the elementwise product and d is a vector of independent Bernoulli, Ber(θ ), distributed random variables. The overall objective function is closely related to ridge regression with a g-prior (Heaton et al. 2017).

6 Bayesian Neural Networks

149

6 Bayesian Neural Networks* Bayesian deep learning (Neal 1990; Saul et al. 1996; Frey and Hinton 1999; Lawrence 2005; Adams et al. 2010; Mnih and Gregor 2014; Kingma and Welling 2013; Rezende et al. 2014) provides a powerful and general framework for statistical modeling. Such a framework allows for a completely new approach to data modeling and solves a number of problems that conventional models cannot address: (i) DLs (deep learners) permit complex dependencies between variables to be explicitly represented which are difficult, if not impossible, to model with copulas; (ii) they capture correlations between variables in high-dimensional datasets; and (iii) they characterize the degree of uncertainty in predicting large-scale effects from large datasets relevant for quantifying uncertainty. Uncertainty refers to the statistically unmeasurable situation of Knightian uncertainty, where the event space is known but the probabilities are not (Chen et al. 2017). Oftentimes, a forecast may be shrouded in uncertainty arising from noisy data or model uncertainty, either through incorrect modeling assumptions or parameter error. It is desirable to characterize this uncertainty in the forecast. In conventional Bayesian modeling, uncertainty is used to learn from small amounts of lowdimensional data under parametric assumptions on the prior. The choice of the prior is typically the point of contention and chosen for solution tractability rather than modeling fidelity. Recently, deterministic deep learners have been shown to scale well to large, high-dimensional, datasets. However, the probability vector obtained from the network is often erroneously interpreted as model confidence (Gal 2016). A typical approach to model uncertainty in neural network models is to assume that model parameters (weights and biases) are random variables (as illustrated in Fig. 4.16). Then ANN model approaches Gaussian process as the number of weights goes to infinity (Neal 2012; Williams 1997). In the case of finite number of weights, a network with random parameters is called a Bayesian neural network (MacKay 1992b). Recent advances in “variational inference” techniques and software that represent mathematical models as a computational graph (Blundell et al. 2015a) enable probabilistic deep learning models to be built, without having to worry about how to perform testing (forward propagation) or inference (gradient- based optimization, with back-propagation and automatic differentiation). Variational inference is an approximate technique which allows multi-modal likelihood functions to be extremized with standard stochastic gradient descent algorithm. An alternative to variational and MCMC algorithms was recently proposed by Gal (2016) and builds on efficient dropout regularization technique. All of the current techniques rely on approximating the true posterior over the model parameters p(w | X, Y ) by another distribution qθ (w) which can be evaluated in a computationally tractable way. Such a distribution is chosen to be as close as possible to the true posterior and is found by minimizing the Kullback– Leibler (KL) divergence

150

4 Feedforward Neural Networks 3

0.90

2

0.75

1 0.60

x2

0 0.45

-1 0.30

-2

-3 -3

Probability of class (label=0)

1.05

0.15

-2

-1

x1

1

2

3

3

0.00

0.40 0.35

2 1

x2

0.25 0.20

0.15

Uncertainty

0.30

-1 0.10

-2

-3 -3

0.05

-2

-1

x1

1

2

3

0.00

Fig. 4.16 Bayesian classification of the half-moon problem with neural networks. (top) The posterior mean and (bottom) the posterior std. dev.

θ ∗ ∈ arg min θ

( qθ (w) log

qθ (w) dw. p(w | X, Y )

There are numerous approaches to Bayesian deep learning for uncertainty quantification including MCMC (Markov chain Monte Carlo) methods. These are known to scale poorly with the number of observations and recent studies have

6 Bayesian Neural Networks

151

developed SG-MCMC (stochastic gradient MCMC) and related methods such as PX-MCMC (parameter expansion MCMC) to ease the computational burden. A Bayesian extension of feedforward network architectures has been considered by several authors (Neal 1990; Saul et al. 1996; Frey and Hinton 1999; Lawrence 2005; Adams et al. 2010; Mnih and Gregor 2014; Kingma and Welling 2013; Rezende et al. 2014). Recent results show how dropout regularization can be used to represent uncertainty in deep learning models. In particular, Gal (2015) shows that dropout provides uncertainty estimates for the predicted values. The predictions generated by the deep learning models with dropout are nothing but samples from the predictive posterior distribution. A classical example of using neural networks to model a vector of binary variables is the Boltzmann machine (BM), with two layers. The first layer encodes latent variables and the second layer encodes the observed variables. Both conditional distributions p(data | latent variables) and p(latent variables | data) are specified using logistic functions parameterized by weights and offset vectors. The size of the joint distribution table grows exponentially with the number of variables and (Hinton and Sejnowski 1983) proposed using Gibbs sampler to calculate updates to model weights on each iteration. The multi-modal nature of the posterior distribution leads to prohibitive computational times required to learn models of a practical size. Tieleman (2008) proposed a variational approach that replaces the posterior p(latent variables | data) and approximates it with another easy to calculate distribution and was considered in Salakhutdinov (2008). Several extensions to the BMs have been proposed. Exponential family extensions have been considered by (Smolensky 1986; Salakhutdinov 2008; Salakhutdinov and Hinton 2009; Welling et al. 2005). There have also been multiple approaches to building inference algorithms for deep learning models (MacKay 1992a; Hinton and Van Camp 1993; Neal 1992; Barber and Bishop 1998). Performing Bayesian inference on a neural network calculates the posterior distribution over the weights given the observations. In general, such a posterior cannot be calculated analytically, or even efficiently sampled from. However, several recently proposed approaches address the computational problem for some specific deep learning models (Graves 2011; Kingma and Welling 2013; Rezende et al. 2014; Blundell et al. 2015b; Hernández-Lobato and Adams 2015; Gal and Ghahramani 2016). The recent successful approaches to develop efficient Bayesian inference algorithms for deep learning networks are based on the reparameterization techniques for calculating Monte Carlo gradients while performing variational inference. Such an approach has led to an explosive development in the application of stochastic variational inference. Given the data D = (X, Y ), the variational inference relies on approximating the posterior p(θ | D) with a variation distribution q(θ | D, φ), where θ = (W, b). Then q is found by minimizing the Kullback–Leibler divergence between the approximate distribution and the posterior, namely ( KL(q || p) =

q(θ | D, φ) log

q(θ | D, φ) dθ. p(θ | D)

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4 Feedforward Neural Networks

Since p(θ | D) is not necessarily tractable, we replace minimization of KL(q || p) with maximization of the evidence lower bound (ELBO) ( ELBO(φ) =

q(θ | D, φ) log

p(Y | X, θ )p(θ ) dθ q(θ | D, φ)

The log of the total probability (evidence) is then log p(D) = ELBO(φ) + KL(q || p). The sum does not depend on φ, thus minimizing KL(q || p) is the same as maximizing ELBO(q). Also, since KL(q || p) ≥ 0, which follows from Jensen’s inequality, we have log p(D) ≥ ELBO(φ). Thus, the evidence lower bound name. The resulting maximization problem ELBO(φ) → maxφ is solved using stochastic gradient descent. To calculate the gradient, it is convenient to write the ELBO as ( ELBO(φ) =

( q(θ | D, φ) log p(Y | X, θ )dθ −

q(θ | D, φ) log

q(θ | D, φ) dθ p(θ )

% The gradient of the first term ∇φ q(θ | D, φ) log p(Y | X, θ )dθ = ∇φ Eq log p(Y | X, θ ) is not an expectation and thus cannot be calculated using Monte Carlo methods. The idea is to represent the gradient ∇φ Eq log p(Y | X, θ ) as an expectation of some random variable, so that Monte Carlo techniques can be used to calculate it. There are two standard methods to do it. First, the logderivative trick uses the following identity ∇x f (x) = f (x)∇x log f (x) to obtain ∇φ Eq log p(Y | θ ). Thus, if we select q(θ | φ) so that it is easy to compute its derivative and generate samples from it, the gradient can be efficiently calculated using Monte Carlo techniques. Second, we can use the reparameterization trick by representing θ as a value of a deterministic function, θ = g(, x, φ), where ∼ r() does not depend on φ. The derivative is given by ( ∇φ Eq log p(Y | X, θ ) =

r()∇φ log p(Y | X, g(, x, φ))d

= E [∇g log p(Y | X, g(, x, φ))∇φ g(, x, φ)]. The reparameterization is trivial when q(θ | D, φ) = N(θ | μ(D, φ), (D, φ)), and θ = μ(D, φ) + (D, φ), ∼ N(0, I ). Kingma and Welling (2013) propose using (D, φ) = I and representing μ(D, φ) and as outputs of a neural network (multilayer perceptron), the resulting approach was called a variational autoencoder. A generalized reparameterization has been proposed by Ruiz et al. (2016) and combines both log-derivative and reparameterization techniques by assuming that can depend on φ.

8 Exercises

153

7 Summary In this chapter we have introduced some of the theory of function approximation and out-of-sample estimation with neural networks when the observation points are i.i.d. Such a case is not suitable for times series data and shall be the subject of later chapters. We restricted our attention to feedforward neural networks in order to explore some of the theoretical arguments which help us reason scientifically about architecture design. We have seen that feedforward networks use hidden units, or perceptrons, to partition the input space into regions bounded with manifolds. In the case of ReLU activated units, each manifold is a hyperplane and the hidden units form a hyperplane arrangement. We have introduced various approaches to reason about the effect of the number of units in each layer in addition to reasoning about the effect of hidden layers. We also introduced various concepts and methods necessary for understanding and applying neural networks to i.i.d. data including – Fat shattering, VC dimension, and the empirical risk measure (ERM) as the basis for characterizing the learnability of a class of MLPs; – The construction of neural networks as splines and their pointwise approximation error bound; – The reason for composing layers in deep learning; – Stochastic gradient descent and back-propagation as techniques for training neural networks; and – Imposing constraints on the network needed for approximating financial derivatives and other constrained optimization problems in finance.

8 Exercises Exercise 4.1 Show that substituting ∇ij Ik =

Xj , i = k, 0,

i = k,

into Eq. 4.47 gives ∇ij σk ≡

∂σk = ∇i σk Xj = σk (δki − σi )Xj . ∂wij

Exercise 4.2 Show that substituting the derivative of the softmax function w.r.t. wij into Eq. 4.52 gives for the special case when the output is Yk = 1, k = i, and Yk = 0, ∀k = i:

154

4 Feedforward Neural Networks

∇ij L(W, b) := [∇W L(W, b)]ij =

(σi − 1)Xj , Yi = 1, Yk = 0, ∀k = i.

0,

Exercise 4.3 Consider feedforward neural networks constructed using the following two types of activation functions: – Identity I d(x) := x – Step function (a.k.a. Heaviside function) 2 H (x) :=

1 if x ≥ 0, 0 otherwise.

1. Consider a feedforward neural network with one input x ∈ R, a single hidden layer with K units having step function activations, H (x), and a single output with identity (a.k.a. linear) activation, I d(x). The output can be written as , K (2) (1) (1) (2) ˆ wk H (bk + wk x) . f (x) = I d b + k=1

Construct neural networks using these activation functions. a. Consider the step function 2 u(x; a) := yH (x − a) =

y, if x ≥ a, 0, otherwise.

Construct a neural network with one input x and one hidden layer, whose response is u(x; a). Draw the structure of the neural network, specify the activation function for each unit (either I d or H ), and specify the values for all weights (in terms of a and y). b. Now consider the indicator function 2 1, if x ∈ [a, b), 1[a,b) (x) = 0, otherwise. Construct a neural network with one input x and one hidden layer, whose response is y1[a,b) (x), for given real values y, a and b. Draw the structure of the neural network, specify the activation function for each unit (either I d or H ), and specify the values for all weights (in terms of a, b and y).

8 Exercises

155

Exercise 4.4 A neural network with a single hidden layer can provide an arbitrarily close approximation to any 1-dimensional bounded smooth function. This question will guide you through the proof. Let f (x) be any function whose domain is [C, D), for real values C < D. Suppose that the function is Lipschitz continuous, that is, ∀x, x ∈ [C, D), |f (x ) − f (x)| ≤ L|x − x|, for some constant L ≥ 0. Use the building blocks constructed in the previous part to construct a neural network with one hidden layer that approximates this function within > 0, that is, ∀x ∈ [C, D), |f (x) − fˆ(x)| ≤ , where fˆ(x) is the output of your neural network given input x. Your network should use only the identity or the Heaviside activation functions. You need to specify the number K of hidden units, the activation function for each unit, and a formula for calculating each weight w0 , (k) (k) wk , w0 , and w1 , for each k ∈ {1, . . . , K}. These weights may be specified in terms of C, D, L, and , as well as the values of f (x) evaluated at a finite number of x values of your choosing (you need to explicitly specify which x values you use). You do not need to explicitly write the fˆ(x) function. Why does your network attain the given accuracy ? Exercise 4.5 Consider a shallow neural network regression model with n tanh activated units in (2) the hidden layer and d outputs. The hidden-outer weight matrix Wij = n1 and the input-hidden weight matrix W (1) = 1. The biases are zero. If the features, X1 , . . . , Xp are i.i.d. Gaussian random variables with mean μ = 0, variance σ 2 , show that a. Yˆ ∈ [−1, 1]. b. Yˆ is independent of the number of hidden units, n ≥ 1. c. The expectation, E[Yˆ ] = 0, and the variance V[Yˆ ] ≤ 1. Exercise 4.6 Determine the VC dimension of the sum of indicator functions where = [0, 1] Fk (x) = {f : → {0, 1}, f (x) =

k

1x∈[t2i ,t2i+1 ) , 0 ≤ t0 < · · · < t2k+1 ≤ 1, k ≥ 1}.

i=0

Exercise 4.7 Show that a feedforward binary classifier with two Heaviside activated units shatters the data {0.25, 0.5, 0.75}. Exercise 4.8 Compute the weight and bias updates of W (2) and b(2) given a shallow binary classifier (with one hidden layer) with unit weights, zero biases, and ReLU activation of two hidden units for the labeled observation (x = 1, y = 1).

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8.1 Programming Related Questions* Exercise 4.9 Consider the following dataset (taken from Anscombe’s quartet): (x1 , y1 ) = (10.0, 9.14), (x2 , y2 ) = (8.0, 8.14), (x3 , y3 ) = (13.0, 8.74), (x4 , y4 ) = (9.0, 8.77), (x5 , y5 ) = (11.0, 9.26), (x6 , y6 ) = (14.0, 8.10), (x7 , y7 ) = (6.0, 6.13), (x8 , y8 ) = (4.0, 3.10), (x9 , y9 ) = (12.0, 9.13), (x10 , y10 ) = (7.0, 7.26), (x11 , y11 ) = (5.0, 4.74).

a. Use a neural network library of your choice to show that a feedforward network with one hidden layer consisting of one unit and a feedforward network with no hidden layers, each using only linear activation functions, do not outperform linear regression based on ordinary least squares (OLS). b. Also demonstrate that a neural network with a hidden layer of three neurons using the tanh activation function and an output layer using the linear activation function captures the non-linearity and outperforms the linear regression. Exercise 4.10 Review the Python notebook deep_classifiers.ipynb. This notebook uses Keras to build three simple feedforward networks applied to the half-moon problem: a logistic regression (with no hidden layer); a feedforward network with one hidden layer; and a feedforward architecture with two hidden layers. The halfmoons problem is not linearly separable in the original coordinates. However you will observe—after plotting the fitted weights and biases—that a network with many hidden neurons gives a linearly separable representation of the classification problem in the coordinates of the output from the final hidden layer. Complete the following questions in your own words. a. Did we need more than one hidden layer to perfectly classify the half-moons dataset? If not, why might multiple hidden layers be useful for other datasets? b. Why not use a very large number of neurons since it is clear that the classification accuracy improves with more degrees of freedom? c. Repeat the plotting of the hyperplane, in Part 1b of the notebook, only without the ReLU function (i.e., activation=“linear”). Describe qualitatively how the decision surface changes with increasing neurons. Why is a (non-linear) activation function needed? The use of figures to support your answer is expected. Exercise 4.11 Using the EarlyStopping callback in Keras, modify the notebook Deep_Classifiers.ipynb to terminate training under the following stopping criterion |L(k+1) − L(k) | ≤ δ with δ = 0.1.

Appendix

157

Exercise 4.12*** Consider a feedforward neural network with three inputs, two units in the first hidden layer, two units in the second hidden layer, and three units in the output layer. The activation function for hidden layer 1 is ReLU, for hidden layer 2 is sigmoid, and for the output layer is softmax. The initial weights are given by the matrices

W (1)

⎞ ⎛ ) ) * * 0.5 0.6 0.1 0.3 0.7 0.4 0.3 = , W (2) = , W (3) = ⎝0.6 0.7⎠ , 0.9 0.4 0.4 0.7 0.2 0.3 0.2

and all the biases are unit vectors. ' & ' & Assuming that the input 0.1 0.7 0.3 corresponds to the output 1 0 0 , manually compute the updated weights and biases after a single epoch (forward + backward pass), clearly stating all derivatives that you have used. You should use a learning rate of 1. As a practical exercise, you should modify the implementation of a stochastic gradient descent routine in the back-propagation Python notebook. Note that the notebook example corresponds to the example in Sect. 5, which uses sigmoid activated hidden layers only. Compare the weights and biases obtained by TensorFlow (or your ANN library of choice) with those obtained by your procedure after 200 epochs.

Appendix Answers to Multiple Choice Questions Question 1 Answer: 1, 2, 3, 4. All answers are found in the text. Question 2 Answer: 1,2. A feedforward architecture is always convex w.r.t. each input variable if every activation function is convex and the weights are constrained to be either all positive or all negative. Simply using convex activation functions is not sufficient, since the composition of a convex function and the affine transformation of a convex function do not preserve the convexity. For example, if σ (x) = x 2 , w = −1, and b = 1, then σ (wσ (x) + b) = (−x 2 + 1)2 is not convex in x. A feedforward architecture with positive weights is a monotonically increasing function of the input for any choice of monotonically increasing activation function. The weights of a feedforward architecture need not be constrained for the output of a feedforward network to be bounded. For example, activating the output with a softmax function will bound the output. Only if the output is not activated, should the weights and bias in the final layer be bounded to ensure bounded output.

158

4 Feedforward Neural Networks

The bias terms in a network shift the output but also effect the derivatives of the output w.r.t. to the input when the layer is activated. Question 3 Answer: 1,2,3,4. The training of a neural network involves minimizing a loss function w.r.t. the weights and biases over the training data. L1 regularization is used during model selection to penalize models with too many parameters. The loss function is augmented with a Lagrange penalty for the number of weights. In deep learning, regularization can be applied to each layer of the network. Therefore each layer has an associated regularization parameter. Back-propagation uses the chain rule to update the weights of the network but is not guaranteed to convergence to a unique minimum. This is because the loss function is not convex w.r.t. the weights. Stochastic gradient descent is a type of optimization method which is implemented with back-propagation. There are variants of SGD, however, such as adding Nestov’s momentum term, ADAM , or RMSProp.

Back-Propagation Let us consider a feedforward architecture with an input layer, L − 1 hidden layers, and one output layer, with K units in the output layer for classification of K categories. As a result, we have L sets of weights and biases (W () , b() ) for = 1, . . . , L, corresponding to the layer inputs Z (−1) and outputs Z () for = 1, . . . , L. Recall that each layer is an activation of a semi-affine transformation, I () (Z (−1) ) := W (L) Z (−1) + b(L) . The corresponding activation functions are denoted as σ () . The activation function for the output layer is a softmax function, σs (x). Here we use the cross-entropy as the loss function, which is defined as L := −

K

Yk log Yˆk .

k=1

The relationship between the layers, for ∈ {1, . . . , L} are Yˆ (X) = Z (L) = σs (I (L) ) ∈ [0, 1]K ,

Z () = σ () I () , = 1, . . . , L − 1, Z (0) = X. The update rules for the weights and biases are W () = −γ ∇W () L, b() = −γ ∇b() L.

Appendix

159

We now begin the back-propagation, tracking the intermediate calculations carefully using Einstein summation notation. For the gradient of L w.r.t. W (L) we have ∂L (L)

∂wij

=

K (L) ∂L ∂Zk (L)

k=1

=

∂Zk

(L)

∂wij

K K (L) (L) ∂L ∂Zk ∂Im (L)

k=1

∂Zk

(L)

m=1

∂Im

(L)

∂wij

But ∂L (L) ∂Zk

=−

(L)

∂Zk

(L) ∂Im

= =

=

(L)

Zk ∂

(L)

∂Im

[σ (I (L) )]k (L)

exp[Ik ]

∂

(L) K (L) ∂Im n=1 exp[In ] ⎧ (L) (L) exp[I ] ⎪ ⎨− K k (L) Kexp[Im ](L)

⎪ ⎩

=

Yk

n=1 exp[In (L) exp[Ik ] K (L) n=1 exp[In ]

]

−

n=1 exp[In ] (L) exp[Ik ] K (L) n=1 exp[In ]

−σk σm

if k = m

σk (1 − σm )

otherwise

= σk (δkm − σm )

if k = m (L) exp[Im ] K (L) n=1 exp[In ]

where δkm is the Kronecker’s Delta

(L)

∂Im

(L) ∂wij

⇒

∂L (L)

∂wij

(L−1)

= δmi Zj

=−

K K Yk (L)

k=1

Zk

= −Zj(L−1)

Yk (δki − Zi(L) )

k=1

=

(L−1) (L) Zj (Zi

(L−1)

(L) (L) Zm (δkm − Zm )δmi Zj

m=1 K

otherwise

− Yi ),

160

4 Feedforward Neural Networks

K

where we have used the fact that b(L) , we have ∂L (L)

∂bi

=

= 1 in the last equality. Similarly for

k=1 Yk

K K (L) (L) ∂L ∂Zk ∂Im (L)

k=1

∂Zk

(L)

m=1

∂Im

(L)

∂bi

= Zi(L) − Yi It follows that ∇b(L) L = Z (L) − Y ∇W (L) L = ∇b(L) L ⊗ Z (L−1) , where ⊗ denotes the outer product. Now for the gradient of L w.r.t. W (L−1) we have ∂L (L−1) ∂wij

=

K ∂L k=1

(L)

∂Zk

(L−1) ∂Zk(L) ∂wij (L−1)

=

K K (L) n ∂L ∂Zk (L)

k=1

∂Zk

(L)

m=1

∂Im

(L)

∂Im

(L−1) n

∂Zn

p=1

∂Ip

(L−1)

n=1

∂Zn

∂Ip

(L−1)

∂wij

If we assume that σ () (x) = sigmoid(x), ∈ {1, . . . , L − 1}, then (L)

∂Im

(L−1) ∂Zn

(L) = wmn

(L−1)

∂Zn

(L−1)

∂Ip

= =

∂

)

(L−1)

*

1 (L−1)

1 + exp(−In

∂Ip

) (L−1)

1

exp(−In

(L−1) 1 + exp(−In )

)

δnp (L−1) 1 + exp(−In )

= Zn(L−1) (1 − Zn(L−1) ) δnp = σn(L−1) (1 − σn(L−1) )δnp ∂Ip(L−1) (L−1)

∂wij ⇒

∂L (L) ∂wij

= δpi Zj(L−2)

=−

K K Yk (L) k=1 Zk m=1

(L)

(L) Zk (δkm − Zm )

(L−1)

(L−1)

(L−1)

Appendix

161 (L−1) n

(L) wmn

(L−1) n

n=1

=

(L−2)

Zn(L−1) (1 − Zn(L−1) ) δnp δpi Zj

p=1

(L−1) n (L−2) (L) (L) (L−1) − Yk (δkm − Zm ) wmn Zn (1 − Zn(L−1) ) δni Zj k=1 m=1 n=1

K

=−

K

K

Yk

k=1

K

(L) (L−2) (L) (δkm − Zm )wmi Zi (1 − Zi(L−1) )Zj(L−2)

m=1

(L−2)

= −Zj

(L−1)

Zi

(L−1)

(1 − Zi

K

)

(L)

wmi

m=1 (L−2)

= Zj

(L−1)

Zi

(L−1)

(1 − Zi

K

)

K (L) (δkm Yk − Zm Yk ) k=1

(L)

(L) wmi (Zm − Ym )

m=1

=

(L−2) (L−1) (L−1) Zi (1 − Zi )(Z (L) Zj

(L)

− Y )T w,i

Similarly we have ∂L

(L−1)

(L−1) ∂bi

= Zi

(L−1)

(1 − Zi

(L)

)(Z (L) − Y )T w,i .

It follows that we can define the following recursion relation for the loss gradient: T

∇b(L−1) L = Z (L−1) ◦ (1 − Z (L−1) ) ◦ (W (L) ∇b(L) L) ∇W (L−1) L = ∇b(L−1) L ⊗ Z (L−2) T

= Z (L−1) ◦ (1 − Z (L−1) ) ◦ (W (L) ∇W (L) L), where ◦ denotes the Hadamard product (element-wise multiplication). This recursion relation can be generalized for all layers and choice of activation functions. To see this, let the back-propagation error δ () := ∇b() L, and since

∂σ () ∂I ()

= ij

∂σi() ()

∂Ij

= σi() (1 − σi() )δij or equivalently in matrix–vector form

162

4 Feedforward Neural Networks

∇I () σ () = diag(σ () ◦ (1 − σ () )), we can write, in general, for any choice of activation function for the hidden layer, δ () = ∇I () σ () (W (+1) )T δ (+1) , and ∇W () L = δ () ⊗ Z (−1) .

Proof of Theorem 4.2 Using the same deep structure shown in Fig. 4.9, Liang and Srikant (2016) find the binary expansion sequence {x0 , . . . , xn }. In this step, they used n binary steps units in total. Then they rewrite gm+1 ( ni=0 2xni ), gm+1

, n xi i=0

2i

=

n j =0

=

n j =0

, n - xi 1 xj · j gm 2 2i

i=0

, n - xi 1 max 2(xj − 1) + j gm ,0 . 2 2i

(4.57)

i=0

Clearly Eq. 4.57 defines iterations between the outputs of neighboring layers. Defin

p xj n ˆ ing the output of the multilayer neural network as f (x) = i=0 ai gi j =0 2j . For this multilayer network, the approximation error is ⎛ ⎞ p n p xj i ⎠− |f (x) − fˆ(x)| = ai gi ⎝ a x i j 2 i=0 j =0 i=0 ⎛ ⎤ ⎡ ⎞ p n xj i ⎣|ai | · gi ⎝ ⎠−x ⎦≤ p . ≤ j 2 2n−1 i=0

j =0

3 4 This indicates, to achieve ε-approximation error, one should choose n = log pε + 1. Besides, since O(n + p) layers with O(n) binary step units& and O(pn)' ReLU units are used in total, this multilayer neural network thus has O p + log pε layers, & ' & ' O log pε binary step units, and O p log pε ReLU units.

Appendix

163

Table 4.2 Definitions of the functions f (x) and g(x)

f (x) := max(x − 14 , 0), cIf = {[0, 14 ], ( 14 , 1]},

g(x) := max(x − 12 , 0) cIg = {[0, 12 ], ( 12 , 1]}.

Proof of Lemmas from Telgarsky (2016) Proof (Proof of 4.1) Let cIf denote the partition of R corresponding to f , and cIg denote the partition of R corresponding to g. First consider f + g, and moreover any intervals Uf ∈ cIf and Ug ∈ cIg . Necessarily, f + g has a single slope along Uf ∩ Ug . Consequently, f + g is |cI|-sawtooth, where cI is the set of all intersections of intervals from cIf and cIg , meaning cI := {Uf ∩ Ug : Uf ∈ cIf , Ug ∈ cIg }. By sorting the left endpoints of elements of cIf and cIg , it follows that |cI| ≤ k + l (the other intersections are empty). For example, consider the example in Fig. 4.11 with partitions given in Table 4.2. The set of all intersections of intervals from cIf and cIg contains 3 elements: 1 1 1 1 1 1 cI = {[0, ] ∩ [0, ], ( , 1] ∩ [0, ], ( , 1] ∩ ( , 1]} 4 2 4 2 4 2

(4.58)

Now consider f ◦ g, and in particular consider the image f (g(Ug )) for some interval Ug ∈ cIg . g is affine with a single slope along Ug ; therefore, f is being considered along a single unbroken interval g(Ug ). However, nothing prevents g(Ug ) from hitting all the elements of cIf ; since Ug was arbitrary, it holds that f ◦ g is (|cIf | · |cIg |)-sawtooth. 1 Proof Recall the notation f˜(x) := [f (x) ≥ 1/2], whereby E(f ) := n i [yi = f˜(xi )]. Since f is piecewise monotonic with a corresponding partition R having at most t pieces, then f has at most 2t − 1 crossings of 1/2: at most one within each interval of the partition, and at most 1 at the right endpoint of all but the last interval. Consequently, f˜ is piecewise constant, where the corresponding partition of R is into at most 2t intervals. This means n points with alternating labels must land in 2t buckets, thus the total number of points landing in buckets with at least three points is at least n − 4t.

Python Notebooks The notebooks provided in the accompanying source code repository are designed to gain insight in toy classification datasets. They provide examples of deep feedforward classification, back-propagation, and Bayesian network classifiers. Further details of the notebooks are included in the README.md file.

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Chapter 5

Interpretability

This chapter presents a method for interpreting neural networks which imposes minimal restrictions on the neural network design. The chapter demonstrates techniques for interpreting a feedforward network, including how to rank the importance of the features. An example demonstrating how to apply interpretability analysis to deep learning models for factor modeling is also presented.

1 Introduction Once the neural network has been trained, a number of important issues surface around how to interpret the model parameters. This aspect is a prominent issue for practitioners in deciding whether to use neural networks in favor of other machine learning and statistical methods for estimating factor realizations, sometimes even if the latter’s predictive accuracy is inferior. In this section, we shall introduce a method for interpreting multilayer perceptrons which imposes minimal restrictions on the neural network design. Chapter Objectives By the end of this chapter, the reader should expect to accomplish the following: – Apply techniques for interpreting a feedforward network, including how to rank the importance of the features. – Learn how to apply interpretability analysis to deep learning models for factor modeling.

© Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_5

167

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2 Background on Interpretability There are numerous techniques for interpreting machine learning methods which treat the model as a black-box. A good example are Partial Dependence Plots (PDPs) as described by Greenwell et al. (2018). Other approaches also exist in the literature. Garson (1991) partitions hidden-output connection weights into components associated with each input neuron using absolute values of connection weights. Olden and Jackson (2002) determine the relative importance, [R]ij , of the ith output to the j th predictor variable of the model as a function of the weights, according to a simple linear expression. We seek to understand the limitations on the choice of activation functions and understand the effect of increasing layers and numbers of neurons on probabilistic interpretability. For example, under standard Gaussian i.i.d. data, how robust are the model’s estimate of the importance of each input variable with variable number of neurons?

2.1 Sensitivities We shall therefore turn to a “white-box” technique for determining the importance of the input variables. This approach generalizes Dimopoulos et al. (1995) to a deep neural network with interaction terms. Moreover, the method is directly consistent with how coefficients are interpreted in linear regression—they are model sensitivities. Model sensitivities are the change of the fitted model output w.r.t. input. As a control, we shall use this property to empirically evaluate how reliably neural networks, even deep networks, learn data from a linear model. Such an approach is appealing to practitioners who are evaluating the comparative performance of linear regression with neural networks and need the assurance that a neural network model is at least able to reproduce and match the coefficients on a linear dataset. We also offset the common misconception that the activation functions must be deactivated for a neural network model to produce a linear output. Under linear data, any non-linear statistical model should be able to reproduce a statistical linear model under some choice of parameter values. Irrespective of whether data is linear or non-linear in practice - the best control experiment for comparing a neural network estimator with an OLS estimator is to simulate data under a linear regression model. In this scenario, the correct model coefficients are known and the error in the coefficient estimator can be studied. To evaluate fitted model sensitivities analytically, we require that the function Yˆ = f (X) is continuous and differentiable everywhere. Furthermore, for stability of

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the interpretation, we shall require that f (x) is Lipschitz continuous.1 That is, there is a positive real constant K s.t. ∀x1 , x2 ∈ Rp , |F (x1 ) − F (x2 )| ≤ K|x1 − x2 |. Such a constraint is necessary for the first derivative to be bounded and hence amenable to the derivatives, w.r.t. to the inputs, providing interpretability. Fortunately, provided that the weights and biases are finite, each semi-affine function is Lipschitz continuous everywhere. For example, the function tanh(x) is continuously differentiable and its derivative 1−tanh2 (x) is globally bounded. With finite weights, the composition of tanh(x) with an affine function is also Lipschitz. Clearly ReLU(x) := max(·, 0) is not continuously differentiable and one cannot use the approach described here. Note that for the following examples, we are indifferent to the choice of homoscedastic or heteroscedastic error, since the model sensitivities are independent of the error.

3 Explanatory Power of Neural Networks In a linear regression model Yˆ = Fβ (X) := β0 + β1 X1 + · · · + βK XK ,

(5.1)

the model sensitivities are ∂Xi Yˆ = βi .

(5.2)

In a feedforward neural network, we can use the chain rule to obtain the model sensitivities ∂Xi Yˆ = ∂Xi FW,b (X) = ∂Xi σW (L) ,b(L) ◦ · · · ◦ σW (1) ,b(1) (X). (L)

(1)

(5.3)

(1) For example, with one hidden layer, σ (x) := tanh(x) and σW (1) ,b(1) (X) :=

σ (I (1) ) := σ (W (1) X + b(1) ): ∂Xj Yˆ =

(2)

(1)

(1)

w,i (1 − σ 2 (Ii ))wij

where ∂x σ (x) = (1 − σ 2 (x)).

(5.4)

i

In matrix form, with general σ , the Jacobian2 of σ w.r.t X is J = D(I (1) )W (1) of σ , ∂X Yˆ = W (2) J (I (1) ) = W (2) D(J (1) )W (1) ,

1 If

(5.5)

Lipschitz continuity is not imposed, then a small change in one of the input values could result in an undesirable large variation in the derivative. 2 When σ is an identity function, the Jacobian J (I (1) ) = W (1) .

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where Dii (I ) = σ (Ii ), Dij = 0, i = j is a diagonal matrix. Bounds on the sensitivities are given by the product of the weight matrices min(W (2) W (1) , 0) ≤ ∂X Yˆ ≤ max(W (2) W (1) , 0).

(5.6)

3.1 Multiple Hidden Layers The model sensitivities can be readily generalized to an L layer deep network by evaluating the Jacobian matrix: ∂X Yˆ = W (L) J (I (L−1) ) = W (L) D(I (L−1) )W (L−1) . . . D(I (1) )W (1) .

(5.7)

3.2 Example: Step Test To illustrate our interpretability approach, we shall consider a simple example. The model is trained to the following data generation process where the coefficients of the features are stepped and the error, here, is i.i.d. uniform: Yˆ =

10

iXi , Xi ∼ U(0, 1).

(5.8)

i=1

Figure 5.1 shows the ranked importance of the input variables in a neural network with one hidden layer. Our interpretability method is compared with well-known black-box interpretability methods such as Garson’s algorithm (Garson 1991) and Olden’s algorithm (Olden and Jackson 2002). Our approach is the only technique to interpret the fitted neural network which is consistent with how a linear regression model would interpret the input variables.

4 Interaction Effects The previous example is too simplistic to illustrate another important property of our interpretability method, namely the ability to capture pairwise interaction terms. The pairwise interaction effects are readily available by evaluating the elements of the Hessian matrix. For example, with one hidden layer, the Hessian takes the form: (1) (1) ∂X2 i Xj Yˆ = W (2) diag(Wi )D (I (1) )Wj ,

(5.9)

where it is assumed that the activation function is at least twice differentiable everywhere, e.g. tanh(x).

Variable

4 Interaction Effects

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0.00 0.05 0.10 0.15 0.20 Importance (Garson’s algorithm)

0 10 20 30 40 Importance (Olden’s algorithm)

Fig. 5.1 Step test: This figure shows the ranked importance of the input variables in a neural network with one hidden layer. (left) Our sensitivity based approach for input interpretability. (Center) Garson’s algorithm and (Right) Olden’s algorithm. Our approach is the only technique to interpret the fitted neural network which is consistent with how a linear regression model would interpret the input variables

4.1 Example: Friedman Data To illustrate our input variable and interaction effect ranking approach, we will use a classical nonlinear benchmark regression problem. The input space consists of ten i.i.d. uniform U (0, 1) random variables; however, only five out of these ten actually appear in the true model. The response is related to the inputs according to the formula

Variable

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0.0 2.5 5.0 7.5 10.0 Importance (Sensitivity)

0.0 0.1 0.2 0.3 Importance (Garson's algorithm)

−10 0 10 20 30 Importance (Olden's algorithm)

Fig. 5.2 Friedman test: Ranked model sensitivities of the fitted neural network to the input. (left) Our sensitivity based approach for input interpretability. (Center) Garson’s algorithm and (Right) Olden’s algorithm

Y = 10 sin (π X1 X2 ) + 20 (X3 − 0.5)2 + 10X4 + 5X5 + , ' & using white noise error, ∼ N 0, σ 2 . We fit a NN with one hidden layer containing eight units and a weight decay of 0.01 (these parameters were chosen using 5-fold cross-validation) to 500 observations simulated from the above model with σ = 1. The cross-validated R 2 value was 0.94. Figures 5.2 and 5.3, respectively, compare the ranked model sensitivities and ranked interaction terms of the fitted neural network with Garson’s and Olden’s algorithm.

5 Bounds on the Variance of the Jacobian General results on the bound of the variance of the Jacobian for any activation function are difficult to derive. However, we derive the following result for a ReLU activated single-layer feedforward network. In matrix form, with σ (x) = max(x, 0), the Jacobian, J , can be written as a linear combination of Heaviside functions: J := J (X) = ∂X Yˆ (X) = W (2) J (I (1) ) = W (2) H (W (1) X + b(1) )W (1) ,

(5.10)

5 Bounds on the Variance of the Jacobian

173

x_12 x_24

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x_25 x_14 x_45 x_15 x_310 x_28 x_34 x_23 0

1

2 Importance (Sensitivity)

3

4

Fig. 5.3 Friedman test: Ranked pairwise interaction terms in the fitted neural network to the input. (Left) Our sensitivity based approach for ranking interaction terms. (Center) Garson’s algorithm and (Right) Olden’s algorithm

(1)

where Hii (Z) = H (Ii ) = 1{I (1) >0} , Hij = 0, j ≥ i. We assume that the mean i of the Jacobian is independent of the number of hidden units, μij := E[Jij ]. Then we can state the following bound on the Jacobian of the network for the special case when the input is one-dimensional. Theorem (Dixon and Polson 2019) If X ∈ Rp is i.i.d. and there are n hidden units, with ReLU activation i, then the variance of a single-layer feedforward network with K outputs is bounded by μij V[Jij ] = μij

n−1 < μij , ∀i ∈ {1, . . . , K} and ∀j ∈ {1, . . . , p}. n

(5.11)

See Appendix “Proof of Variance Bound on Jacobian” for the proof. Remark 5.1 The theorem establishes a negative result for a ReLU activated shallow network—increasing the number of hidden units, increases the bound on the variance of the Jacobian, and hence reduces interpretability of the sensitivities. Note that if we do not assume that the mean of the Jacobian is fixed under varying n, then we have the more general bound: V[Jij ] ≤ μij ,

(5.12)

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and hence the effect of network architecture on the bound of the variance of the Jacobian is not clear. Note that the theorem holds without (i) distributional assumptions on X other than i.i.d. data and (ii) specifying the number of data points. Remark 5.2 This result also suggests that the inputs should be rescaled so that each μij , the expected value of the Jacobian, is a small positive value, although it may not be possible to find such a scaling for all (i, j ) pairs.

5.1 Chernoff Bounds We can derive probabilistic bounds on the Jacobians for any choice of activation function. Let δ > 0 and a1 , . . . , an−1 be reals in (0, 1]. Let X1 , . . . , Xn−1 be independent Bernoulli trials with E[Xk ] = pk so that E[J ] =

n−1

ak pk = μ.

(5.13)

k=1

The Chernoff-type bound exists on deviations of J above the mean

eδ Pr(J > (1 + δ)μ) = (1 + δ)1+δ

μ (5.14)

.

A similar bound exists for deviations of J below the mean. For γ ∈ (0, 1]:

eγ Pr(J − μ < −γ μ) < (1 + γ )1+γ

μ .

(5.15)

These bounds are generally weak and are suited to large deviations, i.e. the tail regions. The bounds are shown in the Fig. 5.4 for different values of μ. Here, μ is increasing towards the upper right-hand corner of the plot.

5.2 Simulated Example In this section, we demonstrate the estimation properties of neural network sensitivities applied to data simulated from a linear model. We show that the sensitivities in a neural network are consistent with the linear model, even if the neural network model is non-linear. We also show that the confidence intervals, estimated by sampling, converge with increasing hidden units. We generate 400 simulated training samples from the following linear model with i.i.d. Gaussian error:

5 Bounds on the Variance of the Jacobian

0.85 0.70

0.75

0.80

Pr[J> μ(1+δ)]

0.90

0.95

1.00

Fig. 5.4 The Chernoff-type bounds for deviations of J above the mean, μ. Various μ are shown in the plot, with μ increasing towards the upper right-hand corner of the plot

175

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0.2

0.4

0.6

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1.0

δ

Table 5.1 This table compares the functional form of the variable sensitivities and values with an OLS estimator. NN0 is a zero hidden layer feedforward network and NN1 is a one hidden layer feedforward network with 10 hidden neurons and tanh activation functions Model OLS NN0 NN1

Intercept Sensitivity of X1 Sensitivity of X2 0.011 βˆ1 1.015 βˆ2 1.018 βˆ0 (1) (1) bˆ (1) 0.020 Wˆ 1 1.018 Wˆ 2 1.021 Wˆ (2) σ (bˆ (1) ) + bˆ (2) 0.021 E[Wˆ (2) D(I (1) )Wˆ 1(1) ] 1.014 E[Wˆ (2) D(I (1) )Wˆ 2(1) ] 1.022

Y = β1 X1 + β2 X2 + , X1 , X2 , ∼ N(0, 1),

β1 = 1, β2 = 1.

(5.16)

Table 5.1 compares an OLS estimator with a zero hidden layer feedforward network (NN0 ) and a one hidden layer feedforward network with 10 hidden neurons and tanh activation functions (NN1 ). The functional form of the first two regression models is equivalent, although the OLS estimator has been computed using a matrix solver, whereas the zero layer hidden network parameters have been fitted with stochastic gradient descent. The fitted parameters values will vary slightly with each optimization as the stochastic gradient descent is randomized. However, the sensitivity terms are given in closed form and easily mapped to the linear model. In an industrial setting, such a one-to-one mapping is useful for migrating to a deep factor model where, for model validation purposes, compatibility with linear models should be recovered in a limiting case. Clearly, if the data is not generated from a linear model, then the parameter values would vary across models.

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Fig. 5.5 This figure shows the empirical distribution of the sensitivities βˆ1 and βˆ2 . The sharpness of the distribution is observed to converge with the number of hidden units. (a) Density of βˆ1 . (b) Density of βˆ2 Table 5.2 This table shows the moments and 99% confidence interval of the empirical distribution of the sensitivity βˆ1 . The sharpness of the distribution is observed to converge monotonically with the number of hidden units Hidden Units 2 10 50 100 200

Mean 0.980875 0.9866159 0.99183553 1.0071343 1.0152218

Median 1.0232913 1.0083131 1.0029879 1.0175397 1.0249312

Std.dev 0.10898393 0.056483902 0.03123002 0.028034585 0.026156902

1% C.I. 0.58121675 0.76814914 0.8698967 0.89689034 0.9119074

99% C.I. 1.0729908 1.0322522 1.0182846 1.0296803 1.0363332

Table 5.3 This table shows the moments and the 99% confidence interval of the empirical distribution of the sensitivity βˆ2 . The sharpness of the distribution is observed to converge monotonically with the number of hidden units Hidden Units 2 10 50 100 200

Mean 0.98129386 0.9876832 0.9903236 0.9842479 0.9976638

Median 1.0233982 1.0091512 1.0020974 0.9946766 1.0074166

Std.dev 0.10931312 0.057096474 0.031827927 0.028286876 0.026751818

1% C.I. 0.5787732 0.76264584 0.86471796 0.87199813 0.8920307

99% C.I. 1.073728 1.0339714 1.0152498 1.0065105 1.0189484

Figure 5.5 and Tables 5.2 and 5.3 show the empirical distribution of the fitted sensitivities using the single hidden layer model with increasing hidden units. The sharpness of the distributions is observed to converge monotonically with the number of hidden units. The confidence intervals are estimated under a nonparametric distribution. In general, provided the weights and biases of the network are finite, the variances of the sensitivities are bounded for any input and choice of activation function.

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We do not recommend using ReLU activation because it does not permit identification of the interaction terms and has provably non-convergent sensitivity variances as a function of the number of hidden units (see Appendix “Proof of Variance Bound on Jacobian”).

6 Factor Modeling Rosenberg and Marathe (1976) introduced a cross-sectional fundamental factor model to capture the effects of macroeconomic events on individual securities. The choice of factors are microeconomic characteristics—essentially common factors, such as industry membership, financial structure, or growth orientation (Nielsen and Bender 2010). The BARRA fundamental factor model expresses the linear relationship between K fundamental factors and N asset returns: rt = Bt ft + t , t = 1, . . . , T ,

(5.17)

where Bt = [1 | β 1&(t) |'· · · | β K (t)] is the N ×K +1 matrix of known factor loadings (betas): βi,k (t) := β k i (t) is the exposure of asset i to factor k at time t. The factors are asset specific attributes such as market capitalization, industry classification, style classification. ft = [αt , f1,t , . . . , fK,t ] is the K + 1 vector of unobserved factor realizations at time t, including αt . rt is the N -vector of asset returns at time t. The errors are assumed independent 2 ] = σ 2. of the factor realizations ρ(fi,t , j,t ) = 0, ∀i, j, t with Gaussian error, E[j,t

6.1 Non-linear Factor Models We can extend the linear model to a non-linear cross-sectional fundamental factor model of the form rt = Ft (Bt ) + t ,

(5.18)

where rt are asset returns, Ft : RK → R is a differentiable non-linear function that maps the ith row of B to the ith asset return at time t. The map is assumed to incorporate a bias term so that Ft (0) = αt . In the special case when Ft (Bt ) is linear, the map is Ft (Bt ) = Bt ft . A key feature is that we do not assume that t is described by a parametric distribution, such as a Gaussian distribution. In our example, we shall treat t as i.i.d., however, we can extend the methodology to non-i.i.d. idea as in Dixon and Polson (2019). In our setup, the model shall just be used to predict the next period returns only and stationarity of the factor realizations is not required.

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We approximate a non-linear map, Ft (Bt ), with a feedforward neural network cross-sectional factor model: rt = FWt ,bt (Bt ) + t ,

(5.19)

where FWt ,bt is a deep neural network with L layers.

6.2 Fundamental Factor Modeling This section presents an application of deep learning to a toy fundamental factor model. Factor models in practice include significantly more fundamental factors that used here. But the purpose, here, is to illustrate the application of interpretable deep learning to financial data. We define the universe as the top 250 stocks from the S&P 500 index, ranked by market cap. Factors are given by Bloomberg and reported monthly over a hundred month period beginning in February 2008. We remove stocks with too many missing factor values, leaving 218 stocks. The historical factors are inputs to the model and are standardized to enable model interpretability. These factors are (i) current enterprise value; (ii) Priceto-Book ratio; (iii) current enterprise value to trailing 12 month EBITDA; (iv) Price-to-Sales ratio; (v) Price-to-Earnings ratio; and (vi) log market cap. The responses are the monthly asset returns for each stock in our universe based on the daily adjusted closing prices of the stocks. We use Tensorflow (Abadi et al. 2016) to implement a two hidden layer feedforward network and develop a custom implementation for the least squares error and variable sensitivities and is available in the deep factor models notebook. The OLS regression is implemented by the Python StatsModels module. All deep learning results are shown using L1 regularization and tanh activation functions. The number of hidden units and regularization parameters are found by three-fold cross-validation and reported alongside the results. Figure 5.6 compares the performance of an OLS estimator with the feedforward neural network with 10 hidden units in the first hidden layer and 10 hidden units in the second layer and λ1 = 0.001. Figure 5.7 shows the in-sample MSE as a function of the number of hidden units in the hidden layer. The neural networks are trained here without L1 regularization to demonstrate the effect of solely increasing the number of hidden units in the first layer. Increasing the number of hidden units reduces the bias in the model. Figure 5.8 shows the effect of L1 regularization on the MSE errors for a network with 10 units in each of the two hidden layers. Increasing the level of L1 regularization increases the in-sample bias but reduces the out-of-sample bias, and hence the variance of the estimation error. Figure 5.9 compares the distribution of sensitivities to each factor over the entire 100 month period using the neural network (top) and OLS regression (bottom). The

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179

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Fig. 5.6 This figures compares the in-sample and out-of-sample performances of an OLS estimator (OLS) with a feedforward neural network (NN), as measured by the mean squared error (MSE). The neural network is observed to always exhibit slightly lower out-of-sample MSE, although the effect of deep networks on this problem is marginal because the dataset is too simplistic. (a) Insample error. (b) Out-of-sample error

0.0060 MSE In-Sample

Fig. 5.7 This figure shows the in-sample MSE as a function of the number of hidden units in the hidden layer. Increasing the number of hidden units reduces the bias in the model

0.0055 0.0050 0.0045 0.0040 Zero

1

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Fig. 5.8 These figures show the effect of L1 regularization on the MSE errors for a network with 10 neurons in each of the two hidden layers. (a) In-sample. (b) Out-of-sample

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Fig. 5.9 The distribution of sensitivities to each factor over the entire 100 month period using the neural network (top). The sensitivities are sorted in ascending order from left to right by their median values. The same sensitivities using OLS linear regression (bottom)

sensitivities are sorted in ascending order from left to right by their median values. We observe that the OLS regression is much more sensitive to the factors than the NN. We further note that the NN ranks the top sensitivities differently to OLS. Clearly, the above results are purely illustrative of the interpretability methodology and not intended to be representative of a real-world factor model. Such a choice of factors is observed to provide little benefit on the information ratios of a simple stock selection strategy. Larger Dataset For completeness, we provide evidence that our neural network factor model generates positive and higher information ratios than OLS when used to sort portfolios from a larger universe, using up to 50 factors (see Table 5.4 for a description of the factors). The dataset is not provided due to data licensing restrictions. We define the universe as 3290 stocks from the Russell 3000 index. Factors are given by Bloomberg and reported monthly over the period from November 2008 to November 2018. We train a two-hidden layer deep network with 50 hidden units using ReLU activation. Figure 5.10 compares the out-of-sample performance of neural networks and OLS regression by the MSE (left) and the information ratios of a portfolio selection strategy which selects the n stocks with the highest predicted monthly returns (right). The information ratios are evaluated for various size portfolios, using the Russell 3000 index as the benchmark. Also shown, for control, are randomly selected portfolios.

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Table 5.4 A short description of the factors used in the Russell 3000 deep learning factor model demonstrated at the end of this chapter B/P CF/P E/P S/EV

EB/EV FE/P

DIV

MC S TA TrA

EaV/TA

CFV/TA

SV/TA

Value factors Book to price Cash flow to price Earning to price Sales to enterprise value (EV). EV is given by EV=Market Cap + LT Debt + max(ST Debt-Cash,0), where LT (ST) stands for long (short) term EBIDTA to EV Forecasted E/P. Forecast earnings are calculated from Bloomberg earnings consensus estimates data For coverage reasons, Bloomberg uses the 1-year and 2-year forward earnings Dividend yield. The exposure to this factor is just the most recently announced annual net dividends divided by the market price Stocks with high dividend yields have high exposures to this factor Size factors Log (Market capitalization) Log (sales) Log (total assets) Trading activity factors Trading activity is a turnover based measure Bloomberg focuses on turnover which is trading volume normalized by shares outstanding This indirectly controls for the Size effect The exponential weighted average (EWMA) of the ratio of shares traded to shares outstanding In addition, to mitigate the impacts of those sharp short-lived spikes in trading volume, Bloomberg winsorizes the data first daily trading volume data is compared to the long-term EWMA volume(180 day half-life), then the data is capped at 3 standard deviations away from the EWMA average Earnings variability factors Earnings volatility to total assets Earnings volatility is measured over the last 5 years/median total assets over the last 5 years Cash flow volatility to total assets Cash flow volatility is measured over the last 5 years/median total assets over the last 5 years Sales volatility to total assets Sales volatility over the last 5 years/median total assets over the last 5 year (continued)

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Table 5.4 (continued) Volatility factors Rolling volatility which is the return volatility over the latest 252 trading days CB Rolling CAPM beta which is the regression coefficient from the rolling window regression of stock returns on local index returns Growth factors TAG Total asset growth is the 5-year average growth in total assets divided by the average total assets over the last 5 years EG Earnings growth is the 5-year average growth in earnings divided by the average total assets over the last 5 years GSIC sectorial codes (I)ndustry {10, 20, 30, 40, 50, 60, 70} (S)ub-(I)ndustry {10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80} (S)ector {10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60} (I)ndustry (G)roup {10, 20, 30, 40, 50} RV

Fig. 5.10 (a) The out-of-sample MSE is compared between OLS and a two-hidden layer deep network applied to a universe of 3290 stocks from the Russell 3000 index over the period from November 2008 to November 2018. (b) The information ratios of a portfolio selection strategy which selects the n stocks from the universe with the highest predicted monthly returns. The information ratios are evaluated for various size portfolios. The information ratios are based on outof-sample predicted asset returns using OLS regression, neural networks, and randomized selection with no predictive model

Finally, Fig. 5.11 compares the distribution of sensitivities to each factor over the entire 100 month period using the neural network (top) and OLS regression (bottom). The sensitivities are sorted in ascending order from left to right by their median values. We observe that the NN ranking of the factors differs substantially from the OLS regression.

7 Summary

183

Fig. 5.11 The distribution of factor model sensitivities to each factor over the entire ten-year period using the neural network applied to the Russell 3000 asset factor loadings (top). The sensitivities are sorted in ascending order from left to right by their median values. The same sensitivities using OLS linear regression (bottom). See Table 5.4 for a short description of the fundamental factors

7 Summary An important aspect in adoption of neural networks in factor modeling is the existence of a statistical framework which provides the transparency and statistical interpretability of linear least squares estimation. Moreover, one should expect to use such a framework applied to linear data and obtain similar results to linear regression, thus isolating the effects of non-linearity versus the effect of using different optimization algorithms and model implementation environments. In this chapter, we introduce a deep learning framework for interpretable cross-sectional modeling and demonstrate its application to a simple fundamental factor model. Deep learning generalizes the linear fundamental factor models by capturing non-linearity, interaction effects, and non-parametric shocks in financial econometrics. This framework provides interpretability, with confidence intervals, and ranking of the factor importance and interaction effects. In the case when the network contains no hidden layers, our approach recovers a linear fundamental factor model. The framework allows the impact of non-linearity and non-parametric treatment of the error on the factors over time and forms the basis for generalized interpretability of fundamental factors.

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8 Exercises Exercise 5.1* Consider the following data generation process Y = X1 + X2 + , X1 , X2 , ∼ N (0, 1), i.e. β0 = 0 and β1 = β2 = 1. a. For this data, write down the mathematical expression for the sensitivities of the fitted neural network when the network has – – – – –

zero hidden layers; one hidden layer, with n unactivated hidden units; one hidden layer, with n tanh activated hidden units; one hidden layer, with n ReLU activated hidden units; and two hidden layers, each with n tanh activated hidden units.

Exercise 5.2** Consider the following data generation process Y = X1 + X2 + X1 X2 + , X1 , X2 ∼ N(0, 1), ∼ N (0, σn2 ), i.e. β0 = 0 and β1 = β2 = β12 = 1, where β12 is the interaction term. σn2 is the variance of the noise and σn = 0.01. a. For this data, write down the mathematical expression for the interaction term (i.e., the off-diagonal components of the Hessian matrix) of the fitted neural network when the network has – – – – –

zero hidden layers; one hidden layer, with n unactivated hidden units; one hidden layer, with n tanh activated hidden units; one hidden layer, with n ReLU activated hidden units; and two hidden layers, each with n tanh activated hidden units.

Why is the ReLU activated network problematic for estimating interaction terms?

8.1 Programming Related Questions* Exercise 5.3* For the same problem in the previous exercise, use 5000 simulations to generate a regression training set dataset for the neural network with one hidden layer. Produce a table showing how the mean and standard deviation of the sensitivities βi behave as the number of hidden units is increased. Compare your result with tanh and ReLU activation. What do you conclude about which

Appendix

185

activation function to use for interpretability? Note that you should use the notebook Deep-Learning-Interpretability.ipynb as the starting point for experimental analysis. Exercise 5.4* Generalize the sensitivities function in Exercise 5.3 to L layers for either tanh or ReLU activated hidden layers. Test your function on the data generation process given in Exercise 5.1. Exercise 5.5** Fixing the total number of hidden units, how do the mean and standard deviation of the sensitivities βi behave as the number of layers is increased? Your answer should compare using either tanh or ReLU activation functions. Note, do not mix the type of activation functions across layers. What you conclude about the effect of the number of layers, keeping the total number of units fixed, on the interpretability of the sensitivities? Exercise 5.6** For the same data generation process as the previous exercise, use 5000 simulations to generate a regression training set for the neural network with one hidden layer. Produce a table showing how the mean and standard deviation of the interaction term behave as the number of hidden units is increased, fixing all other parameters. What do you conclude about the effect of the number of hidden units on the interpretability of the interaction term? Note that you should use the notebook Deep-Learning-Interaction.ipynb as the starting point for experimental analysis.

Appendix Other Interpretability Methods Partial Dependence Plots (PDPs) evaluate the expected output w.r.t. the marginal density function of each input variable, and allow the importance of the predictors to be ranked. More precisely, partitioning the data X into an interest set, Xs , and its complement, Xc = X \ Xs , then the “partial dependence” of the response on Xs is defined as ( (5.20) fs (Xs ) = EXc f5(Xs , X c ) = f5(Xs , X c ) pc (Xc ) dXc , where pc (Xc ) is the marginal probability density of X c : pc (Xc ) = Equation (5.20) can be estimated from a set of training data by n ' 1 5& f¯s (Xs ) = f X s , X i,c , n i=1

%

p (x) dx s .

(5.21)

186

5 Interpretability

where Xi,c (i = 1, 2, . . . , n) are the observations of Xc in the training set; that is, the effects of all the other predictors in the model are averaged out. There are a number of challenges with using PDPs for model interpretability. First, the interaction effects are ignored by the simplest version of this approach. While Greenwell et al. (2018) propose a methodology extension to potentially address the modeling of interactive effects, PDPs do not provide a 1-to-1 correspondence with the coefficients in a linear regression. Instead, we would like to know, under strict control conditions, how the fitted weights and biases of the MLP correspond to the fitted coefficients of linear regression. Moreover in the context of neural networks, by treating the model as a black-box, it is difficult to gain theoretical insight in to how the choice of the network architecture affects its interpretability from a probabilistic perspective. Garson (1991) partitions hidden-output connection weights into components associated with each input neuron using absolute values of connection weights. Garson’s algorithm uses the absolute values of the connection weights when calculating variable contributions, and therefore does not provide the direction of the relationship between the input and output variables. Olden and Jackson (2002) determines the relative importance, rij = [R]ij , of the ith output to the j th predictor variable of the model as a function of the weights, according to the expression (2)

(1)

rij = Wj k Wki .

(5.22)

The approach does not account for non-linearity introduced into the activation, which is the most critical aspects of the model. Furthermore, the approach presented was limited to a single hidden layer.

Proof of Variance Bound on Jacobian Proof The Jacobian can be written in matrix element form as Jij = [∂X Yˆ ]ij =

n

(2)

(1)

(1)

wik wkj H (Ik ) =

k=1

n

ck Hk (I )

(5.23)

k=1

(2) (1) wkj and Hk (I ) := H (Ik(1) ) is the Heaviside function. As where ck := cij k := wik a linear combination of indicator functions, we have

Jij =

n−1 k=1

ak 1{I (1) >0,I (1) ≤0} + an 1{I (1) >0} , ak := k

k+1

k

n

ci .

(5.24)

i=1

Alternatively, the Jacobian can be expressed in terms of a weighted sum of independent Bernoulli trials involving X:

Appendix

Jij =

187 n−1

ak 1{w(1) X>−b(1) ,w(1)} k,

k=1

(1) k+1, X≤−bk+1 }

k

+ an 1{w(1) X>−b(1) } . n,

n

(5.25)

Without loss of generality, consider the case when p = 1, the dimension of the input space is one. Then Eq. 5.25 simplifies to: Jij =

n−1

ak 1xk 0} ] = ≤0}

n

n

ak pk (1 − pk ).

k=1

(5.28) Under the assumption that the mean of the Jacobian is invariant to the number of hidden units, or if the weights are constrained so that the mean is constant, then the μ weights are ak = npijk . Then the variance is bounded by the mean: V[Jij ] = μij

n−1 < μij . n

(5.29)

If we relax the assumption that μij is independent of n then, under the original weights ak := ki=1 ci : V[Jij ] =

n

ak pk (1 − pk )

k=1

≤

n

ak pk

k=1

= μij ≤

n

ak .

k=1

188

5 Interpretability

Russell 3000 Factor Model Description Python Notebooks The notebooks provided in the accompanying source code repository are designed to gain familiarity with how to implement interpretable deep networks. The examples include toy simulated data and a simple factor model. Further details of the notebooks are included in the README.md file.

References Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., et al. (2016). Tensor flow: A system for large-scale machine learning. In Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation, OSDI’16 (pp. 265–283). Dimopoulos, Y., Bourret, P., & Lek, S. (1995, Dec). Use of some sensitivity criteria for choosing networks with good generalization ability. Neural Processing Letters, 2(6), 1–4. Dixon, M. F., & Polson, N. G. (2019). Deep fundamental factor models. Garson, G. D. (1991, April). Interpreting neural-network connection weights. AI Expert, 6(4), 46– 51. Greenwell, B. M., Boehmke, B. C., & McCarthy, A. J. (2018, May). A simple and effective modelbased variable importance measure. arXiv e-prints, arXiv:1805.04755. Nielsen, F., & Bender, J. (2010). The fundamentals of fundamental factor models. Technical Report 24, MSCI Barra Research Paper. Olden, J. D., & Jackson, D. A. (2002). Illuminating the “black box”: a randomization approach for understanding variable contributions in artificial neural networks. Ecological Modelling, 154(1), 135–150. Rosenberg, B., & Marathe, V. (1976). Common factors in security returns: Microeconomic determinants and macroeconomic correlates. Research Program in Finance Working Papers 44, University of California at Berkeley.

Part II

Sequential Learning

Chapter 6

Sequence Modeling

This chapter provides an overview of the most important modeling concepts in financial econometrics. Such methods form the conceptual basis and performance baseline for more advanced neural network architectures presented in the next chapter. In fact, each type of architecture is a generalization of many of the models presented here. This chapter is especially useful for students from an engineering or science background, with little exposure to econometrics and time series analysis.

1 Introduction More often in finance, the data consists of observations of a variable over time, e.g. stock prices, bond yields, etc. In such a case, the observations are not independent over time, rather observations are often strongly related to their recent histories. For this reason, the ordering of the data matters (unlike cross-sectional data). This is in contrast to most methods of machine learning which assume that the data is i.i.d. Moreover algorithms and techniques for fitting machine learning models, such as back-propagation for neural networks and cross-validation for hyperparameter tuning, must be modified for use on time series data. “Stationarity” of the data is a further important delineation necessary to successfully apply models to time series data. If the estimated moments of the data change depending on the window of observation, then the modeling problem is much more difficult. Neural network approaches to addressing these challenges are presented in the next chapter. An additional consideration is the data frequency—the frequency at which the data is observed assuming that the timestamps are uniform. In general, the frequency of the data governs the frequency of the time series model. For example, support that we seek to predict the week ahead stock price from daily historical adjusted close prices on business days. In such a case, we would build a model from daily prices © Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_6

191

192

6 Sequence Modeling

and then predict 5 daily steps ahead, rather than building a model using only weekly intervals of data. In this chapter we shall primarily consider applications of parametric, linear, and frequentist models to uniform time series data. Such methods form the conceptual basis and performance baseline for more advanced neural network architectures presented in the next chapter. In fact, each type of architecture is a generalization of many of the models presented here. Please note that the material presented in this chapter is not intended as a substitute for a more comprehensive and rigorous treatment of econometrics, but rather to provide enough background for Chap. 8. Chapter Objectives By the end of this chapter, the reader should expect to accomplish the following: – Explain and analyze linear autoregressive models; – Understand the classical approaches to identifying, fitting, and diagnosing autoregressive models; – Apply simple heteroscedastic regression techniques to time series data; – Understand how exponential smoothing can be used to predict and filter time series; and – Project multivariate time series data onto lower dimensional spaces with principal component analysis.

Note that this chapter can be skipped if the reader is already familiar with econometrics. This chapter is especially useful for students from an engineering or physical sciences background, with little exposure to econometrics and time series analysis.

2 Autoregressive Modeling We begin by considering a single variable Yt indexed by t to indicate the variable changes over time. This variable may depend on other variables Xt ; however, we shall simply consider the case when the dependence of Yt is on past observations of itself—this is known as univariate time series analysis.

2.1 Preliminaries Before we can build a model to predict Yt , we recall some basic definitions and terminology, starting with a continuous time setting and then continuing thereafter solely in a discrete-time setting.

2 Autoregressive Modeling

193

•

> Stochastic Process

A stochastic process is a sequence of random variables, indexed by continuous time: {Yt }∞ t=−∞ .

•

> Time Series

A time series is a sequence of observations of a stochastic process at discrete times over a specific interval: {yt }nt=1 .

•

> Autocovariance

The j th autocovariance of a time series is γj t := E[(yt − μt )(yt−j − μt−j )], where μt := E[yt ].

•

> Covariance (Weak) Stationarity

A time series is weak (or wide-sense) covariance stationary if it has time constant mean and autocovariances of all orders: μt = μ,

∀t

γj t = γj ,

∀t.

As we have seen, this implies that γj = γ−j : the autocovariances depend only on the interval between observations, but not the time of the observations.

•

> Autocorrelation

The j th autocorrelation, τj is just the j th autocovariance divided by the variance: τj =

•

γj . γ0

> White Noise

White noise, t , is i.i.d. error which satisfies all three conditions:

(6.1)

194

6 Sequence Modeling

a. E[t ] = 0, ∀t; b. V[t ] = σ 2 , ∀t; and c. t and s are independent, t = s, ∀t, s. Gaussian white noise just adds a normality assumption to the error. White noise error is often referred to as a “disturbance,” “shock,” or “innovation” in the financial econometrics literature.

With these definitions in place, we are now ready to define autoregressive processes. Tacit in our usage of these models is that the time series exhibits autocorrelation.1 If this is not the case, then we would choose to use cross-sectional models seen in Part I of this book.

2.2 Autoregressive Processes Autoregressive models are parametric time series models describing yt as a linear combination of p past observations and white noise. They are referred to as “processes” as they are representative of random processes which are dependent on one or more past values.

•

> AR(p) Process

The pth order autoregressive process of a variable Yt depends only on the previous values of the variable plus a white noise disturbance term yt = μ +

p

φi yt−i + t ,

(6.2)

i=1 p

where t is independent of {yt−1 }i=1 . We refer to μ as the drift term. p is referred to as the order of the model. Defining the polynomial function φ(L) := (1 − φ1 L − φ2 L2 − · · · − φp Lp ), where yt−j is a j th lagged observation of yt given by the Lag operator or Backshift operator, yt−j = Lj [yj ] . The AR(p) process can be expressed in the more compact form φ(L)[yt ] = μ + t .

1 We

shall identify statistical tests for establishing autocorrelation later in this chapter.

(6.3)

2 Autoregressive Modeling

195

This compact form shall be conducive to analysis describing the properties of the AR(p) process. We mention in passing that the identification of the parameter p from data, i.e. the number of lags in the model rests on the data being weakly covariance stationary.2

2.3 Stability An important property of AR(p) processes is whether past disturbances exhibit an inclining or declining impact on the current value of y as the lag increases. For example, think of the impact of a news event about a public company on the stock price movement over the next minute versus if the same news event had occurred, say, six months in the past. One should expect that the latter is much less significant than the former. To see this, consider the AR(1) process and write yt in terms of the inverse of (L) yt =

−1

(L)[μ + t ],

(6.4)

so that for an AR(1) process ∞

yt =

1 φ j Lj [μ + t ], [μ + t ] = 1 − φL

(6.5)

j =0

and the infinite sum will be stable, i.e. the φ j terms do not grow with j , provided that |φ| < 1. Conversely, unstable AR(p) processes exhibit the counter-intuitive behavior that the error disturbance terms become increasingly influential as the lag increases. t We can calculate the Impulse Response Function (IRF), ∂∂yt−j ∀j , to characterize the influence of past disturbances. For the AR(p) model, the IRF is given by φ j and hence is geometrically decaying when the model is stable.

2.4 Stationarity Another desirable property of AR(p) models is that their autocorrelation function convergences to zero as the lag increases. A sufficient condition for convergence is stationary. From the characteristic equation

2 Statistical

tests for identifying the order of the model will be discussed later in the chapter.

196

6 Sequence Modeling

(z) = (1 −

z z z ) · (1 − ) · . . . · (1 − ) = 0, λ1 λ2 λp

(6.6)

it follows that a AR(p) model is strictly stationary and ergodic if all the roots lie outside the unit sphere in the complex plane C. That is |λi | > 1, i ∈ {1, . . . , p} and | · | is the modulus of a complex number. Note that if the characteristic equation has at least one unit root, with all other roots lying outside the unit sphere, then this is a special case of non-stationarity but not strict stationarity.

•

> Stationarity of Random Walk

We can show that the following random walk (zero mean AR(1) process) is not strictly stationary: yt = yt−1 + t

(6.7)

Written in compact form gives (L)[yt ] = t ,

(L) = 1 − L,

(6.8)

and the characteristic polynomial, (z) = 1 − z = 0, implies that the real root z = 1. Hence the root is on the unit circle and the model is a special case of nonstationarity.

Finding roots of polynomials is equivalent to finding eigenvalues. The Cayley– Hamilton theorem states that the roots of any polynomial can be found by turning it into a matrix and finding the eigenvalues. Given the p degree polynomial3 : q(z) = c0 + c1 z + . . . + cp−1 zp−1 + zp ,

(6.9)

we define the p × p companion matrix ⎛ ⎜ ⎜ ⎜ C := ⎜ ⎜ ⎜ ⎝

3 Notice

that the zp coefficient is 1.

0 0 .. .

1

0 .. . 0 0 −c0 −c1

...

1 0 .. .. . . 0 0 . . . −cp−2

⎞ 0 .. ⎟ . ⎟ ⎟ .. ⎟ , . ⎟ ⎟ 1 ⎠ −cp−1

(6.10)

2 Autoregressive Modeling

197

then the characteristic polynomial det (C − λI ) = q(λ), and so the eigenvalues of C are the roots of q. Note that if the polynomial does not have a unit leading coefficient, then one can just divide the polynomial by that coefficient to arrive at the form of Eq. 6.9, without changing its roots. Hence the roots of any polynomial can be found by computing the eigenvalues of a companion matrix. The AR(p) has a characteristic polynomial of the form (z) = 1 − φ1 z − · · · − φp zp

(6.11)

and dividing by −φp gives q(z) = −

(z) 1 φ1 =− + z + · · · + zp φp φp φp

(6.12)

and hence the companion matrix is of the form ⎛ ⎜ ⎜ ⎜ ⎜ C := ⎜ ⎜ ⎜ ⎝

1

...

0 .. . .. . 1

⎞

⎟ ⎟ 0 1 0 ⎟ ⎟ .. .. .. ⎟. . . . ⎟ ⎟ 0 0 0 0 ⎠ φ φ φ1 p−1 p−1 1 − . . . − − φp φp φp φp 0 .. .

(6.13)

2.5 Partial Autocorrelations Autoregressive models carry a signature which allows its order, p, to be determined from time series data provided the data is stationary. This signature encodes the memory in the model and is given by “partial autocorrelations.” Informally each partial autocorrelation measures the correlation of a random variable, yt , with its lag, yt−h , while controlling for intermediate lags. The formal definition of the partial autocorrelation is now given.

•

> Partial Autocorrelation

A partial autocorrelation at lag h ≥ 2 is a conditional autocorrelation between a variable, yt , and its hth lag, yt−h under the assumption that the values of the intermediate lags, yt−1 , . . . , yt−h+1 are controlled: τ˜h := τ˜t,t−h := .

γ˜h . , γ˜t,h γ˜t−h,h

198

6 Sequence Modeling

where γ˜h := γ˜t,t−h := E[yt − P (yt | yt−1 , . . . , yt−h+1 ), yt−h − P (yt−h | yt−1 , . . . , yt−h+1 )] is the lag-h partial autocovariance, P (W | Z) is an orthogonal projection of W onto the set Z and γ˜t,h := E[(yt − P (yt | yt−1 , . . . , yt−h+1 ))2 ].

(6.14)

The partial autocorrelation function τ˜h : N → [−1, 1] is a map h :→ τ˜h . The plot of τ˜h against h is referred to as the partial correlogram.

AR(p) Processes Using the property that a linear orthogonal projection yˆt = P (yt | yt−1 , . . . , yt−h+1 ) is given by the OLS estimator as yˆt = φ1 yt−1 + · · · + φh−1 yt−h+1 , gives the Yule-Walker equations for an AR(p) process, relating the partial autocorrelations T˜ p := [τ˜1 , . . . , τ˜p ] to the autocorrelations Tp := [τ1 , . . . , τp ]: ⎡

1

⎢ ⎢ τ1 ˜ R p T p = T p , Rp = ⎢ ⎢ .. ⎣ . τp−1

⎤ τ1 . . . τp−1 . ⎥ .. .. . . .. ⎥ ⎥ .. ⎥ . .. .. . . . ⎦ τp−2 . . . 1

(6.15)

For h ≤ p, we can solve for the hth lag partial autocorrelation by writing τ˜h =

|Rh∗ | , |Rh |

(6.16)

where | · | is the matrix determinant and the j th column of [Rh∗ ],j = [Rh ],j , j = h and the hth column is [Rh∗ ],h = Th . For example, the lag-1 partial autocorrelation is τ˜1 = τ1 and the lag-2 partial autocorrelation is 1 τ1 τ1 τ2 τ2 − τ12 . (6.17) τ˜2 = = 1 τ1 1 − τ12 τ1 1 We note, in particular, that the lag-2 partial autocorrelation of an AR(1) process, with autocorrelation T2 = [τ1 , τ12 ] is τ˜2 =

τ12 − τ12 1 − τ12

= 0,

(6.18)

2 Autoregressive Modeling

199

and this is true for all lag orders greater than the order of the AR process. We can reason about this property from another perspective—through the partial autocovariances. The lag-2 partial autocovariance of an AR(1) process is γ˜2 := γ˜t,t−2 := E[yt − yˆt , yt−2 − yˆt−2 ],

(6.19)

where yˆ = P (yt | yt−1 ) and yˆt−2 = P (yt−2 | yt−1 ). When P is a linear orthogonal projection, we have from the property of an orthogonal projection P (W | Z) = μW +

Cov(W, Z) (Z − μZ ) V[Z]

(6.20)

t−1 ) and P (yt | yt−1 ) = φ V(y V(yt−1 ) = φ so that yˆt = φyt−1 , yˆt−2 = φyt−1 and hence t = yt − yˆt so the lag-2 partial autocovariance is

γ˜2 = E[t , yt−2 − φyt−1 ] = 0.

(6.21)

Clearly the lag-1 partial autocovariance of an AR(1) process is γ˜1 = E[yt − μ, yt−1 − μ] = γ1 = φγ0 .

(6.22)

2.6 Maximum Likelihood Estimation The exact likelihood when the density of the data is independent of (φ, σn2 ) is L(y, x; φ, σn2 ) =

T

fYt |Xt (yt |xt ; φ, σn2 )fXt (xt ).

(6.23)

t=1

Under this assumption, the exact likelihood is proportional to the conditional likelihood function: L(y, x; φ, σn2 ) ∝ L(y|x; φ, σn2 ) =

T

fYt |Xt (yt |xt ; φ, σn2 )

t=1

= (σn2 2π )−T /2 exp{−

T 1 (yt − φ T xt )2 }. 2σn2 t=1

200

6 Sequence Modeling

In many cases such an assumption about the independence of the density of the data and the parameters is not warranted. For example, consider the zero mean AR(1) with unknown noise variance: yt = φyt−1 + t , t ∼ N(0, σn2 )

(6.24)

Yt |Yt−1 ∼ N(φyt−1 , σn2 ) Y1 ∼ N(0,

σn2 ). 1 − φ2

The exact likelihood is L(x; φ, σn2 ) = fYt |Yt−1 (yt |yt−1 ; φ, σn2 )fY1 (y1 ; φ, σn2 ) ) *−1/2 T −1 σn2 1 − φ2 2 2 = 2π exp{− y1 }(σn 2π )− 2 2 2 1−φ 2σn exp{−

T 1 (yt − φyt−1 )2 }, 2σn2 t=2

where we made use of the moments of Yt —a result which is derived in Sect. 2.8. Despite the dependence of the density of the data on the parameters, there may be practically little advantage of using exact maximum likelihood against the conditional likelihood method (i.e., dropping the fY1 (y1 ; φ, σn2 ) term). This turns out to be the case for linear models. Maximizing the conditional likelihood is equivalent to ordinary least squares estimation.

2.7 Heteroscedasticity The AR model assumes that the noise is i.i.d. This may be an overly optimistic assumption which can be relaxed by assuming that the noise is time dependent. Treating the noise as time dependent is exemplified by a heteroscedastic AR(p) model yt = μ +

p

2 φi yt−i + t , t ∼ N(0, σn,t ).

(6.25)

i=1

There are many tests for heteroscedasticity in time series models and one of them, the ARCH test, is summarized in Table 6.3. The estimation procedure for heteroscedastic models is more complex and involves two steps: (i) estimation of the errors from the maximum likelihood function which treats the errors as independent and (ii) estimation of model parameters under a more general maximum likelihood

2 Autoregressive Modeling

201

estimation which treats the errors as time-dependent. Note that such a procedure could be generalized further to account for correlation in the errors but requires the inversion of the covariance matrix, which is computationally intractable with large time series. The conditional likelihood is L(y|X; φ, σn2 ) =

T

fYt |Xt (yt |xt ; φ, σn2 )

t=1

1 = (2π )−T /2 det (D)−1/2 exp{− (y − φ T X)T D −1 (y − φ T X)}, 2 2 is the diagonal covariance matrix and X ∈ RT ×p is the data where Dtt = σn,t matrix defined as [X]t = xt . The advantage of this approach is its relative simplicity. The treatment of noise variance as time dependent in finance has long been addressed by more sophisticated econometrics models and the approach presented here brings AR models into line with the specifications of more realistic models. On the other hand, the use of the sample variance of the residuals is only appropriate when the sample size is sufficient. In practice, this translates into the requirement for a sufficiently large historical period before a prediction can be made. Another disadvantage is that the approach does not explicitly define the relationship between the variances. We shall briefly revisit heteroscedastic models and explore a model for regressing the conditional variance on previous conditional variances in Sect. 2.9.

2.8 Moving Average Processes The Wold representation theorem (a.k.a. Wold decomposition) states that every covariance stationary time series can be written as the sum of two time series, one deterministic and one stochastic. In effect, we have already considered the deterministic component when choosing an AR process.4 The stochastic component can be represented as a “moving average process” or MA(q) process which expresses yt as a linear combination of current and q past disturbances. Its definition is as follows:

•

> MA(q) Process

The qth order moving average process is the linear combination of the white noise q process {t−i }t=0 , ∀t 4 This

is an overly simplistic statement because the AR(1) process can be expressed as a MA(∞) process and vice versa.

202

6 Sequence Modeling

yt = μ +

q

θi t−i + t .

(6.26)

i=1

2 It turns out that yt−1 depends on {t−1 , t−2 , . . . }, but not t and hence γt,t−2 = 0. It should be apparent that this property holds even when P is a non-linear projection provided that the errors are independent (but not necessarily identical). Another brief point of discussion is that an AR(1) process can be rewritten as a MA(∞) process. Suppose that the AR(1) process has a mean μ and the variance of the noise is σn2 , then by a binomial expansion of the operator (1 − φL)−1 we have ∞

μ φ j t−j , yt = + 1−φ

(6.27)

j =0

where the moments can be easily found and are E[yt ] = V[yt ] =

μ 1−φ ∞

2 φ 2j E[t−j ]

j =0

= σn2

∞ j =0

φ 2j =

σn2 . 1 − φ2

AR and MA models are important components of more complex models which are known as ARMA or, more generally, ARIMA models. The expression of a pattern as a linear combination of past observations and past innovations turns out to be more flexible in time series modeling than any single component. These are by no means the only useful techniques and we briefly turn to another technique which smooths out shorter-term fluctuations and consequently boosts the signal to noise ratio in longer term predictions.

2.9 GARCH Recall from Sect. 2.7 that heteroscedastic time series models treat the error as time dependent. A popular parametric, linear, and heteroscedastic method used in financial econometrics is the Generalized Autoregressive Conditional Heteroscedastic (GARCH) model (Bollerslev and Taylor) . A GARCH(p,q) model specifies that the conditional variance (i.e., volatility) is given by an ARMA(p,q) model—there are p lagged conditional variances and q lagged squared noise terms:

2 Autoregressive Modeling

203

σt2 := E[t2 |t−1 ] = α0 +

q

2 αi t−i +

p

i=1

2 βi σt−i .

i=1

This model gives an explicit relationship between the current volatility and previous volatilities. Such a relationship is useful for predicting volatility in the model, with obvious benefits for volatility modeling in trading and risk management. This simple relationship enables us to characterize the behavior of the model, as we shall see shortly. A necessary condition for model stationarity is the following constraint: (

q

αi +

i=1

p

βi ) < 1.

i=1

When the model is stationary, the long-run volatility converges to the unconditional variance of t : σ 2 := var(t ) =

1−(

α0 . p i=1 αi + i=1 βi )

q

To see this, let us consider the l-step ahead forecast using a GARCH(1,1) model: 2 2 + β1 σt−1 σt2 = α0 + α1 t−1 2 σˆ t+1

=

α0 + α1 E[t2 |t−1 ] + β1 σt2

= σ 2 + (α1 + β1 )(σt2 − σ 2 ) 2 2 2 σˆ t+2 = α0 + α1 E[t+1 |t−1 ] + β1 E[σt+1 |t−1 ]

= σ 2 + (α1 + β1 )2 (σt2 − σ 2 ) 2 2 2 σˆ t+l = α0 + α1 E[t+l−1 |t−1 ] + β1 E[σt+l−1 |t−1 ]

= σ 2 + (α1 + β1 )l (σt2 − σ 2 ),

(6.28) (6.29) (6.30) (6.31) (6.32) (6.33) (6.34)

where we have substituted for the unconditional variance, σ 2 = α0 /(1 − α1 − β1 ). 2 → σ 2 as l → ∞ so as the forecast From the above equation we can see that σˆ t+1 horizon goes to infinity, the variance forecast approaches the unconditional variance of t . From the l-step ahead variance forecast, we can see that (α1 + β1 ) determines how quickly the variance forecast converges to the unconditional variance. If the variance sharply rises during a crisis, the number of periods, K, until it is halfway between the first forecast and the unconditional variance is (α1 + β1 )K = 0.5, so the half-life5 is given by K = ln(0.5)/ ln(α1 + β1 ).

5 The

half-life is the lag k at which its coefficient is equal to a half.

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For example, if (α1 + β1 ) = 0.97 and steps are measured in days, the half-life is approximately 23 days.

2.10 Exponential Smoothing Exponential smoothing is a type of forecasting or filtering method that exponentially decreases the weight of past and current observations to give smoothed predictions y˜t+1 . It requires a single parameter, α, also called the smoothing factor or smoothing coefficient. This parameter controls the rate at which the influence of the observations at prior time steps decay exponentially. α is often set to a value between 0 and 1. Large values mean that the model pays attention mainly to the most recent past observations, whereas smaller values mean more of the history is taken into account when making a prediction. Exponential smoothing takes the forecast for the previous period y˜t and adjusts with the forecast error, yt − y˜t . The forecast for the next period becomes y˜t+1 = y˜t + α(yt − y˜t ),

(6.35)

y˜t+1 = αyt + (1 − α)y˜t .

(6.36)

or equivalently

Writing this as a geometric decaying autoregressive series back to the first observation: y˜t+1 = αyt + α(1 − α)yt−1 + α(1 − α)2 yt−2 + α(1 − α)3 yt−3 + · · · + α(1 − α)t−1 y1 + α(1 − α)t y˜1 , hence we observe that smoothing introduces a long-term model of the entire observed data, not just a sub-sequence used for prediction in a AR model, for example. For geometrically decaying models, it is useful to characterize it by the half-life—the lag k at which its coefficient is equal to a half: 1 , 2

(6.37)

ln(2α) . ln(1 − α)

(6.38)

α(1 − α)k = or k=−

The optimal amount of smoothing, α, ˆ is found by maximizing a likelihood function.

3 Fitting Time Series Models: The Box–Jenkins Approach

205

3 Fitting Time Series Models: The Box–Jenkins Approach While maximum likelihood estimation is the approach of choice for fitting the ARMA models described in this chapter, there are many considerations beyond fitting the model parameters. In particular, we know from earlier chapters that the bias–variance tradeoff is a central consideration which is not addressed in maximum likelihood estimation without adding a penalty term. Machine learning achieves generalized performance through optimizing the bias–variance tradeoff, with many of the parameters being optimized through crossvalidation. This is both a blessing and a curse. On the one hand, the heavy reliance on numerical optimization provides substantial flexibility but at the expense of computational cost and, often-times, under-exploitation of structure in the time series. There are also potential instabilities whereby small changes in hyperparameters lead to substantial differences in model performance. If one were able to restrict the class of functions represented by the model, using knowledge of the relationship and dependencies between variables, then one could in principle reduce the complexity and improve the stability of the fitting procedure. For some 75 years, econometricians and statisticians have approached the problem of time series modeling with ARIMA in a simple and intuitive way. They follow a three-step process to fit and assess AR(p). This process is referred to as a Box–Jenkins approach or framework. The three basic steps of the Box–Jenkins modeling approach are: a. (I)dentification—determining the order of the model (a.k.a. model selection); b. (E)stimation—estimation of model parameters; and c. (D)iagnostic checking—evaluating the fit of the model. This modeling approach is iterative and parsimonious—it favors models with fewer parameters.

3.1 Stationarity Before the order of the model can be determined, the time series must be tested for stationarity. A standard statistical test for covariance stationarity is the Augmented Dickey-Fuller (ADF) test which often accounts for the (c)onstant drift and (t)ime trend. The ADF test is a unit root test—the Null hypothesis is that the characteristic polynomial exhibits at least a unit root and hence the data is non-stationary. If the Null can be rejected at a confidence level, α, then the data is stationary. Attempting to fit a time series model to non-stationary data will result in dubious interpretations of the estimated partial autocorrelation function and poor predictions and should therefore be avoided.

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3.2 Transformation to Ensure Stationarity Any trending time series process is non-stationary. Before we can fit an AR(p) model, it is first necessary to transform the original time series into a stationary form. In some instances, it may be possible to simply detrend the time series (a transformation which works in a limited number of cases). However this is rarely full proof. To the potential detriment of the predictive accuracy of the model, we can however systematically difference the original time series one or more times until we arrive at a stationary time series. To gain insight, let us consider a simple example. Suppose we are given the following linear model with a time trend of the form: yt = α + βt + t , t ∼ N(0, 1).

(6.39)

We first observe that the mean of yt is time dependent: E[yt ] = α + βt,

(6.40)

and thus this is model is non-stationary. Instead we can difference the process to give yt − yt−1 = (α + βt + t ) − (α + β(t − 1) + t−1 ) = β + t − t−1 ,

(6.41)

and hence the mean and the variance of this difference process are constant and the difference process is stationary: E[yt − yt−1 ] = β

(6.42)

E[(yt − yt−1 − β) ] = 2σ . 2

2

(6.43)

Any difference process can be written as an ARIMA(p,d,q) process, where here d = 1 is the order of differencing to achieve stationarity. There is, in general, no guarantee that first order differencing yields a stationary difference process. One can apply higher order differencing, d > 1, to the detriment of recovering the original signal, but one must often resort to non-stationary time series methods such as Kalman filters, Markov-switching models, and advanced neural networks for sequential data covered later in this part of the book.

3.3 Identification A common approach for determining the order of a AR(p) from a stationary time series is to estimate the partial autocorrelations and determine the largest lag which is significant. Figure 6.1 shows the partial correlogram, the plot of the estimated

3 Fitting Time Series Models: The Box–Jenkins Approach

207

Fig. 6.1 This plot shows the partial correlogram, the plot of the estimated partial autocorrelations against the lag. The solid horizontal lines define the 95% confidence interval. We observe that all but the first lag are approximately within the envelope and hence we may determine the order of the AR(p) model as p = 1

partial autocorrelations against the lag. The solid horizontal lines define the 95% confidence interval which can be constructed for each coefficient using 1 ± 1.96 × √ , T

(6.44)

where T is the number of observations. Note that we have assumed that T is sufficiently large that the autocorrelation coefficients are assumed to be normally distributed with zero mean and standard error of √1 .6 T We observe that all but the first lag are approximately within the envelope and hence we may determine the order of the AR(p) model as p = 1. The properties of the partial autocorrelation and autocorrelation plots reveal the orders of the AR and MA models. In Fig. 6.1, there is an immediate cut-off in the partial autocorrelation (acf) plot after 1 lag indicating an AR(1) process. Conversely, the location of a sharp cut-off in the estimated autocorrelation function determines the order, q, of a MA process. It is often assumed that the data generation process is a combination of an AR and MA model—referred to as an ARMA(p,q) model. Information Criterion While the partial autocorrelation function is useful for determining the AR(p) model order, in many cases there is an undesirable element of subjectively in the choice. It is often preferable to use the Akaike Information Criteria (AIC) to measure the quality of fit. The AIC is given by AI C = ln(σˆ 2 ) +

6 This

2k , T

assumption is admitted by the Central Limit Theorem.

(6.45)

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where σˆ 2 is the residual variance (the residual sums of squares divided by the number of observations T ) and k = p + q + 1 is the total number of parameters estimated. This criterion expresses a bias–variance tradeoff between the first term, the quality of fit, and the second term, a penalty function proportional to the number of parameters. The goal is to select the model which minimizes the AIC by first using maximum likelihood estimation and then adding the penalty term. Adding more parameters to the model reduces the residuals but increases the right-hand term, thus the AIC favors the best fit with the fewest number of parameters. On the surface, the overall approach has many similarities with regularization in machine learning where the loss function is penalized by a LASSO penalty (L1 norm of the parameters) or a ridge penalty (L2 norm of the parameters). However, we emphasize that AIC is estimated post-hoc, once the maximum likelihood function is evaluated, whereas in machine learning models, the penalized loss function is directly minimized.

3.4 Model Diagnostics Once the model is fitted we must assess whether the residual exhibits autocorrelation, suggesting the model is underfitting. The residual of fitted time series model should be white noise. To test for autocorrelation in the residual, Box and Pierce propose the Portmanteau statistic Q∗ (m) = T

m

ρˆl2 ,

l=1

as a test statistic for the Null hypothesis H0 : ρˆ1 = · · · = ρˆm = 0 against the alternative hypothesis Ha : ρˆi = 0 for some i ∈ {1, . . . , m}. ρˆi are the sample autocorrelations of the residual. The Box-Pierce statistic follows an asymptotically chi-squared distribution with m degrees of freedom. The closely related Ljung–Box test statistic increases the power of the test in finite samples: m ρˆl2 . Q(m) = T (T + 2) T −l l=1

(6.46)

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209

Fig. 6.2 This plot shows the results of applying a Ljung–Box test to the residuals of an AR(p) model. (Top) The standardized residuals are shown against time. (Center) The estimated ACF of the residuals is shown against the lag index. (Bottom) The p-values of the Ljung–Box test statistic are shown against the lag index

This statistic also follows asymptotically a chi-squared distribution with m degrees of freedom. The decision rule is to reject H0 if Q(m) > χα2 where χα2 denotes the 100(1 − α)th percentile of a chi-squared distribution with m degrees of freedom and is the significance level for rejecting H0 . For a AR(p) model, the Ljung–Box statistic follows asymptotically a chisquared distribution with m − p degrees of freedom. Figure 6.2 shows the results of applying a Ljung–Box test to the residuals of an AR(p) model. (Top) The standardized residuals are shown against time. (Center) The estimated ACF of the residuals is shown against the lag index. (Bottom) The p-values of the Ljung–Box test statistic are shown against the lag index. The figure shows that if the maximum lag in the model is sufficiently large, then the p-value is small and the Null is rejected in favor of the alternative hypothesis. Failing the test requires repeating the Box–Jenkins approach until the model no longer under-fits. The only mild safe-guard against over-fitting is the use of AIC for model selection, but in general there is no strong guarantee of avoiding over-fitting as the performance of the model is not assessed out-of-sample in this framework. Assessing the bias variance tradeoff by cross-validation has arisen as the better approach for generalizing model performance. Therefore any model that has been fitted under a Box–Jenkins approach needs to be assessed out-of-sample by time series cross-validation—the topic of the next section. There are many diagnostic tests which have been developed for time series modeling which we have not discussed here. A small subset has been listed in Table 6.3. The readers should refer to a standard financial econometrics textbook

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such as Tsay (2010) for further details of these tests and elaboration on their application to linear models.

4 Prediction While the Box–Jenkins approach is useful in identifying, fitting, and critiquing models, there is no guarantee that such a model shall exhibit strong predictive properties of course. We seek to predict the value of yt+h given information t up to and including time t. Producing a forecast is simply a matter of taking the conditional expectation of the data under the model. The h-step ahead forecast from a AR(p) model is given by yˆt+h = E[yt+h | t ] =

p

φi yˆt+h−i ,

(6.47)

i=1

where yˆt+h = yt+h , h ≤ 0 and E[t+h | t ] = 0, h > 0. Note that conditional expectations of observed variables are not equal to the unconditional expectations. In particular E[t+h | t ] = t + h, h ≤ 0, whereas E[t+h ] = 0. The quality of the forecast is measured over the forecasting horizon from either the MSE or the MAE.

4.1 Predicting Events If the output is categorical, rather than continuous, then the ARMA model is used to predict the log-odds ratio of the binary event rather than the conditional expectation of the response. This is analogous to using a logit function as a link in logistic regression. Other general metrics are also used to assess model accuracy such as a confusion matrix, the F1-score and Receiver Operating Characteristic (ROC) curves. These metrics are not specific to time series data and could be applied to cross-section models to. The following example will illustrate a binary event prediction problem using time series data. Example 6.1

Predicting Binary Events

Suppose we have conditionally i.i.d. Bernoulli r.v.s Xt with pt := P(Xt = 1 | t ) representing a binary event and conditional moments given by (continued)

4 Prediction

Example 6.1

211

(continued)

• E[Xt | ] = 0 · (1 − pt ) + 1 · pt = pt • V[Xt | ] = pt (1 − pt ) The log-odds ratio shall be assumed to follow an ARMA model, )

pt ln 1 − pt

*

= φ −1 (L)(μ + θ (L)t ).

(6.48)

and the category of the model output is determined by a threshold, e.g. pt >= 0.5 corresponds to a positive event. If the number of out-of-sample observations is 24, we can compare the prediction with the observed event and construct a truth table (a.k.a. confusion matrix) as illustrated in Table 6.1. In this example, the accuracy is (12 + 2)/24—the ratio of the sum of the diagonal terms to the set size. The type I (false positive) and type II (false negative) errors, shown by the off-diagonal elements as 8 and 2, respectively.

Table 6.1 The confusion matrix for the above example

Actual 1 0 Sum

Predicted 1 0 12 2 8 2 20 4

Sum 14 10 24

In this example, the accuracy is (12+2)/24—the ratio of the sum of the diagonal terms to the set size. Of special interest are the type I (false positive) and type II (false negative) errors, shown by the off-diagonal elements as 8 and 2, respectively. In practice, careful consideration must be given as to whether there is equal tolerance for type 1 and type 2 errors. The significance of the classifier can be estimated from a chi-squared statistic with one degree of freedom under a Null hypothesis that the classifier is a white noise. In general, Chi-squared testing is used to determine whether two variables are independent of one another. In this case, if the Chi-squared statistic is above a given critical threshold value, associated with a significance level, then we can say that the classifier is not white noise. Let us label the elements of the confusion matrix as in Table 6.2 below. The column and row sums of the confusion matrices and the total number of test samples, m, are also shown. The chi-squared statistic with one degree of freedom is given by the squared difference of the expected result (i.e., a white noise model where the prediction is independent of the observations) and the model prediction, Yˆ , relative to the expected result. When normalized by the number of observations, each element of the confusion matrix is the joint probability [P(Y, Yˆ )]ij . Under a white noise model,

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Table 6.2 The confusion matrix of a binary classification is shown together with the column and row sums and the total number of test samples, m

Actual 1 0 Sum

Predicted 1 0 m11 m12 m21 m22 m,1 m,2

Sum m1, m1, m

the observed outcome, Y , and the predicted outcome, Yˆ , are independent and so [P(Y, Yˆ )]ij = [P(Y )]i [P(Yˆ )]j which is the ith row sum, mi, , multiplied by the j th column sum, m,j , divided by m. Since mi,j is based on the model prediction, the chi-squared statistic is thus χ2 =

2 2 (mij − mi, m,j /m)2 . mi, m,j /m

(6.49)

i=1 j =1

Returning to the example above, the chi-squared statistic is χ 2 = (12 − (14 × 20)/24)2 /(14 × 20)/24 + (2 − (14 × 4)/24)2 /(14 × 4)/24 + (8 − (10 × 20)/24)2 /(10 × 20)/24 + (2 − (10 × 4)/24)2 /(10 × 4)/24 = 0.231. This value is far below the threshold value of 6.635 for a chi-squared statistic with one degree of freedom to be significant. Thus we cannot reject the Null hypothesis. The predicted model is not sufficiently indistinguishable from white noise. The example classification model shown above used a threshold of pt >= 0.5 to classify an event as positive. This choice of threshold is intuitive but arbitrary. How can we measure the performance of a classifier for a range of thresholds? A ROC-Curve contains information about all possible thresholds. The ROCCurve plots true positive rates against false positive rates, where these terms are defined as follows: – True Positive Rate (TPR) is T P /(T P + F N): fraction of positive samples which the classifier correctly identified. This is also known as Recall or Sensitivity. Using the confusion matrix in Table 6.1, the TPR=12/(12 + 2) = 6/7. – False Positive Rate (FPR) is F P /(F P +T N): fraction of positive samples which the classifier misidentified. In the example confusion matrix, the FPR=8/(8 + 2) = 4/5. – Precision is T P /(T P + F P ): fraction of samples that were positive from the group that the classifier predicted to be positive. From the example confusion matrix, the precision is 12/(12 + 8) = 3/5.

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213

Fig. 6.3 The ROC curve for an example model shown by the green line

Each point in a ROC curve is a (TPR, FPR) pair for a particular choice of the threshold in the classifier. The straight dashed black line in Fig. 6.3 represents a random model. The green line shows the ROC curve of the model—importantly it is should always be above the line. The perfect model would exhibit a TPR of unity for all FPRs, so that there is no area above the curve. The advantage of this performance measure is that it is robust to class imbalance, e.g. rare positive events. This is not true of classification accuracy which leads to misinterpretation of the quality of the fit when the data is imbalanced. For example, a constant model Yˆ = f (X) = 1 would be x% accurate if the data consists of x% positive events. Additional related metrics can be derived. Common ones include Area Under the Curve (AUC), which is the area under the green line in Fig. 6.3. The F1-score is the harmonic mean of the precision and recall and is also frequently used. The F1-score reaches its best value at unity and worst at zero and is 2×3/5×6/7 given by F 1 = 2·precision·recall precision+recall . From the example above F1 = 3/5+6/7 = 0.706.

4.2 Time Series Cross-Validation Cross-validation—the method of hyperparameter tuning by rotating through K folds (or subsets) of training-test data—differs for time series data. In prediction models over time series data, no future observations can be used in the training set. Instead, a sliding window must be used to train and predict out-of-sample over multiple repetitions to allow for parameter tuning as illustrated in Fig. 6.4. One frequent challenge is whether to fix the length of the window or allow it to “telescope” by including the ever extending history of observations as the window is “walked forward.” In general, the latter has the advantage of including more observations in the training set but can lead to difficulty in interpreting the confidence of the parameters, due to the loss of control of the sample size.

5 Principal Component Analysis The final section in this chapter approaches data modeling from quite a different perspective, with the goal being to reduce the dimension of multivariate time series

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Verification: tuning hyperparameters

Out-ofsample period

In-sample period

split

Historical period Aggregate test period

Fig. 6.4 Times series cross-validation, also referred to as “walk forward optimization,” is used instead of standard cross-validation for cross-sectional data to preserve the ordering of observations in time series data. This experimental design avoids look-ahead bias in the fitted model which occurs when one or more observations in the training set are from the future

data. The approach is widely used in finance, especially when the dimensionality of the data presents barriers to computational tractability or practical risk management and trading challenges such as hedging exposure to market risk factors. For example, it may be advantageous to monitor a few risk factors in a large portfolio rather than each instrument. Moreover such factors should provide economic insight into the behavior of the financial markets and be actionable from an investment management perspective. 6 7N Formally, let yi i=1 be a set of N observation vectors, each of dimension n. We 6 7N assume that n ≤ N . Let Y ∈ Rn×N be a matrix whose columns are yi i=1 , ⎡

⎤ | | Y = ⎣ y1 · · · yN ⎦ . | | The element-wise average of the N observations is an n dimensional signal which may be written as: 1 1 yi = Y1N , N N N

y¯ =

i=1

where 1N ∈ RN ×1 is a column vector of all-ones. Denote Y0 as a matrix whose columns are the demeaned observations (we center each observation yi by subtracting y¯ from it):

5 Principal Component Analysis

215

Y0 = Y − y¯ 1TN .

Projection A linear projection from Rm to Rn is a linear transformation of a finite dimensional vector given by a matrix multiplication: xi = WT yi , where yi ∈ Rn , xi ∈ Rm , and W ∈ Rn×m . Each element j in the vector xi is an inner product between yi and the j -th column of W, which we denote by wj . Let X ∈ Rm×N be a matrix whose columns are the set of N vectors of N xi = N1 X1N be the element-wise average, transformed observations, let x¯ = N1 i=1

and X0 = X − x¯ 1TN the demeaned matrix. Clearly, X = WT Y and X0 = WT Y0 .

5.1 Principal Component Projection When the matrix WT represents the transformation that applies principal component 7 analysis, 6 7n we denote W = P, and the columns of the orthonormal matrix, P, denoted pj j =1 , are referred to as loading vectors. The transformed vectors {xi }N i=1 are referred to as principal components or scores. The first loading vector is defined as the unit vector with which the inner products of the observations have the greatest variance: p1 = max wT1 Y0 YT0 w1 s.t. wT1 w1 = 1. w1

(6.50)

The solution to Eq. 6.50 is known to be the eigenvector of the sample covariance matrix Y0 YT0 corresponding to its largest eigenvalue.8 Next, p2 is the unit vector which has the largest variance of inner products between it and the observations after removing the orthogonal projections of the observations onto p1 . It may be found by solving:

T p2 = max wT2 Y0 − p1 pT1 Y0 Y0 − p1 pT1 Y0 w2 s.t. wT2 w2 = 1. w2

is, P−1 = PT . normalize the eigenvector and disregard its sign.

7 That 8 We

(6.51)

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The solution to Eq. 6.51 is known to be the eigenvector corresponding to the largest eigenvalue under the constraint that it is not collinear with p1 . Similarly, the remaining loading vectors are equal to the remaining eigenvectors of Y0 YT0 corresponding to descending eigenvalues. The eigenvalues of Y0 YT0 , which is a positive semi-definite matrix, are nonnegative. They are not necessarily distinct, but since it is a symmetric matrix it has n eigenvectors that are all orthogonal, and it is always diagonalizable. Thus, the matrix P may be computed by diagonalizing the covariance matrix: Y0 YT0 = P"P−1 = P"PT , where " = X0 XT0 is a diagonal matrix whose diagonal elements {λi }ni=1 are sorted in descending order. The transformation back to the observations is Y = PX. The fact that the covariance matrix of X is diagonal means that PCA is a decorrelation transformation and is often used to denoise data.

5.2 Dimensionality Reduction PCA is often used as a method for dimensionality reduction, the process of reducing the number of variables in a model in order to avoid the curse of dimensionality. PCA gives the first m principal components (m < n) by applying the truncated transformation Xm = PTm Y, where each column of Xm ∈ Rm×N is a vector whose elements are the first m principal components, and Pm is a matrix whose columns are the first m loading vectors, ⎡

⎤ | | Pm = ⎣ p1 · · · pm ⎦ ∈ Rn×m . | | Intuitively, by keeping only m principal components, we are losing information, and we minimize this loss of information by maximizing their variances.

6 Summary

217

An important concept in measuring the amount of information lost is the total reconstruction error Y − Yˆ F , where F denotes the Frobenius matrix norm. Pm is also a solution to the minimum total squared reconstruction min

W∈Rn×m

8 82 8 8 8Y0 − WWT Y0 8 s.t. WT W = Im×m . F

(6.52)

The m leading loading vectors form an orthonormal basis which spans the m dimensional subspace onto which the projections of the demeaned observations have the minimum squared difference from the original demeaned observations. In other words, Pm compresses each demeaned vector of length n into a vector of length m (where m ≤ n) in such a way that minimizes the sum of total squared reconstruction errors. The minimizer of Eq. 6.52 is not unique: W = Pm Q is also a solution, where Q ∈ Rm×m is any orthogonal matrix, QT = Q−1 . Multiplying Pm from the right by Q transforms the first m loading vectors into a different orthonormal basis for the same subspace.

6 Summary This chapter has reviewed foundational material in time series analysis and econometrics. Such material is not intended to substitute more comprehensive and formal treatment of methodology, but rather provide enough background for Chap. 8 where we shall develop neural networks analogues. We have covered the following objectives: – Explain and analyze linear autoregressive models; – Understand the classical approaches to identifying, fitting, and diagnosing autoregressive models; – Apply simple heteroscedastic regression techniques to time series data; – Understand how exponential smoothing can be used to predict and filter time series; and – Project multivariate time series data onto lower dimensional spaces with principal component analysis. It is worth noting that in industrial applications the need to forecast more than a few steps ahead often arises. For example, in algorithmic trading and electronic market making, one needs to forecast far enough into the future, so as to make the forecast economically realizable either through passive trading (skewing of the price) or through aggressive placement of trading orders. This economic realization of the trading signals takes time, whose actual duration is dependent on the frequency of trading.

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We should also note that in practice linear regressions predicting the difference between a future and current price, taking as inputs various moving averages, are often used in preference to parametric models, such as GARCH. These linear regressions are often cumbersome, taking as inputs hundreds or thousands of variables.

7 Exercises Exercise 6.1 Calculate the mean, variance, and autocorrelation function (acf) of the following zero-mean AR(1) process: yt = φ1 yt−1 + t , where φ1 = 0.5. Determine whether the process is stationary by computing the root of the characteristic equation (z) = 0. Exercise 6.2 You have estimated the following ARMA(1,1) model for some time series data yt = 0.036 + 0.69yt−1 + 0.42ut−1 + ut , where you are given the data at time t − 1, yt−1 = 3.4 and uˆ t−1 = −1.3. Obtain the forecasts for the series y for times t, t + 1, t + 2 using the estimated ARMA model. If the actual values for the series are −0.032, 0.961, 0.203 for t, t + 1, t + 2, calculate the out-of-sample Mean Squared Error (MSE) and Mean Absolute Error (MAE). Exercise 6.3 Derive the mean, variance, and autocorrelation function (ACF) of a zero mean MA(1) process. Exercise 6.4 Consider the following log-GARCH(1,1) model with a constant for the mean equation yt = μ + ut , ut ∼ N(0, σt2 ) 2 ln(σt2 ) = α0 + α1 u2t−1 + β1 lnσt−1

– What are the advantages of a log-GARCH model over a standard GARCH model?

Appendix

219

– Estimate the unconditional variance of yt for the values α0 = 0.01, α1 = 0.1, β1 = 0.3. – Derive an algebraic expression relating the conditional variance with the unconditional variance. – Calculate the half-life of the model and sketch the forecasted volatility. Exercise 6.5 Consider the simple moving average (SMA) St =

Xt + Xt−1 + Xt+2 + . . . + Xt−N +1 , N

and the exponential moving average (EMA), given by E1 = X1 and, for t ≥ 2, Et = αXt + (1 − α)Et−1 , where N is the time horizon of the SMA and the coefficient α represents the degree of weighting decrease of the EMA, a constant smoothing factor between 0 and 1. A higher α discounts older observations faster. a. Suppose that, when computing the EMA, we stop after k terms, instead of going after the initial value. What fraction of the total weight is obtained? b. Suppose that we require 99.9% of the weight. What k do we require? c. Show that, by picking α = 2/(N + 1), one achieves the same center of mass in the EMA as in the SMA with the time horizon N . d. Suppose that we have set α = 2/(N + 1). Show that the first N points in an EMA represent about 87.48% of the total weight. Exercise 6.6 Suppose that, for the sequence of random variables {yt }∞ t=0 the following model holds: yt = μ + φyt−1 + t ,

|φ| ≤ 1,

t ∼ i.i.d.(0, σ 2 ).

Derive the conditional expectation E[yt | y0 ] and the conditional variance Var[yt | y0 ].

Appendix Hypothesis Tests

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Table 6.3 A short summary of some of the most useful diagnostic tests for time series modeling in finance Name Chi-squared test

t-test

Mariano-Diebold test ARCH test

Portmanteau test

Description Used to determine whether the confusion matrix of a classifier is statistically significant, or merely white noise Used to determine whether the output of two separate regression models are statistically different on i.i.d. data Used to determine whether the output of two separate time series models are statistically different The ARCH Engle’s test is constructed based on the property that if the residuals are heteroscedastic, the squared residuals are autocorrelated. The Ljung–Box test is then applied to the squared residuals A general test for whether the error in a time series model is auto-correlated Example tests include the Box-Ljung and the Box-Pierce test

Python Notebooks Please see the code folder of Chap. 6 for e.g., implementations of ARIMA models applied to time series prediction. An example, applying PCA to decompose stock prices is also provided in this folder. Further details of these notebooks are included in the README.md file for Chap. 6.

Reference Tsay, R. S. (2010). Analysis of financial time series (3rd ed.). Wiley.

Chapter 7

Probabilistic Sequence Modeling

This chapter presents a powerful class of probabilistic models for financial data. Many of these models overcome some of the severe stationarity limitations of the frequentist models in the previous chapters. The fitting procedure demonstrated is also different—the use of Kalman filtering algorithms for state-space models rather than maximum likelihood estimation or Bayesian inference. Simple examples of hidden Markov models and particle filters in finance and various algorithms are presented.

1 Introduction So far we have seen how sequences can be modeled using autoregressive processes, moving averages, GARCH, and similar methods. There exists another school of thought, which gave rise to hidden Markov models, Baum–Welch and Viterbi algorithms, Kalman and particle filters. In this school of thought, one assumes the existence of a certain latent process (say X), which evolves over time (so we may write Xt ). This unobservable, latent process drives another, observable process (say Yt ), which we may observe either at all times or at some subset of times. The evolution of the latent process Xt , as well as the dependence of the observable process Yt on Xt , may be driven by random factors. We therefore talk about a stochastic or probabilistic model. We also refer to such a model as a statespace model. The state-space model consists in a description of the evolution of the latent state over time and the dependence of the observables on the latent state. We have already seen probabilistic methods presented in Chaps. 2 and 3. These methods primarily assume that the data is i.i.d. On the other hand, the time series methods presented in the previous chapter are designed for time series data but are not probabilistic. This chapter shall build on these earlier chapters by considering a © Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_7

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powerful class of models for financial data. Many of these models overcome some of the severe stationarity limitations of the frequentist models in the previous chapters. The fitting procedure is also different—we will see the use of Kalman filtering algorithms for state-space models rather than maximum likelihood estimation or Bayesian inference. Chapter Objectives By the end of this chapter, the reader should expect to accomplish the following: – Formulate hidden Markov models (HMMs) for probabilistic modeling over hidden states; – Gain familiarity with the Baum–Welch algorithmic for fitting HMMs to time series data; – Use the Viterbi algorithm to find the most likely path; – Gain familiarity with state-space models and the application of Kalman filters to fit them; and – Apply particle filters to financial time series.

2 Hidden Markov Modeling A hidden Markov model (HMM) is a probabilistic model representing probability distributions over sequences of observations. HMMs are the simplest “dynamic” Bayesian network1 and have proven a powerful model in many applied fields including finance. So far in this book, we have largely considered only i.i.d. observations.2 Of course, financial modeling is often seated in a Markovian setting where observations depend on, and only on, the previous observation. We shall briefly review HMMs in passing as they encapsulate important ideas in probabilistic modeling. In particular, they provide intuition for understanding hidden variables and switching. In the next chapter we shall see the examples of switching in dynamic recurrent neural networks, such as GRUs and LSTMs, which use gating. However, this gating is an implicit modeling step and cannot be controlled explicitly as may be needed for regime switching in finance. Let us assume at time t that the discrete state, st , is hidden from the observer. Furthermore, we shall assume that the hidden state is a Markov process. Note this setup differs from mixture models which treat the hidden variable as i.i.d. The time t observation, yt , is assumed to be independent of the state at all other times. By the Markovian property, the joint distribution of the sequence of states, s := {st }Ti=1 ,

1 Dynamic

Bayesian networks models are a graphical model used to model dynamic processes through hidden state evolution. 2 With the exception of heteroscedastic modeling in Chap. 6.

2 Hidden Markov Modeling

223

Fig. 7.1 This figure shows the probabilistic graph representing the conditional dependence relations between the observed and the hidden variables in the HMM

and sequence of observations, y = {yt }Tt=1 is given by the product of transition probability densities p(st | st−1 ) and emission probability densities, p(yt | st ): p(s, y) = p(s1 )p(y1 | s1 )

T

p(st | st−1 )p(yt | st ).

(7.1)

t=2

Figure 7.1 shows the Bayesian network representing the conditional dependence relations between the observed and the hidden variables in the HMM. The conditional dependence relationships define the edges of a graph between parent nodes, Yt , and child nodes St . Example 7.1

Bull or Bear Market?

Suppose that the market is either in a Bear or Bull market regime represented by s = 0 or s = 1, respectively. Such states or regimes are assumed to be hidden. Over each period, the market is observed to go up or down and represented by y = −1 or y = 1. Assume that the emission probability matrix—the conditional dependency matrix between observed and hidden variables—is independent of time and given by ⎡

⎤ y/ s 0 1 P(yt = y | st = s) = ⎣ −1 0.8 0.2⎦ , 1 0.2 0.8

(7.2)

and the transition probability density matrix for the Markov process {St } is given by 0.9 0.1 A= , [A]ij := P(St = si | St−1 = sj ). 0.1 0.9

(7.3)

Given the observed sequence {−1, 1, 1} (i.e., T = 3), we can compute the probability of a realization of the hidden state sequence {1, 0, 0} using Eq. 7.1. Assuming that P(s1 = 0) = P(s1 = 1) = 12 , the computation is (continued)

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7 Probabilistic Sequence Modeling

Example 7.1

(continued)

P(s, y) = P(s1 = 1)P(y1 = −1 | s1 = 1)P(s2 = 0 | s1 = 1)P(y2 = 1 | s2 = 0) P(s3 = 0 | s2 = 0)P(y3 = 1 | s3 = 0), = 0.5 · 0.2 · 0.1 · 0.2 · 0.9 · 0.2 = 0.00036. We first introduce the so-called forward and backward quantities, respectively, defined for all states st ∈ {1, . . . , K} and over all times Ft (s) := P(st = s, y1:t )

and

Bt (s) := p(yt+1:T | st = s)

(7.4)

with the convention that BT (s) = 1. For all t ∈ {1, . . . , T } and for all r, s ∈ {1, . . . , K} we have P(st = s, y) = Ft (s)Bt (s),

(7.5)

and combining the forward and backward quantities gives P(st−1 = r, st = s, y) = Ft−1 (r)P(st = s | st−1 = r)p(yt | st = s)Bt (s).

(7.6)

The forward–backward algorithm, also known as the Baum–Welch algorithm, is an unsupervised learning algorithm for fitting HMMs which belongs to the class of EM algorithms.

2.1 The Viterbi Algorithm In addition to finding the probability of the realization of a particularly hidden state sequence, we may also seek the most likely sequence realization. This sequence can be estimated using the Viterbi algorithm. Suppose once again that we observe a sequence of T observations, y = {y1 , . . . , yT }. However, for each 1 ≤ t ≤ T , yt ∈ O, where O = {o1 , o2 , . . . , oN }, N ∈ N, is now in some observation space. We suppose that, for each 1 ≤ t ≤ T , the observation yt is driven by a (hidden) state st ∈ S, where S := {∫1 , . . . , ∫K }, K ∈ N, is some state space. For example, yt might be the credit rating of a corporate bond and st might indicate some latent variable such as the overall health of the relevant industry sector. Given y, what is the most likely sequence of hidden states, x = {x1 , x2 , . . . , xT }?

2 Hidden Markov Modeling

225

To answer this question, we need to introduce a few more constructs. First, the set of initial probabilities must be given: π = {π1 , . . . , πK }, so that πi is the probability that s1 = ∫i , 1 ≤ i ≤ K. We also need to specify the transition matrix A ∈ RK×K , such that the element Aij , 1 ≤ i, j ≤ K is the transition probability of transitioning from state ∫i to state ∫j . Finally, we need the emission matrix B ∈ RK×N , such that the element Bij , 1 ≤ i ≤ K, 1 ≤ j ≤ N is the probability of observing oj from state ∫i . Let us now consider a simple example to fix ideas. Example 7.2

The Crooked Dealer

A dealer has two coins, a fair coin, with P(Heads) = 12 , and a loaded coin, with P(Heads) = 45 . The dealer starts with the fair coin with probability 35 . The dealer then tosses the coin several times. After each toss, there is a 25 probability of a switch to the other coin. The observed sequence is Heads, Tails, Heads, Tails, Heads, Heads, Heads, Tails, Heads. In this case, the state space and observation space are, respectively, S = {∫1 = Fair, ∫2 = Loaded},

O = {o1 = Heads, o2 = Tails},

with initial probabilities π = {π1 = 0.6, π2 = 0.4}, transition probabilities ) * 0.6 0.4 A= , 0.4 0.6 and the emission matrix is B=

) * 0.5 0.5 . 0.8 0.2

Given the sequence of observations y = (Heads, Tails, Heads, Tails, Heads, Heads, Heads, Tails, Heads), we would like to find the most likely sequence of hidden states, s = {s1 , . . . , sT }, i.e., determine which of the two coins generated which of the coin tosses.

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7 Probabilistic Sequence Modeling

One way to answer this question is by applying the Viterbi algorithm as detailed in the notebook Viterbi.ipynb. We note that the most likely state sequence s, which produces the observation sequence y = {y1 , . . . , yT }, satisfies the recurrence relations V1,k = P(y1 | s1 = ∫k ) · πk , & ' Vt,k = max P(yt | st = ∫k ) · Aik · Vt−1,i , 1≤i≤K

where Vt,k is the probability of the most probable state sequence {s1 , . . . , st } such that st = ∫k , Vt,k = P(s1 , . . . , st , y1 , . . . , yt | st = ∫k ). The actual Viterbi path can be obtained by, at each step, keeping track of which state index i was used in the second equation. Let ξ(k, t) be the function that returns the value of i that was used to compute Vt,k if t > 1, or k if t = 1. Then sT = ∫arg

max (VT ,k ) ,

1≤i≤K

st−1 = ∫ξ(st ,t) . We leave the application of the Viterbi algorithm to our example as an exercise for the reader. Note that the Viterbi algorithm determines the most likely complete sequence of hidden states given the sequence of observations and the model specification, including the known transition and emission matrices. If these matrices are known, there is no reason to use the Baum–Welch algorithm. If they are unknown, then the Baum–Welch algorithm must be used.

2.1.1

Filtering and Smoothing with HMMs

Financial data is typically noisy and we need techniques which can extract the signal from the noise. There are many techniques for reducing the noise. Filtering is a general term for extracting information from a noisy signal. Smoothing is a particular kind of filtering in which low-frequency components are passed and highfrequency components are attenuated. Filtering and smoothing produce distributions of states at each time step. Whereas maximum likelihood estimation chooses the state with the highest probability at the “best” estimate at each time step, this may not lead to the best path in HMMs. We have seen that the Baum–Welch algorithm can be deployed to find the optimal state trajectory, not just the optimal sequence of “best” states.

3 Particle Filtering

227

2.2 State-Space Models HMMs belong in the same class as linear Gaussian state-space models. These are known as “Kalman filters” which are continuous latent state analogues of HMMs. Note that we have already seen examples of continuous state-space models, which are not necessarily Gaussian, in our exposition on RNNs in Chap. 8. The state transition probability p(st | st−1 ) can be decomposed into deterministic and noise: st = Ft (st−1 ) + t ,

(7.7)

for some deterministic function and t is zero-mean i.i.d. noise. Similarly, the emission probability p(yt | st ) can be decomposed as: yt = Gt (st ) + ξt ,

(7.8)

with zero-mean i.i.d. observation noise. If the functions Ft and Gt are linear and time independent, then we have st = Ast−1 + t , yt = Cst + ξt ,

(7.9) (7.10)

where A is the state transition matrix and C is the observation matrix. For completeness, we contrast the Kalman filter with a univariate RNN, as described in Chap. 8. When the observations are predictors, xt , and the hidden variables are st we have st = F (st−1 , yt ) := σ (Ast−1 + Byt ),

(7.11)

yt = Cst + ξt ,

(7.12)

where we have ignored the bias terms for simplicity. Hence, the RNN state equation differs from the Kalman filter in that (i) it is a non-linear function of both the previous state and the observation; and (ii) it is noise-free.

3 Particle Filtering A Kalman filter maintains its state as moments of the multivariate Gaussian distribution, N(m, P). This approach is appropriate when the state is Gaussian, or when the true distribution can be closely approximated by the Gaussian. What if the distribution is, for example, bimodal?

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7 Probabilistic Sequence Modeling

Arguably the simplest way to approximate more or less any distribution, including a bimodal distribution, is by data points sampled from that distribution. We refer to those data points as “particles.” The more particles we have, the more closely we can approximate the target distribution. The approximate, empirical distribution is then given by the histogram. Note that the particles need not be univariate, as in our example. They may be multivariate if we are approximating a multivariate distribution. Also, in our example the particles all have the same weight. More generally, we may consider weighted particles, whose weights are unequal. This setup gives rise to the family of algorithms known as particle filtering algorithms (Gordon et al. 1993; Kitagawa 1993). One of the most common of them is the Sequential Importance Resampling (SIR) algorithm:

3.1 Sequential Importance Resampling (SIR) a. Initialization step: At time t = 0, draw M i.i.d. samples from the initial distribution τ0 . Also, initialize M normalized (to 1) weights to an identical value 1 ˆ (i) 0 | 0 and the normalized M . For i = 1, . . . , M, the samples will be denoted x (i)

weights λ0 . (i) b. Recursive step: At time t = 1, . . . , T , let (ˆxt−1 | t−1 )i=1,...,M be the particles generated at time t − 1. – Importance sampling: For i = 1, . . . M, sample xˆ (i) t | t−1 from the Markov (i) transition kernel τt (· | xˆ t−1 | t−1 ). For i = 1, . . . , M, use the observation density to compute the non-normalized weights (i)

(i)

(i)

ωt := λt−1 · p(yt | xˆ t | t−1 ) and the values of the normalized weights before resampling (“br”) (i)

br (i) λt

ω := M t

(k) k=1 ωt

.

– Resampling (or selection): For i = 1, . . . , M, use an appropriate resampling (i) algorithm (such as multinomial resampling—see below) to sample xt | t from the mixture M k=1

br (k) λt δ(xt

(k)

− xt | t−1 ),

3 Particle Filtering

229

where δ(·) denotes the Dirac delta generalized function, and set the normalized weights after resampling, λ(i) t , appropriately (for most common resampling (i) 1 algorithms this means λt := M ). Informally, SIR shares some of the characteristics of genetic algorithms; based (i) on the likelihoods p(yt | xˆ t | t−1 ), we increase the weights of the more “successful” particles, allowing them to “thrive” at the resampling step. The resampling step was introduced to avoid the degeneration of the particles, with all the weight concentrating on a single point. The most common resampling scheme is the so-called multinomial resampling which we now review.

3.2 Multinomial Resampling Notice, from above, that we are using with the normalized weights computed before br (M) : resampling, br λ(1) t , . . . , λt a. For i = 1, . . . , M, compute the cumulative sums br

"(i) t

=

i

br (k) λt ,

k=1 (M)

so that, by construction, br "t = 1. b. Generate M random samples from U(0, 1), u1 , u2 , . . . , uM . (j ) (i) c. For each i = 1, . . . , M, choose the particle xˆ t | t = xˆ t | t−1 with j ∈ " ! (j ) (j +1) {1, 2, . . . , M − 1} such that ui ∈ br "t , br "t . (1)

(M)

Thus we end up with M new particles (children), xt | t , . . . , xt | t sampled from (1)

(M)

the existing set xt | t−1 , . . . , xt | t−1 , so that some of the existing particles may disappear, while others may appear multiple times. For each i = 1, . . . , M the (i) number of times xt | t−1 appears in the resampled set of particles is known as the

particle’s replication factor, Nt(i) . (i) 1 We set the normalized weights after resampling: λt := M . We could view (1) (M) this algorithm as the sampling of the replication factors Nt , . . . Nt from the (1) (M) br br multinomial distribution with probabilities λt , . . . , λt , respectively. Hence the name of the method.

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7 Probabilistic Sequence Modeling

3.3 Application: Stochastic Volatility Models Stochastic Volatility (SV) models have been studied extensively in the literature, often as applications of particle filtering and Markov chain Monte Carlo (MCMC). Their broad appeal in finance is their ability to capture the “leverage effect”—the observed tendency of an asset’s volatility to be negatively correlated with the asset’s returns (Black 1976). In particular, Pitt, Malik, and Doucet apply the particle filter to the stochastic volatility with leverage and jumps (SVLJ) (Malik and Pitt 2009, 2011a,b; Pitt et al. 2014). The model has the general form of Taylor (1982) with two modifications. For t ∈ N ∪ {0}, let yt denote the log-return on an asset and xt denote the log-variance of that return. Then yt = t ext /2 + Jt #t ,

(7.13)

xt+1 = μ(1 − φ) + φxt + σv ηt ,

(7.14)

where μ is the mean log-variance, φ is the persistence parameter, σv is the volatility of log-variance. The first modification to Taylor’s model is the introduction of correlation between t and ηt : )

t ηt

*

) ∼ N(0, ),

=

1ρ ρ 1

* .

The correlation ρ is the leverage parameter. In general, ρ < 0, due to the leverage effect. The second change is the introduction of jumps. Jt ∈ {0, 1} is a Bernoulli counter with intensity p (thus p is the jump intensity parameter), #t ∼ N(0, σJ2 ) determines the jump size (thus σJ is the jump volatility parameter). We obtain a stochastic volatility with leverage (SVL), but no jumps, if we delete the Jt #t term or, equivalently, set p to zero. Taylor’s original model is a special case of SVLJ with p = 0, ρ = 0. This, then, leads to the following adaptation of SIR, developed by Doucet, Malik, and Pitt, for this special case & with nonadditive, ' correlated noises. The initial distribution of x0 is taken to be N 0, σv2 /(1 − φ 2 ) . a. Initialization step: At time t = 0, draw M i.i.d. particles from the initial distribution N(0, σv2 /(1 − φ 2 )). Also, initialize M normalized (to 1) weights to 1 an identical value of M . For i = 1, 2, . . . , M, the samples will be denoted xˆ0(i)| 0 (i)

and the normalized weights λ0 . (i) b. Recursive step: At time t ∈ N, let (xˆt−1 | t−1 )i=1,...,M be the particles generated at time t − 1.

4 Point Calibration of Stochastic Filters

231

i Importance sampling: – First, (i)

(i)

– For i = 1, . . . , M, sample ˆt−1 from p(t−1 | xt−1 = xˆt−1 | t−1 , yt−1 ). (If no (i)

yt−1 is available, as at t = 1, sample from p(t−1 | xt−1 = xˆt−1 | t−1 )). (i) | t−1

– For i = 1, . . . , M, sample xˆt

(i)

(i)

from p(xt | xt−1 = xˆt−1 | t−1 , yt−1 , ˆt−1 ).

– For i = 1, . . . , M, compute the non-normalized weights: (i)

(i)

(i) | t−1 ),

ωt := λt−1 · pγt (yt | xˆt

(7.15)

using the observation density ) p(yt |

(i) xˆt | t−1 , p, σJ2 )

) p

2π(e

= (1 − p)

(i) | t−1

xˆt

2πe

(i) | t−1

xˆt

*−1/2

+ σJ2 )

) exp

)

*−1/2 exp

(i) | t−1

−yt2 /(2exˆt

(i) | t−1

−yt2 /(2exˆt

* ) +

* + 2σJ2 )

,

and the values of the normalized weights before resampling (‘br’): (i)

br (i) λt

ω := M t

(k) k=1 ωt

.

ii Resampling (or selection): For i = 1, . . . , M, use an appropriate resampling (i) algorithm (such as multinomial resampling) sample xˆt | t from the mixture M

br (k) λt δ(xt

(k) | t−1 ),

− xˆt

k=1

where δ(·) denotes the Dirac delta generalized function, and set the normalized (i) weights after resampling, λt , according to the resampling algorithm.

4 Point Calibration of Stochastic Filters We have seen in the example of the stochastic volatility model with leverage and jumps (SVLJ) that the state-space model may be parameterized by a parameter vector, θ ∈ Rdθ , dθ ∈ N. In that particular case,

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7 Probabilistic Sequence Modeling

⎞ μ ⎜φ ⎟ ⎜ ⎟ ⎜ 2⎟ ⎜σ ⎟ θ = ⎜ η ⎟. ⎜ρ ⎟ ⎜ 2⎟ ⎝ σJ ⎠ p ⎛

We may not know the true value of this parameter. How do we estimate it? In other words, how do we calibrate the model, given a time series of either historical or generated observations, y1 , . . . , yT , T ∈ N. The frequentist approach relies on the (joint) probability density function of the observations, which depends on the parameters, p(y1 , y2 , . . . , yT | θ). We can regard this as a function of θ with y1 , . . . , yT fixed, p(y1 , . . . , yT | θ ) =: L(θ )— the likelihood function. This function is sometimes referred to as marginal likelihood, since the hidden states, x1 , . . . , xT , are marginalized out. We seek a maximum likelihood estimator (MLE), θˆ ML , the value of θ that maximizes the likelihood function. Each evaluation of the objective function, L(θ ), constitutes a run of the stochastic filter over the observations y1 , . . . , yT . By the chain rule (i), and since we use a Markov chain (ii), (i)

p(y1 , . . . , yT ) =

T

(ii)

p(yt | y0 , . . . , yt−1 ) =

t=1

T (

p(yt | xt )p(xt | y0 , . . . , yt−1 ) dxt .

t=1

Note that, for ease of notation, we have omitted the dependence of all the probability densities on θ , e.g., instead of writing p(y1 , . . . , yT ; θ ). For the particle filter, we can estimate the log-likelihood function from the nonnormalized weights: p(y1 , . . . , yT ) =

T (

p(yt | xt )p(xt | y0 , . . . , yt−1 ) dxt ≈

t=1

T t=1

,

M 1 (k) ωt , M k=1

whence ln(L(θ )) = ln

T t=1

,

M 1 (k) ωt M k=1

-+ =

T t=1

,

M 1 (k) ln ωt . M

(7.16)

k=1

This was first proposed by Kitagawa (1993, 1996) for the purposes of approximating θˆ ML . In most practical applications one needs to resort to numerical methods, perhaps quasi-Newton methods, such as Broyden–Fletcher–Goldfarb–Shanno (BFGS) (Gill et al. 1982), to find θˆ ML .

5 Bayesian Calibration of Stochastic Filters

233

Pitt et al. (2014) point out the practical difficulties which result when using the above as an objective function in an optimizer. In the resampling (or selection) step of the particle filter, we are sampling from a discontinuous empirical distribution function. Therefore, ln(L(θ)) will not be continuous as a function of θ. To remedy this, they rely on an alternative, continuous, resampling procedure. A quasi-Newton method is then used to find θˆ ML for the parameters θ = (μ, φ, σv2 , ρ, p, σJ2 ) of the SVLJ model. We note in passing that Kalman filters can also be calibrated using a similar maximum likelihood approach.

5 Bayesian Calibration of Stochastic Filters Let us briefly discuss how filtering methods relate to Markov chain Monte Carlo methods (MCMC)—a vast subject in its own right; therefore, our discussion will be cursory at best. The technique takes its origin from Metropolis et al. (1953). Following Kim et al. (1998) and Meyer and Yu (2000); Yu (2005), we demonstrate how MCMC techniques can be used to estimate the parameters of the SVL model. They calibrate the parameters to the time series of observations of daily mean-adjusted log-returns, y1 , . . . , yT to obtain the joint prior density p(θ , x0 , . . . , xT ) = p(θ )p(x0 | θ )

T

p(xt | xt−1 , θ )

t=1

by successive conditioning. Here θ := (μ, φ, σv2 , ρ) is, as before, the vector of the model parameters. We assume prior independence of the parameters and choose the same priors (as in Kim et al. (1998)) for μ, φ, and σv2 , and a uniform prior for ρ. The observation model and the conditional independence assumption give the likelihood p(y1 , . . . , yT | θ , x0 , . . . , xT ) =

T

p(yt | xt ),

t=1

and the joint posterior distribution of the unobservables (the parameters θ and the hidden states x0 , . . . , xT ; in the Bayesian perspective these are treated identically and estimated in a similar manner) follows from Bayes’ theorem; for the SVL model, this posterior satisfies p(θ , x0 , . . . , xT | y1 , . . . , yT ) ∝ p(μ)p(φ)p(σv2 )p(ρ) T t=1

p(xt+1 | xt , μ, φ, σv2 )

T t=1

p(yt | xt+1 , xt , μ, φ, σv2 , ρ),

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7 Probabilistic Sequence Modeling

where p(μ), p(φ), p(σv2 ), p(ρ) are the appropriately chosen priors,

xt+1 | xt , μ, φ, σv2 ∼ N μ(1 − φ) + φxt , σv2 , ) * ρ xt /2 2 xt 2 e yt | xt+1 , xt , μ, φ, σv , ρ ∼ N (xt+1 − μ(1 − φ) − φxt ) , e (1 − ρ ) . σv Meyer and Yu use the software package BUGS3 (Spiegelhalter et al. 1996; Lunn et al. 2000) represent the resulting Bayesian model as a directed acyclic graph (DAG), where the nodes are either constants (denoted by rectangles), stochastic nodes (variables that are given a distribution, denoted by ellipses), or deterministic nodes (logical functions of other nodes); the arrows either indicate stochastic dependence (solid arrows) or logical functions (hollow arrows). This graph helps visualize the conditional (in)dependence assumptions and is used by BUGS to construct full univariate conditional posterior distributions for all unobservables. It then uses Markov chain Monte Carlo algorithms to sample from these distributions. The algorithm based on the original work (Metropolis et al. 1953) is now known as the Metropolis algorithm. It has been generalized by Hastings (1930–2016) to obtain the Metropolis–Hastings algorithm (Hastings 1970) and further by Green to obtain what is known as the Metropolis–Hastings–Green algorithm (Green 1995). A popular algorithm based on a special case of the Metropolis–Hastings algorithm, known as the Gibbs sampler, was developed by Geman and Geman (1984) and, independently, Tanner and Wong (1987).4 It was further popularized by Gelfand and Smith (1990). Gibbs sampling and related algorithms (Gilks and Wild 1992; Ritter and Tanner 1992) are used by BUGS to sample from the univariate conditional posterior distributions for all unobservables. As a result we perform Bayesian estimation—obtain estimates of the distributions of the parameters μ, φ, σv2 , ρ— rather than frequentist estimation, where a single value of the parameters vector, which maximizes the likelihood, θˆ ML , is produced. Stochastic filtering, sometimes in combination with MCMC, can be used for both frequentist and Bayesian parameter estimation (Chen 2003). Filtering methods that update estimates of the parameters online, while processing observations in real-time, are referred to as adaptive filtering (see Sayed (2008); Vega and Rey (2013); Crisan and Míguez (2013); Naesseth et al. (2015) and references therein). We note that a Gibbs sampler (or variants thereof) is a highly nontrivial piece of software. In addition to the now classical BUGS/WinBUGS there exist powerful Gibbs samplers accessible via modern libraries, such as Stan, Edward, and PyMC3.

3 An

acronym for Bayesian inference Using Gibbs Sampling. the Gibbs sampler is referred to as data augmentation following this paper.

4 Sometimes

7 Exercises

235

6 Summary This chapter extends Chap. 2 by presenting probabilistic methods for time series data. The key modeling assumption is the existence of a certain latent process Xt , which evolves over time. This unobservable, latent process drives another, observable process. Such an approach overcomes limitations of stationarity imposed on the methods in the previous chapter. The reader should verify that they have achieved the primary learning objectives of this chapter: – Formulate hidden Markov models (HMMs) for probabilistic modeling over hidden states; – Gain familiarity with the Baum–Welch algorithm for fitting HMMs to time series data; – Use the Viterbi algorithm to find the most likely path; – Gain familiarity with state-space models and the application of Kalman filters to fit them; and – Apply particle filters to financial time series.

7 Exercises Exercise 7.1: Kalman Filtering of Autoregressive Moving Average ARMA(p, q) Model The autoregressive moving average ARMA(p, q) model can be written as yt = φ1 yt−1 + . . . + φp yt−p + ηt + θ1 ηt−1 + . . . + θq ηt−q , where ηt ∼ N(0, σ 2 ) and includes as special cases all AR(p) and MA(q) models. Such models are often fitted to financial time series. Suppose that we would like to filter this time series using a Kalman filter. Write down a suitable process and the observation models. Exercise 7.2: The Ornstein–Uhlenbeck Process Consider the one-dimensional Ornstein–Uhlenbeck (OU) process, the stationary Gauss–Markov process given by the SDE dXt = θ (μ − Xt ) dt + σ dWt , where Xt ∈ R, X0 = x0 , and θ > 0, μ, and σ > 0 are constants. Formulate the Kalman process model for this process. Exercise 7.3: Deriving the Particle Filter for Stochastic Volatility with Leverage and Jumps

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7 Probabilistic Sequence Modeling

We shall regard the log-variance xt as the hidden states and the log-returns yt as observations. How can we use the particle filter to estimate xt on the basis of the observations yt ? a. Show that, in the absence of jumps, # xt = μ(1 − φ) + φxt−1 + σv ρyt−1 e−xt−1 /2 + σv 1 − ρ 2 ξt−1 i.i.d.

for some ξt ∼ N(0, 1). b. Show that p(t | xt , yt ) =δ(t − yt e−xt /2 )P[Jt = 0 | xt , yt ] + φ(t ; μt | Jt =1 , σ2t | Jt =1 )P[Jt = 1 | xt , yt ], where μt | Jt =1 =

yt exp(xt /2) exp(xt ) + σJ2

and σ2t | Jt =1 =

σJ2 exp(xt ) + σJ2

.

c. Explain how you could implement random sampling from the probability distribution given by the density p(t | xt , yt ). d. Write down the probability density p(xt | xt−1 , yt−1 , t−1 ). e. Explain how you could sample from this distribution. f. Show that the observation density is given by ) (i) p(yt | xˆt | t−1 , p, σJ2 )

) p

2π(e

= (1 − p)

(i)

xˆt | t−1

+ σJ2 )

(i)

2π e

*−1/2

xˆt | t−1

*−1/2

) * (i) x ˆ exp −yt2 /(2e t | t−1 ) +

) * (i) xˆt | t−1 2 2 exp −yt /(2e + 2σJ ) .

Exercise 7.4: The Viterbi Algorithm and an Occasionally Dishonest Casino The dealer has two coins, a fair coin, with P(Heads) = 12 , and a loaded coin, with P(Heads) = 45 . The dealer starts with the fair coin with probability 35 . The dealer then tosses the coin several times. After each toss, there is a 25 probability of a switch to the other coin. The observed sequence is Heads, Tails, Tails, Heads, Tails, Heads, Heads, Heads, Tails, Heads. Run the Viterbi algorithm to determine which coin the dealer was most likely using for each coin toss.

References

237

Appendix Python Notebooks The notebooks provided in the accompanying source code repository are designed to gain familiarity with how to implement the Viterbi algorithm and particle filtering for stochastic volatility model calibration. Further details of the notebooks are included in the README.md file.

References Black, F. (1976). Studies of stock price volatility changes. In Proceedings of the Business and Economic Statistics Section. Chen, Z. (2003). Bayesian filtering: From Kalman filters to particle filters, and beyond. Statistics, 182(1), 1–69. Crisan, D., & Míguez, J. (2013). Nested particle filters for online parameter estimation in discretetime state-space Markov models. ArXiv:1308.1883. Gelfand, A. E., & Smith, A. F. M. (1990, June). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85(410), 398–409. Geman, S. J., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741. Gilks, W. R., & Wild, P. P. (1992). Adaptive rejection sampling for Gibbs sampling, Vol. 41, pp. 337–348. Gill, P. E., Murray, W., & Wright, M. H. (1982). Practical optimization. Emerald Group Publishing Limited. Gordon, N. J., Salmond, D. J., & Smith, A. F. M. (1993). Novel approach to nonlinear/nonGaussian Bayesian state estimation. In IEE Proceedings F (Radar and Signal Processing). Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–32. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97–109. Kim, S., Shephard, N., & Chib, S. (1998, July). Stochastic volatility: Likelihood inference and comparison with ARCH models. The Review of Economic Studies, 65(3), 361–393. Kitagawa, G. (1993). A Monte Carlo filtering and smoothing method for non-Gaussian nonlinear state space models. In Proceedings of the 2nd U.S.-Japan Joint Seminar on Statistical Time Series Analysis (pp. 110–131). Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5(1), 1–25. Lunn, D. J., Thomas, A., Best, N. G., & Spiegelhalter, D. (2000). WinBUGS – a Bayesian modelling framework: Concepts, structure and extensibility. Statistics and Computing, 10, 325– 337. Malik, S., & Pitt, M. K. (2009, April). Modelling stochastic volatility with leverage and jumps: A simulated maximum likelihood approach via particle filtering. Warwick Economic Research Papers 897, The University of Warwick, Department of Economics, Coventry CV4 7AL. Malik, S., & Pitt, M. K. (2011a, February). Modelling stochastic volatility with leverage and jumps: A simulated maximum likelihood approach via particle filtering. document de travail 318, Banque de France Eurosystème.

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Malik, S., & Pitt, M. K. (2011b). Particle filters for continuous likelihood evaluation and maximisation. Journal of Econometrics, 165, 190–209. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21. Meyer, R., & Yu, J. (2000). BUGS for a Bayesian analysis of stochastic volatility models. Econometrics Journal, 3, 198–215. Naesseth, C. A., Lindsten, F., & Schön, T. B. (2015). Nested sequential Monte Carlo methods. In Proceedings of the 32nd International Conference on Machine Learning. Pitt, M. K., Malik, S., & Doucet, A. (2014). Simulated likelihood inference for stochastic volatility models using continuous particle filtering. Annals of the Institute of Statistical Mathematics, 66, 527–552. Ritter, C., & Tanner, M. A. (1992). Facilitating the Gibbs sampler: The Gibbs stopper and the Griddy-Gibbs sampler. Journal of the American Statistical Association, 87(419), 861–868. Sayed, A. H. (2008). Adaptive filters. Wiley-Interscience. Spiegelhalter, D., Thomas, A., Best, N. G., & Gilks, W. R. (1996, August). BUGS 0.5: Bayesian inference using Gibbs sampling manual (version ii). Robinson Way, Cambridge CB2 2SR: MRC Biostatistics Unit, Institute of Public Health. Tanner, M. A., & Wong, W. H. (1987, June). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82(398), 528–540. Taylor, S. J. (1982). Time series analysis: theory and practice. Chapter Financial returns modelled by the product of two stochastic processes, a study of daily sugar prices, pp. 203–226. NorthHolland. Vega, L. R., & H. Rey (2013). A rapid introduction to adaptive filtering. Springer Briefs in Electrical and Computer Engineering. Springer. Yu, J. (2005). On leverage in a stochastic volatility model. Journal of Econometrics, 127, 165–178.

Chapter 8

Advanced Neural Networks

This chapter presents various neural network models for financial time series analysis, providing examples of how they relate to well-known techniques in financial econometrics. Recurrent neural networks (RNNs) are presented as nonlinear time series models and generalize classical linear time series models such as AR(p). They provide a powerful approach for prediction in financial time series and generalize to non-stationary data. This chapter also presents convolution neural networks for filtering time series data and exploiting different scales in the data. Finally, this chapter demonstrates how autoencoders are used to compress information and generalize principal component analysis.

1 Introduction The universal approximation theorem states that a feedforward network is capable of approximating any function. So why do other types of neural networks exist? One answer to this is efficiency. In this chapter, different architectures shall be explored for their ability to exploit the structure in the data, resulting in fewer weights. Hence the main motivation for different architectures is often parsimony of parameters and therefore less propensity to overfit and reduced training time. We shall see that other architectures can be used, in particular ones that change their behavior over time, without the need to retrain the networks. And we will see how neural networks can be used to compress data, analogously to principal component analysis. There are other neural network architectures which are used in financial applications but are too esoteric to list here. However, we shall focus on three other classes of neural networks which have proven to be useful in the finance industry. The first two are supervised learning techniques and the latter is an unsupervised learning technique.

© Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_8

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8 Advanced Neural Networks

Recurrent neural networks (RNNs) are non-linear time series models and generalize classical linear time series models such as AR(p). They provide a powerful approach for prediction in financial time series and share parameters across time. Convolution neural networks are useful as spectral transformations of spatial and temporal data and generalize techniques such as wavelets, which use fixed basis functions. They share parameters across space. Finally, autoencoders are used to compress information and generalize principal component analysis. Chapter Objectives By the end of this chapter, the reader should expect to accomplish the following: – Characterize RNNs as non-linear autoregressive models and analyze their stability; – Understand how gated recurrent units and long short-term memory architectures give a dynamic autoregressive model with variable memory; – Characterize CNNs as regression, classification, and time series regression of filtered data; – Understand principal component analysis for dimension reduction; – Formulate a linear autoencoder and extract the principal components; and – Understand how to build more complex networks by aggregating these different concepts. The notebooks provided in the accompanying source code repository demonstrate many of the methods in this chapter. See Appendix “Python Notebooks” for further details.

2 Recurrent Neural Networks Recall that if the data D := {xt , yt }N t=1 is auto-correlated observations of X and Y at times t = 1, . . . , N, then the prediction problem can be expressed as a sequence prediction problem: construct a non-linear times series predictor, yˆt+h , of a response, yt+h , using a high-dimensional input matrix of T length sub-sequences Xt : yˆt+h = f (Xt ) where Xt := seqT ,t (X) = (xt−T +1 , . . . , xt ), where xt−j is a j th lagged observation of xt , xt−j = Lj [xt ], for j = 0, . . . , T − 1. Sequence learning, then, is just a composition of a non-linear map and a vectorization of the lagged input variables. If the data is i.i.d., then no sequence is needed (i.e., T = 1), and we recover a feedforward neural network.

2 Recurrent Neural Networks

241

Recurrent neural networks (RNNs) are times series methods or sequence learners which have achieved much success in applications such as natural language understanding, language generation, video processing, and many other tasks (Graves 2012). There are many types of RNNs—we will just concentrate on simple RNN models for brevity of notation. Like multivariate structural autoregressive models, (1) RNNs apply an autoregressive function fW (1) ,b(1) (Xt ) to each input sequence Xt , where T denotes the look back period at each time step—the maximum number of lags. However, rather than directly imposing an autocovariance structure, a RNN provides a flexible functional form to directly model the predictor, Yˆ . As illustrated in Fig. 8.1, this simple RNN is an unfolding of a single hidden layer neural network (a.k.a. Elman network (Elman 1991) ) over all time steps in the sequence, j = 0, . . . , T − 1. For each time step, j , this function fW(1)(1) ,b(1) (Xt,j ) generates a hidden state zt−j from the current input xt and the previous hidden state zt−1 and Xt,j = seqT ,t−j (X) ⊂ Xt :

yˆt +h

zt1−5

zt1−4

zt1−3

zt1−2

zt1−1

zt1

zti −5

zti −4

zti −3

zti −2

zti −1

zti

ztH−5

ztH−4

ztH−3

ztH−2

ztH−1

ztH

xt −5

xt −4

xt −3

xt −2

xt −1

xt

Fig. 8.1 An illustrative example of a recurrent neural network with one hidden layer, “unfolded” over a sequence of six time steps. Each input xt is in the sequence Xt . The hidden layer contains H units and the ith output at time step t is denoted by zti . The connections between the hidden units (1) are recurrent and are weighted by the matrix Wz . At the last time step t, the hidden units connect to a single unit output layer with continuous yˆt+h

242

8 Advanced Neural Networks

response:

(2)

yˆt+h = fW (2) ,b(2) (zt ) := σ (2) (W (2) zt + b(2) ), (1)

hidden states: zt−j = fW (1) ,b(1) (Xt,j ) := σ (1) (Wz(1) zt−j −1 + Wx(1) xt−j + b(1) ), j ∈ {T − 1, . . . , 0}, where σ (1) is an activation function such as tanh(x), and σ (2) is either a softmax function or identity map depending on whether the response is categorical or continuous, respectively. The connections between the extremal inputs xt and the H (1) hidden units are weighted by the time invariant matrix Wx ∈ RH ×P . The recurrent connections between the H hidden units are weighted by the time invariant matrix (1) Wz ∈ RH ×H . Without such a matrix, the architecture is simply a single-layered feedforward network without memory—each independent observation xt is mapped to an output yˆt using the same hidden layer. (1) (1) The sets W (1) = (Wx , Wz ) refer to the input and recurrence weights. W (2) denotes the weights tied to the output of the H hidden units at the last time step, zt , and the output layer. If the response is a continuous vector, Y ∈ RM , then W (2) ∈ RM×H . If the response is categorical, with K states, then W (2) ∈ RK×H . The number of hidden units determines the degree of non-linearity in the model and must be at least the dimensionality of the input p. In our experiments the hidden layer is generally under a hundred units, but can increase to thousands in higher dimensional datasets. There are a number of issues in the RNN design. How many times should the network being unfolded? How many hidden neurons H in the hidden layer? How to perform “variable selection”? The answer to the first question lies in tests for autocorrelation of the data—the sequence length needed in a RNN can be determined by the largest significant lag in an estimated “partial autocorrelation” function. The answer to the second is no different to the problem of how to choose the number of hidden neurons in a MLP—the bias–variance tradeoff being the most important consideration. And indeed the third question is also closely related to the problem of choosing features in a MLP. One can take a principled approach to feature selection, first identifying a subset of features using a Granger-causality test, or the “laissez-machine” approach of including all potentially relevant features and allowing auto-shrinkage to determine the most important weights are more aligned with contemporary experimental design in machine learning. One important caveat on feature selection for RNNs is that each feature must be a time series and therefore exhibit autocorrelation. We shall begin with a simple, univariate, example to illustrate how a RNN, without activation, is a AR(p) time series model.

2 Recurrent Neural Networks

Example 8.1

243

RNNs as Non-linear AR(p) Models

Consider the simplest case of a RNN with one hidden unit, H = 1, no activation function, and the dimensionality of the input vector is P = 1. (1) (1) Suppose further that Wz = φz , |φz | < 1, Wx = φx , Wy = 1, bh = 0 and (1) by = μ. Then we can show that fW (1) ,b(1) (Xt ) is of an autoregressive, AR(p), model of order p with geometrically decaying autoregressive coefficients φi = φx φzi−1 : zt−p = φx xt−p zt−T +2 = φz zt−T +1 + φx xt−T +2 ... = ... zt−1 = φz zt−2 + φx xt−1 xˆt = zt−1 + μ

then p−1 p

xˆt = μ + φx (L + φz L2 + · · · + φz φi xt−i = μ+

L )[xt ]

i=1

This special type of autoregressive model xˆt is “stable” and the order can be identified through autocorrelation tests on X such as the Durbin–Watson, Ljung– Box, or Box–Pierce tests. Note that if we modify the architecture so that the (1) recurrence weights Wz,i = φz,i are lag dependent then the unactivated hidden layer is zt−i = φz,i zt−i−1 + φx xt−i

(8.1)

which gives xˆt = μ + φx (L + φz,1 L + · · · + 2

p−1

φz,i Lp )[xt ],

(8.2)

i=1

$j −1 and thus the weights in this AR(p) model are φj = φx i=1 φz,i which allows a more flexible presentation of the autocorrelation structure than the plain RNN— which is limited to geometrically decaying weights. Note that a linear RNN with infinite number of lags and no bias corresponds to an exponential smoother, zt = αxt + (1 − α)zt−1 when Wz = 1 − α, Wx = α, and Wy = 1.

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8 Advanced Neural Networks

The generalization of a linear RNN from AR(p) to V AR(p) is trivial and can be written as xˆt =μ+

p

φj xt−j , φj :=W (2) (Wz(1) )j −1 Wx(1) , μ := W (2)

j =1

p (Wz(1) )j −1 b(1) +b(2) , j =1

where the square matrix φj ∈ RP ×P and bias vector μ ∈ RP .

(8.3)

2.1 RNN Memory: Partial Autocovariance Generally, with non-linear activation, it is more difficult to describe the RNN as a classical model. However, the partial autocovariance function provides some additional insight here. Let us first consider a RNN(1) process. The lag-1 partial autocovariance is γ˜1 = E[yt − μ, yt−1 − μ] = E[yˆt + t − μ, yt−1 − μ],

(8.4)

and using the RNN(1) model with, for simplicity, a single recurrence weight, φ: yˆt = σ (φyt−1 )

(8.5)

gives γ˜1 = E[σ (φyt−1 ) + t − μ, yt−1 − μ] = E[yt−1 σ (φyt−1 )],

(8.6)

where we have assumed μ = 0 in the second part of the expression. Checking that we recover the AR(1) covariance, set σ := I d so that 2 ] = φV[yt−1 ]. γ˜1 = φE[yt−1

(8.7)

Continuing with the lag-2 autocovariance gives: γ˜2 = E[yt − P (yt | yt−1 ), yt−2 − P (yt−2 | yt−1 )],

(8.8)

and P (yt | yt−1 ) is approximated by the RNN(1): yˆt = σ (φyt−1 ).

(8.9)

Substituting yt = yˆt + t into the above gives γ˜2 = E[t , yt−2 − P (yt−2 | yt−1 )].

(8.10)

2 Recurrent Neural Networks

245

Approximating P (yt−2 | yt−1 ) with the backward RNN(1) yˆt−2 = σ (φ(yˆt−1 + t−1 )),

(8.11)

we see, crucially, that yˆt−2 depends on t−1 but not on t . yt−2 − P (yt−2 | yt−1 ), hence depends on {t−1 , t−2 , . . . }. Thus we have that γ˜2 = 0. As a counterexample, consider the lag-2 partial autocovariance of the RNN(2) process & ' yˆt−2 = σ φσ (φ(yˆt + t ) + t−1 ) ,

(8.12)

which depends on t and hence the lag-2 partial autocovariance is not zero. It is easy to show that the partial autocorrelation τ˜s = 0, s > p and, thus, like the AR(p) process, the partial autocorrelation function for a RNN(p) has a cut-off at p lags. The partial autocorrelation function is independent of time. Such a property can be used to identify the order of the RNN model from the estimated PACF.

2.2 Stability We can generalize the stability constraint on AR(p) models presented in Sect. 2.3 to RNNs by considering the RNN(1) model: yt =

−1

(L)[t ] = (1 − σ (Wz L + b))−1 [t ],

(8.13)

where we have set Wy = 1 and by = 0 without loss of generality, and dropped the superscript ()(1) for ease of notation. Expressing this as an infinite dimensional non-linear moving average model ∞

yt =

1 [t ] = σ j (Wz L + b)[t ], 1 − σ (Wz L + b)

(8.14)

j =0

and the infinite sum will be stable when the σ j (·) terms do not grow with j , i.e. |σ | ≤ 1 for all values of φ and yt−1 . In particular, the choice tanh satisfies the requirement on σ . For higher order models, we follow an induction argument and show first that for a RNN(2) model we obtain yt = =

1 − σ (Wz σ (Wz ∞ j =0

1 [t ] + b) + Wx L + b)

L2

σ j (Wz σ (Wz L2 + b) + Wx L + b)[t ],

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8 Advanced Neural Networks

which again is stable if |σ | ≤ 1 and it follows for any model order that the stability condition holds. It follows that lagged unit impulses of the data strictly decay with the order of the lag when |σ | ≤ 1. Again by induction, at lag 1, the output from the hidden layer is zt = σ (Wz 1 + Wx 0 + b) = σ (Wz 1 + b).

(8.15)

The absolute value of each component of the hidden variable under a unit vector impulse at lag 1 is strictly less than 1: |zt |j = |σ (Wz 1 + b)|j < 1,

(8.16)

if |σ (x)| ≤ 1 and each element of Wz 1 + b is finite. Additionally if σ is strictly monotone increasing, then |zt |j under a lag two unit innovation is strictly less than |zt |j under a lag one unit innovation |σ (Wz 1) + b)j | > |σ (Wz σ (Wz 1 + b) + b)|j .

(8.17)

The implication of this stability result is the reassuring attribute that past random disturbances decay in the model and the effect of lagged data becomes less relevant to the model output with increasing lag.

2.3 Stationarity For completeness, we mention in passing the extension of stationarity analysis in Sect. 2.4 to RNNs. The linear univariate RNN(p), considered above, has a companion matrix of the form ⎛ ⎜ ⎜ ⎜ C := ⎜ ⎜ ⎜ ⎝

0 0 .. .

1

0 .. . 0 0 φ −p −φ −p+1

0 1 .. . 0 ...

⎞ 0 .. ⎟ . ⎟ 0 ⎟ .. ⎟ .. . . ⎟ ⎟ 0 1 ⎠ −φ −2 −φ −1 , ...

(8.18)

and it turns out that for φ = 0 this model is non-stationary. We can hence rule out the choice of a linear activation since this would leave us with a linear RNN. Hence, it appears that some non-linear activation is necessary for the model to be stationary, but we cannot use the Cayley–Hamilton theorem to prove stationarity.

2 Recurrent Neural Networks

247

Half-Life Suppose that the output of the RNN is in Rd . The half-life of the lag is the smallest number of function compositions, k, of σ˜ (x) := σ (Wz x + b) with itself such that the normalized j th output is (Wy σ˜ ◦1 σ˜ ◦2 · · · ◦k−1 σ˜ (1) + by )j ≤ 0.5, k ≥ 2, ∀j ∈ {1, . . . , d}. (Wy σ˜ (1) + by )j (8.19) Note the output has been normalized so that the lag-1 unit impulse ensures that the (1) ratio, rj = 1 for each j . This modified definition exists to account for the effects of the activation function and the semi-affine transformation which are not present in AR(p) model. In general, there is no guarantee that the half-life is finite but we can find parameter values for which the half-life can be found. For example, suppose for simplicity that a univariate RNN is given by xˆt = zt−1 and (k)

rj

=

zt = σ (zt−1 + xt ). Then the lag-1 impulse is xˆt = σ˜ (1) = σ (0+1), the lag-2 impulse is xˆt = σ (σ (1)+ 0) = σ˜ ◦ σ˜ (1), and so on. If σ (x) := tanh(x) and we normalize over the output from the lag-1 impulse to give the values in Table 8.1. Table 8.1 The half-life characterizes the memory decay of the architecture by measuring the number of periods before a lagged unit impulse has at least half of its effect at lag 1. The calculation of the half-life involves nested composition of the recursion relation for the hidden (k) layer until rj is less than a half. The calculations are repeated for each j , hence the half-life may vary depending on the component of the output. In this example, the half-life of the univariate RNN is 9 periods Lag k 1 2 3 4 5 6 7 8 9

r (k) 1.000 0.843 0.744 0.673 0.620 0.577 0.543 0.514 0.489

•

? Multiple Choice Question 1

Which of the following statements are true: a. An augmented Dickey–Fuller test can be applied to time series to determine whether they are covariance stationary.

248

8 Advanced Neural Networks

b. The estimated partial autocorrelation of a covariance stationary time series can be used to identify the design sequence length in a plain recurrent neural network. c. Plain recurrent neural networks are guaranteed to be stable, namely lagged unit impulses decay over time. d. The Ljung–Box test is used to test whether the fitted model residual error is autocorrelated. e. The half-life of a lag-1 unit impulse is the number of lags before the impulse has half its effect on the model output.

2.4 Generalized Recurrent Neural Networks (GRNNs) Classical RNNs, such as those described above, treat the error as homoscedastic— that is, the error is i.i.d. We mention in passing that we can generalize RNNs to heteroscedastic models by modifying the loss function to the squared Mahalanobis length of the residual vector. Such an approach is referred to here as generalized recurrent neural networks (GRNNs) and is mentioned briefly here with the caveat that the field of machine learning in econometrics is nascent and therefore incomplete and such a methodology, while appealing from a theoretic perspective, is not yet proven in practice. Hence the purpose of this subsection is simply to illustrate how more complex models can be developed which mirror some of the developments in parametric econometrics. In its simplest form, we solve a weighted least squares minimization problem using data, Dt : minimize W,b

f (W, b) + λφ(W, b),

L (Y, Yˆ ) := (Y − Yˆ )T −1 (Y − Yˆ ), tt = σt2 , tt = ρtt σt σt , f (W, b) =

T 1 L (yt , yˆt ), T

(8.20) (8.21) (8.22)

t=1

where := E[ T | Xt ] is the conditional covariance matrix of the residual error and φ(W, b) is a regularization penalty term. The conditional covariance matrix of the error must be estimated. This is performed as follows using the notation ()T to denote the transpose of a vector and () to denote model parameters fitted under heteroscedastic error. 1) For each t = 1, . . . , T , estimate the residual error over the training set, t ∈ RN , using the standard (unweighted) loss function to find the weights, Wˆ t , and biases, bˆt where the error is

3 Gated Recurrent Units

249

t = yt − FWˆ t ,bˆt (Xt ).

(8.23)

ˆ is estimated accordingly: 2) The sample conditional covariance matrix ˆ =

1 T t t . T −1 T

(8.24)

i=1

3) Perform the generalized least squares minimization using Eq. 8.20 to obtain a fitted heteroscedastic neural network model, with refined error t = yt − FWˆ ,bˆ (Xt ). t

t

(8.25)

The fitted GRNN FWˆ ,bˆ can then be used for forecasting without any further t t modification. The effect of the sample covariance matrix is to adjust the importance of the observation in the training set, based on the variance of its error and the error correlation. Such an approach can be broadly viewed as a RNN analogue of how GARCH models extend AR models. Of course, GARCH models treat the error distribution as parametric and provide a recurrence relation for forecasting the conditional volatility. In contrast, GRNNs rely on the empirical error distribution and do not forecast the conditional volatility. However, a separate regression could be performed over diagonals of the empirical conditional volatility by using time series cross-validation.

3 Gated Recurrent Units The extension of RNNs to dynamical time series models rests on extending foundational concepts in time series analysis. We begin by considering a smoothed RNN with hidden state hˆ t . Such a RNN is almost identical to a plain RNN, but with an additional scalar smoothing parameter, α, which provides the network with “long memory.”

3.1 α-RNNs Let us consider a univariate α-RNN(p) model in which the smoothing parameter is fixed: yˆt+1 = Wy hˆ t + by , hˆ t = σ (Uh h˜ t−1 + Wh yt + bh ),

(8.26) (8.27)

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8 Advanced Neural Networks

h˜ t = α hˆ t−1 + (1 − α)h˜ t−1 ,

(8.28)

with the starting condition in each sequence, hˆ t−p+1 = yt−p+1 . This model augments the plain RNN by replacing hˆ t−1 in the hidden layer with an exponentially smoothed hidden state h˜ t−1 . The effect of the smoothing is to provide infinite memory when α = 1. For the special case when α = 1, we recover the plain RNN with short memory of length p . We can easily study this model by simplifying the parameterization and considering the unactivated case. Setting by = bh = 0, Uh = Wh = φ and Wy = 1: yˆt+1 = hˆ t ,

(8.29)

= φ(h˜ t−1 + yt ),

(8.30)

= φ(α hˆ t−1 + (1 − α)h˜ t−2 + yt ).

(8.31)

Without loss of generality, consider p = 2 lags in the model so that hˆ t−1 = φyt−1 . Then hˆ t = φ(αφyt−1 + (1 − α)h˜ t−2 + yt )

(8.32)

and the model can be written in the simpler form yˆt+1 = φ1 yt + φ2 yt−1 + φ(1 − α)h˜ t−2 ,

(8.33)

with autoregressive weights φ1 := φ and φ2 := αφ 2 . We now see, in comparison with an AR(2) model, that there is an additional term which vanishes when α = 1 but provides infinite memory to the model since h˜ t−2 depends on y0 , the first observation in the whole time series, not just the first observation in the sequence. The α-RNN model can be trained by treating α as a hyperparameter. The choice to fix α is obviously limited to stationary time series. We can extend the model to non-stationary time series by using a dynamic version of exponential smoothing.

3.1.1

Dynamic αt -RNNs

Dynamic exponential smoothing is a time-dependent, convex, combination of the smoothed output, y˜t , and the observation yt : y˜t+1 = αt yt + (1 − αt )y˜t ,

(8.34)

where αt ∈ [0, 1] denotes the dynamic smoothing factor which can be equivalently written in the one-step-ahead forecast of the form y˜t+1 = y˜t + αt (yt − y˜t ).

(8.35)

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251

Hence the smoothing can be viewed as a form of dynamic forecast error correction; When αt = 0, the forecast error is ignored and the smoothing merely repeats the current hidden state h˜ t to the effect of the model losing its memory. When αt = 1, the forecast error overwrites the current hidden state h˜ t . The smoothing can also be viewed$a weighted sum of the lagged observations, with lower or equal weights, αt−s sr=1 (1 − αt−r+1 ) at the lag s ≥ 1 past observation, yt−s : y˜t+1 = αt yt +

t−1 s=1

αt−s

s

(1 − αt−r+1 )yt−s +

r=1

t−1

(1 − αt−r )y˜1 ,

(8.36)

r=0

where the last term is a time-dependent constant and typically we initialize the exponential smoother with y˜1 = y1 . Note that for any αt−r+1 = 1, the prediction y˜t+1 will have no dependency on all lags {yt−s }s≥r . The model simply forgets the observations at or beyond the rth lag. In the special case when the smoothing is constant and equal to 1 − α, then the above expression simplifies to y˜t+1 = α (L)−1 yt ,

(8.37)

or equivalently written as a AR(1) process in y˜t+1 : (L)y˜t+1 = αyt , for the linear operator

(8.38)

(z) := 1 + (α − 1)z and where L is the lag operator.

3.2 Neural Network Exponential Smoothing Let us suppose now that instead of smoothing the observed time series {ys }s≤1 , we instead smooth a hidden vector hˆ t with αˆ t ∈ [0, 1]H to give a filtered time series h˜ t = αˆ t ◦ hˆ t + (1 − αˆ t ) ◦ h˜ t−1 ,

(8.39)

where ◦ denotes the Hadamard product between vectors. This smoothing is a vectorized form of the above classical setting, only here we note that when (αt )i = 1, the ith component of the hidden variable is unmodified and the past filtered hidden variable is forgotten. On the other hand, when the (αt )i = 0, the ith component of the hidden variable is obsolete, instead setting the current filtered hidden variable to its past value. The smoothing in Eq. 8.39 can be viewed then as updating longterm memory, maintaining a smoothed hidden state variable as the memory through a convex combination of the current hidden variable and the previous smoothed hidden variable.

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The hidden variable is given by the semi-affine transformation: hˆ t = σ (Uh h˜ t−1 + Wh xt + bh )

(8.40)

which in turns depends on the previous smoothed hidden variable. Substituting Eq. 8.40 into Eq. 8.39 gives a function of h˜ t−1 and xt : h˜ t = g(h˜ t−1 , xt ; α) := αˆ t ◦ σ (Uh h˜ t−1 + Wh xt + bh ) + (1 − αˆ t ) ◦ h˜ t−1 .

(8.41)

We see that when αt = 0, the smoothed hidden variable h˜ t is not updated by the input xt . Conversely, when αt = 1, we observe that the hidden variable locally behaves like a non-linear autoregressive series. Thus the smoothing parameter can be viewed as the sensitivity of the smoothed hidden state to the input xt . The challenge becomes how to determine dynamically how much error correction is needed. GRUs address this problem by learning αˆ = F(Wα ,Uα ,bα ) (X) from the input variables with a plain RNN parameterized by weights and biases (Wα , Uα , bα ). The one-step-ahead forecast of the smoothed hidden state, h˜ t , is the filtered output of another plain RNN with weights and biases (Wh , Uh , bh ). Putting this together gives the following α − t model (simple GRU): smoothing : h˜ t = αˆ t ◦ hˆ t + (1 − αˆ t ) ◦ h˜ t−1 smoother update : αˆ t = σ

(1)

(Uα h˜ t−1 + Wα xt + bα )

hidden state update : hˆ t = σ (Uh h˜ t−1 + Wh xt + bh ),

(8.42) (8.43) (8.44)

where σ (1) is a sigmoid or Heaviside function and σ is any activation function. Figure 8.2 shows the response of a αt -RNN when the input consists of two unit impulses. For simplicity, the sequence length is assumed to be 3 (i.e., the RNN has a memory of 3 lags), the biases are set to zero, all the weights are set to one, and σ (x) := tanh(x). Note that the weights have not been fitted here, we are merely observing the effect of smoothing on the hidden state for the simplest choice of parameter values. The RNN loses memory of the unit impulse after three lags, whereas the RNNs with smooth hidden states maintain memory of the first unit impulse even when the second unit impulse arrives. The difference between the dynamically smoothed RNN (the αt -RNN) and α-RNN with a fixed smoothing parameter appears insignificant. Keep in mind however that the dynamical smoothing model has much more flexibility in how it controls the sensitivity of the smoothing to the unit impulses. In the above αt -RNN, there is no means to directly occasionally forget the memory. This is because the hidden variables update equation always depends on the previous smoothed hidden state, unless Uh = 0. However, it can be expected that the fitted recurrence weight Uˆ h will not in general be zero and thus the model is without a “hard reset button.” GRUs also have the capacity to entirely reset the memory by adding an additional reset variable:

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Fig. 8.2 An illustrative example of the response of an αt -RNN and comparison with a plain RNN and a RNN with an exponentially smoothed hidden state, under a constant α (α-RNN). The RNN(3) model loses memory of the unit impulse after three lags, whereas the α-RN N (3) models maintain memory of the first unit impulse even when the second unit impulse arrives. The difference between the αt -RNN (the toy GRU) and the α-RNN appears insignificant. Keep in mind however that the dynamical smoothing model has much more flexibility in how it controls the sensitivity of the smoothing to the unit impulses

smoothing : h˜ t = αˆ t ◦ hˆ t + (1 − αˆ t ) ◦ h˜ t−1 smoother update : αˆ t = σ

(1)

(Uα h˜ t−1 + Wα xt + bα )

hidden state update : hˆ t = σ (Uh rˆt ◦ h˜ t−1 + Wh xt + bh ) reset update : rˆt = σ

(1)

(Ur h˜ t−1 + Wr xt + br ).

(8.45) (8.46) (8.47) (8.48)

The effect of introducing a reset, or switch, rˆt , is to forget the dependence of hˆ t on the smoothed hidden state. Effectively, we turn the update for hˆ t from a plain RNN to a FFN and entirely neglect the recurrence. The recurrence in the update of hˆ t is thus dynamic. It may appear that the combination of a reset and adaptive smoothing is redundant. But remember that αˆ t effects the level of error correction in the update of the smoothed hidden state, h˜ t , whereas rˆt adjusts the level of recurrence in the unsmoothed hidden state hˆ t . Put differently, αˆ t by itself cannot disable the memory in the smoothed hidden state (internal memory), whereas rˆt in combination with αˆ t can. More precisely, when αt = 1 and rˆt = 0, h˜ t = hˆ t = σ (Wh xt + bh ) which is reset to the latest input, xt , and the GRU is just a FFNN. Also, when αt = 1 and rˆt > 0, a GRU acts like a plain RNN. Thus a GRU can be seen as a more general architecture which is capable of being a FFNN or a plain RNN under certain parameter values. These additional layers (or cells) enable a GRU to learn extremely complex longterm temporal dynamics that a plain RNN is not capable of. The price to pay for this flexibility is the additional complexity of the model. Clearly, one must choose whether to opt for a simpler model, such as an αt -RNN, or use a GRU. Lastly, we

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comment in passing that in the GRU, as in a RNN, there is a final feedforward layer to transform the (smoothed) hidden state to a response: yˆt = WY h˜ t + bY .

(8.49)

3.3 Long Short-Term Memory (LSTM) The GRU provides a gating mechanism for propagating a smoothed hidden state—a long-term memory—which can be overridden and even turn the GRU into a plain RNN (with short memory) or even a memoryless FFN. More complex models using hidden units with varying connections within the memory unit have been proposed in the engineering literature with empirical success (Hochreiter and Schmidhuber 1997; Gers et al. 2001; Zheng et al. 2017). LSTMs are similar to GRUs but have a separate (cell) memory, Ct , in addition to a hidden state ht . LSTMs also do not require that the memory updates are a convex combination. Hence they are more general than exponential smoothing. The mathematical description of LSTMs is rarely given in an intuitive form, but the model can be found in, for example, Hochreiter and Schmidhuber (1997). The cell memory is updated by the following expression involving a forget gate, αˆ t , an input gate zˆ t , and a cell gate cˆt ct = αˆ t ◦ ct−1 + zˆ t ◦ cˆt .

(8.50)

In the language of LSTMs, the triple (αˆ t , rˆt , zˆ t ) are, respectively, referred to as the forget gate, output gate, and input gate. Our change of terminology is deliberate and designed to provide more intuition and continuity with GRUs and econometrics. We note that in the special case when zˆ t = 1 − αˆ t we obtain a similar exponential smoothing expression to that used in the GRU. Beyond that, the role of the input gate appears superfluous and difficult to reason with using time series analysis. Likely it merely arose from a contextual engineering model; however, it is tempting to speculate how the additional variable provides the LSTM with a more elaborate representation of complex temporal dynamics. When the forget gate, αˆ t = 0, then the cell memory depends solely on the cell memory gate update cˆt . By the term αˆ t ◦ ct−1 , the cell memory has long-term memory which is only forgotten beyond lag s if αˆ t−s = 0. Thus the cell memory has an adaptive autoregressive structure. The extra “memory,” treated as a hidden state and separate from the cell memory, is nothing more than a Hadamard product: ht = rˆt ◦ tanh(c)t ,

(8.51)

which is reset if rˆt = 0. If rˆt = 1, then the cell memory directly determines the hidden state.

4 Python Notebook Examples

255

Thus the reset gate can entirely override the effect of the cell memory’s autoregressive structure, without erasing it. In contrast, the GRU has one memory, which serves as the hidden state, and it is directly affected by the reset gate. The reset, forget, input, and cell memory gates are updated by plain RNNs all depending on the hidden state ht . Reset gate : rˆt = σ (Ur ht−1 + Wr xt + br )

(8.52)

Forget gate : αˆ t = σ (Uα ht−1 + Wα xt + bα )

(8.53)

Input gate : zˆ t = σ (Uz ht−1 + Wz xt + bz )

(8.54)

Cell memory gate : cˆt = tanh(Uc ht−1 + Wc xt + bc ).

(8.55)

Like the GRU, the LSTM can function as a short memory, plain RNN; just set αt = 0 in Eq. 8.50. However, the LSTM can also function as a coupling of FFNs; just set rˆt = 0 so that ht = 0 and hence there is no recurrence structure in any of the gates. Both GRUs and LSTMs, even if the nomenclature does not suggest it, can model long- and short-term autoregressive memory. The GRU couple these through a smoothed hidden state variable. The LSTM separates out the long memory, stored in the cellular memory, but uses a copy of it, which may additionally be reset. Strictly speaking, the cellular memory has long-short autoregressive memory structure, so it would be misleading in the context of time series analysis to strictly discern the two memories as long and short (as the nomenclature suggests). The latter can be thought of as a truncated version of the former.

•

? Multiple Choice Question 2

Which of the following statements are true: a. A gated recurrent unit uses dynamic exponential smoothing to propagate a hidden state with infinite memory. b. The gated recurrent unit requires that the data is covariance stationary. c. Gated recurrent units are unconditionally stable, for any choice of activation functions and weights. d. A GRU only has one memory, the hidden state, whereas a LSTM has an additional, cellular, memory.

4 Python Notebook Examples The following Python examples demonstrate the application of RNNs and GRUs to financial time series prediction.

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Fig. 8.3 A comparison of out-of-sample forecasting errors produced by a RNN and GRU trained on minute snapshots of Coinbase mid-prices

4.1 Bitcoin Prediction ML_in_Finance-RNNs-Bitcoin.ipynb provides an example of how TensorFlow can be used to train and test RNNs for time series prediction. The example dataset is for predicting minute head mid-prices from minute snapshots of the USD value of Coinbase over 2018. Statistical methods for stationarity and autocorrelation shall be used to characterize the data, identify the sequence length needed in the RNN, and to diagnose the model error. Here we accept the Null as the p-value is larger than 0.01 in absolute value and thus we cannot reject the ADF test at the 99% confidence level. Since plain RNNs are not suited to non-stationary time series modeling, we can use a GRU or LSTM to model non-stationarity data, since these models exhibit dynamic autocorrelation structure. Figure 8.3 compares the out-of-sample forecasting errors produced by a RNN and GRU. See the notebook for further details of the architecture and experiment.

4.2 Predicting from the Limit Order Book The dataset is tick-by-tick, top of the limit order book, data such as mid-prices and volume weighted mid-prices (VWAP) collected from ZN futures. This dataset is heavily truncated for demonstration purposes and consists of 1033492 observations. The data has also been labeled to indicate whether the prices up-tick (1), remain

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257

Fig. 8.4 A comparison of out-of-sample forecasting errors produced by a plain RNN and GRU trained on tick-by-tick smart prices of ZN futures

the same, or down-tick (−1) over the next tick. For demonstration purposes, the timestamps have been removed. In the simple forecasting experiment, we predict VWAPs (a.k.a. “smart prices”) from historical smart prices. Note that a classification experiment is also possible but not shown here. The ADF test is performed over the first 200k observations as it is computationally intensive to apply it to the entire dataset. The ADF test statistic is −3.9706 and the p-value is smaller in absolute value than 0.01 and we thus reject the Null of the ADF test at the 99% confidence level in favor of the data being stationary (i.e., there are no unit roots). The Ljung–Box test is used to identify the number of lags needed in the model. A comparison of out-of-sample VWAP prices produced by a plain RNN and GRU is shown in Fig. 8.4. Because the data is stationary, we observe little advantage in using a GRU over a plain RNN. See ML_in_Finance-RNNs-HFT.ipynb for further details of the network architectures and experiment.

5 Convolutional Neural Networks Convolutional neural networks (CNNs) are feedforward neural networks that can exploit local spatial structures in the input data. Flattening high-dimensional time series, such as limit order book depth histories, would require a very large number of weights in a feedforward architecture. CNNs attempt to reduce the network size by exploiting data locality (Fig. 8.5).

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Fig. 8.5 Convolutional neural networks. Source: Van Veen, F. & Leijnen, S. (2019), “The Neural Network Zoo”, Retrieved from https://www. asimovinstitute.org/neuralnetwork-zoo

Deep CNNs, with multiple consecutive convolutions followed by non-linear functions, have shown to be immensely successful in image processing (Krizhevsky et al. 2012). We can view convolutions as spatial filters that are designed to select a specific pattern in the data, for example, straight lines in an image. For this reason, convolution is frequently used for image processing, such as for smoothing, sharpening, and edge detection of images. Of course, in financial modeling, we typically have different spatial structures, such as the limit order book depths or the implied volatility surface of derivatives. However, the CNN has established its place in time series analysis too.

5.1 Weighted Moving Average Smoothers A common technique in time series analysis and signal processing is to filter the time series. We have already seen exponential smoothing as a special case of a class of smoothers known as “weighted moving average (WMA)” smoothers. WMA smoothers take the form x˜t =

1

i∈I wi i∈I

wi xt−i ,

(8.56)

where x˜t is the local mean of the time series. The weights are specified to emphasize or deemphasize particular observations of xt−i in the span |I|. Examples of wellknown smoothers include the Hanning smoother h(3): x˜t = (xt−1 + 2xt + xt+1 )/4.

(8.57)

Such smoothers have the effect of reducing noise in the time series. The moving average filter is a simple low pass finite impulse response (FIR) filter commonly used for regulating an array of sampled data. It takes |I| samples of input at a time and takes the weighted average of those to produce a single output point. As the

5 Convolutional Neural Networks

259

length of the filter increases, the smoothness of the output increases, whereas the sharp modulations in the data are smoothed out. The moving average filter is in fact a convolution using a very simple filter kernel. More generally, we can write a univariate time series prediction problem as a convolution with a filter as follows. First, the discrete convolution gives the relation between xi and xj : xt−i =

t−1

δij xt−j , i ∈ {0, . . . , t − 1}

(8.58)

j =0

where we have used the Kronecker delta δ. The kernel filtered time series is a convolution x˜t−i = Kj +k+1 xt−i−j , i ∈ {k + 1, . . . , p − k}, (8.59) j ∈J

where J := {−k, . . . , k} so that the span of the filter |J | = 2k + 1, where k is taken as a small integer, and the kernel is K. For simplicity, the ends of the sequence are assumed to be unfiltered but for notational reasons we set x˜t−i = xt−i for i ∈ {1, . . . , k, p − k + 1, . . . , p}. Then the filtered AR(p) model is xˆt = μ +

p

φi x˜t−i

(8.60)

i=1

= μ + (φ1 L + φ2 L2 + · · · + φp Lp )[x˜t ]

(8.61)

= μ + [L, L2 , . . . , Lp ]φ[x˜t ],

(8.62)

with coefficients φ := [φi , . . . , φp ]. Note that there is no look-ahead bias because we do not filter the last k values of the observed data {xs }ts=1 . We have just written our first toy 1D CNN consisting of a feedforward output layer and a non-activated hidden layer with one unit (i.e., kernel): xˆt = Wy zt + by , zt = [x˜t−1 , . . . , x˜t−p ]T , Wy = φ T , by = μ,

(8.63)

where x˜t−i is the ith output from a convolution of the p length input sequence with a kernel consisting of 2k + 1 weights. These weights are fixed over time and hence the CNN is only suited to prediction from stationary time series. Note also, in contrast to a RNN, that the size of the weight matrix Wy increases with the number of lags in the model. The univariate CNN predictor with p lags and H activated hidden units (kernels) is xˆt = Wy vec(zt ) + by

(8.64)

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8 Advanced Neural Networks

[zt ]i,m = σ (

Km,j +k+1 xt−i−j + [bh ]m )

(8.65)

j ∈J

= σ (K ∗ xt + bh ),

(8.66)

where m ∈ {1, . . . , H } denotes the index of the kernel and the kernel matrix K ∈ RH ×2k+1 , hidden bias vector bh ∈ RH and output matrix Wy ∈ R1×pH . Dimension Reduction Since the size of Wy increases with both the number of lags and the number of kernels, it may be preferable to reduce the dimensionality of the weights with an additional layer and hence avoid over-fitting. We will return to this concept later, but one might view it as an alternative to auto-shrinkage or dropout. Non-sequential Models Convolutional neural networks are not limited to sequential models. One might, p for example, sample the past lags non-uniformly so that I = {2i }i=1 then the p maximum lag in the model is 2 . Such a non-sequential model allows a large maximum lag without capturing all the intermediate lags. We will also return to non-sequential models in the section on dilated convolution. Stationarity A univariate CNN predictor, with one kernel and no activation, can be written in the canonical form xˆt = μ + (1 −

(L))[K ∗ xt ] = μ + K ∗ (1 −

(L))[xt ]

(8.67)

= μ + (φ˜ 1 L + . . . φ˜ p L )[xt ]

(8.68)

:= μ + (1 − ˜ (L))[xt ],

(8.69)

p

where, by the linearity of (L) in xt , the convolution commutes and thus we can write φ˜ := K ∗ φ. Finding the roots of the characteristic equation ˜ (z) = 0,

(8.70)

it follows that the CNN is strictly stationary and ergodic if all the roots lie outside the unit circle in the complex plane, |λi | > 1, i ∈ {1, . . . , p}. As before, we would compute the eigenvalues of the companion matrix to find the roots. Provided that ˜ (L)−1 forms a divergent sequence in the noise process {s }t then the model is s=1 stable.

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261

5.2 2D Convolution 2D convolution involves applying a small kernel matrix (a.k.a. a filter), K ∈ R2k+1×2k+1 , over the input matrix (called an image), X ∈ Rm×n , to give a filtered image, Y ∈ Rm−2k×n−2k . In the context of convolutional neural networks, the elements of the filtered image are referred to as the feature map values and are calculated according to the following formula: k

yi,j = [K ∗ X]i,j =

Kk+1+p,k+1+q xi+p+1,j +q+1 ,

p,q=−k

i ∈ {1, . . . , m}, j ∈ {1, . . . .n}.

(8.71)

It is instructive to consider the following example to illustrate the 2D convolution with a small kernel matrix. Example 8.2

2D Convolution

Consider the 4 × 4 input, 3 × 3 kernel, and 2 × 2 output matrices ⎤ ⎡ ⎤ ⎡ 1002 0 −1 1 ⎢0 0 0 3⎥ ⎥ , K = ⎣0 1 0⎦ , Y = 2 1 . The calculation of the X=⎢ ⎣2 0 1 0⎦ −2 5 1 −1 0 0210 outputs for the case when i = j = 1 is yi,j = [K ∗ X]i,j =

k

Kk+1+p,k+1+q xi+p+1,j +q+1 ,

p,q=−k

i ∈ {1, . . . , m}, j ∈ {1, . . . , n} = 0 · 1 + −1 · 0 + 1 · 0 + 0 · 0 + 1 · 0 + 0 · 0 + 1 · 2 + −1 · 0 + 0 · 1 =2 We leave it as an exercise for the reader to compute the output for the remaining values of i and j . As in the example above, when we perform convolution over the 4 × 4 image with a 3 × 3 kernel, we get a 2 × 2 feature map. This is because there are only 4 unique positions where we can place our filter inside this image. As convolutional neural networks were designed for image processing, it is common to represent the color values of the pixels with c color channels. For

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example, RGB values are represented with three channels. The general form of the convolution layer map for a m × n × c input tensor and outputs m × n × H (with stride 1 and padding) is θ : Rm×n×c → Rm×n×H . Writing ⎛ ⎞ f1 ⎜ .. ⎟ f =⎝.⎠

(8.72)

fc we can then write the layer map as θ (f ) = K ∗ f + b,

(8.73)

where K ∈ R[(2k+1)×(2k+1)]×H ×c , b ∈ Rm×n×H given by b := 1m×n ⊗ b and 1m×n is a m × n matrix with all elements being 1. In component form, the operation (8.73) is [θ (f )]j =

c [K]i,j ∗ [f ]i + bj ,

j ∈ {1, . . . , H },

(8.74)

i=1

where [·]i,j contracts the 4-tensor to a 2-tensor by indexing the ith third component and j th fourth component of the tensor and for any g ∈ Rm×n and H ∈ R(2k+1)×(2k+1) [H ∗ g]i,j =

k

Hk+1+p,k+1+q gi+p,j +q , i ∈ {1, . . . , m}, j ∈ {1, . . . .n}.

p,q=−k

(8.75) By analogy to a fully connected feedforward architecture, the weights in the layer are given by the kernel tensor, K, and the biases, b are H -vectors. Instead of a semi-affine transformation, the layer is given by an activated convolution σ (θ (f)). Furthermore, we note that not all neurons in the two consecutive layers are connected to each other. In fact, only the neurons which correspond to inputs within a 2k + 1 × 2k + 1 square connect to the same output neuron. Thus the filter size controls the receptive field of each output. We note, therefore, that some neurons share the same weights. Both of these properties result in far fewer parameters to learn than a fully connected feedforward architecture. Padding is needed to extend the size of the image f so that the filtered image has the same dimensions as the original image. Specifically padding means how to choose fi+p,j +q when (i + p, j + q) is outside of {1, . . . , m} or {1, . . . , n}. The following three choices are often used

5 Convolutional Neural Networks

fi+p,j +q

⎧ ⎪ ⎪ ⎨0, = f(i+p) ⎪ ⎪ ⎩f

263

zero padding, (mod m),(s+q) (mod n) ,

|i−1+p|,|j −1+q| ,

(8.76)

periodic padding, reflected padding,

if i+p ∈ / {1, . . . , m} or j + q ∈ / {1, . . . , n}.

(8.77)

Here d (mod m) ∈ {1, · · · , m} means the remainder when d is divided by m. The operation in Eq. 8.75 is also called a convolution with stride 1. Informally, we performed the convolution by sliding the image area by a unit increment. A common choice in CNNs is to take s = 2. Given an integer s ≥ 1, a convolution with stride s for f ∈ Rm×n is defined as [K ∗s f ]i,j =

k

Kp,q fs(i−1)+1+p,s(j −1)+1+q , i ∈ {1, . . . , %

p,q=−k

n j ∈ {1, . . . , % &}. s Here % ms & denotes the smallest integer greater than

m &}, s (8.78)

m s.

5.3 Pooling Data with high spatial structure often results in observations which have similar values within a neighborhood. Such a characteristic leads to redundancy in data representation and motivates the use of data reduction techniques such as pooling. In addition to a convolution layer, a pooling layer is a map: R¯ +1 : Rm ×n → Rm+1 ×n+1 .

(8.79)

One popular pooling is the so-called average pooling Ravr which can be a convolution with stride 2 or bigger using the kernel K in the form of ⎞ ⎛ 111 1⎝ K= 1 1 1⎠ . 9 111

(8.80)

Non-linear pooling operator is also used, for example, the (2k + 1) × (2k + 1) max-pooling operator with stride s as follows: [Rmax (f )]i,j =

max {fs(i−1)+1+p,s(j −1)+1+q }.

−k≤p,q≤k

(8.81)

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5.4 Dilated Convolution In addition to image processing, CNNs have also been successfully applied to time series. WaveNet, for example, is a CNN developed for audio processing (van den Oord et al. 2016). Time series often displays long-term correlations. Moreover, the dependent variable(s) may exhibit non-linear dependence on the lagged predictors. The WaveNet architecture is a non-linear p-autoregression of the form yt =

p

φi (xt−i ) + t

(8.82)

i=1

where the coefficient functions φi , i ∈ {1, . . . , p} are data-dependent and optimized through the convolutional network. To enable the network to learn these long-term, non-linear, dependencies Borovykh et al. (2017) use stacked layers of dilated convolutions. A dilated convolution effectively allows the network to operate on a coarser scale than with a normal convolution. This is similar to pooling or strided convolutions, but here the output has the same size as the input (van den Oord et al. 2016). In a dilated convolution the filter is applied to every dth element in the input vector, allowing the model to efficiently learn connections between far-apart data points. For an architecture with L layers of dilated convolutions ∈ {1, . . . , L}, a dilated convolution outputs a stack of “feature maps” given by [K () ∗d () f (−1) ]i =

k p=−k

(−1)

Kp() fd () (i−1)+1+p , i ∈ {1, . . . , %

m &}, d ()

(8.83)

where d is the dilation factor and we can choose the dilations to increase by a factor of two: d () = 2−1 . The filters for each layer, K () , are chosen to be of size 1 × (2k + 1) = 1 × 2. An example of a three-layer dilated convolutional network is shown in Fig. 8.6. Using the dilated convolutions instead of regular ones allows the output y to be influenced by more nodes in the input. The input of the network is given by the time series X. In each subsequent layer we apply the dilated convolution, followed by a non-linearity, giving the output feature maps f () , ∈ {1, . . . , L}. Since we are interested in forecasting the subsequent values of the time series, we will train the model so that this output is the forecasted time series Yˆ = {yˆt }N t=1 . The receptive field of a neuron was defined as the set of elements in its input that modifies the output value of that neuron. Now, we define the receptive field r of the model to be the number of neurons in the input in the first layer, i.e. the time series, that can modify the output in the final layer, i.e. the forecasted time series. This then depends on the number of layers L and the filter size 2k + 1, and is given by

5 Convolutional Neural Networks

265

Fig. 8.6 A dilated convolutional neural network with three layers. The receptive field is given by r = 8, i.e. one output value is influenced by eight input neurons. Source: van den Oord et al. (2016)

r := 2L−1 (2k + 1).

(8.84)

In Fig. 8.6, the receptive field is given by r = 8, one output value is influenced by eight input neurons.

•

? Multiple Choice Question 3

Which of the following statements are true: a. CNNs apply a collection of different, but equal width, filters to the data before using a feedforward network for regression or classification. b. CNNs are sparse networks, exploiting locality of the data, to reduce the number of weights. c. A dilated CNN is appropriate for multi-scale time series analysis—it captures a hierarchy of patterns at different resolutions (i.e., dependencies on past lags at different frequencies, e.g. days, weeks, months) d. The number of layers in a CNN is automatically determined during training.

5.5 Python Notebooks ML_in_Finance-1D-CNNs.ipynb demonstrates the application of 1D CNNs to predict the next element in a uniform sequence of integers. The CNN uses a sequence length of 50 and 4 kernels each of width 5. See Exercise 8.7 for a programming challenge involving applying this 1D CNN for time series to the HFT dataset described in the previous section on RNNs. For completeness, ML_in_Finance-2D-CNNs.ipynb demonstrates the application of a 2D CNN to image data from the MNIST dataset. Such an architecture might be appropriate for learning volatility surfaces but is not demonstrated here.

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6 Autoencoders An autoencoder is a self-supervised deep learner which trains the architecture to approximate the identity function, Y = F (Y ), via a bottleneck structure. This means we fit a model Yˆ = FW,b (Y ) which aims to very efficiently concentrate the information required to recreate Y . Put differently, an autoencoder is a form of compression that creates a much more cost-effective representation of Y . Its output layer has the same number of nodes as the input layer, and the cost function is some measure of the reconstruction error, Y − Yˆ . Autoencoders are often used for the purpose of dimensionality reduction and noise reduction. A simple autoencoder that implements dimensionality reduction is a feedforward autoencoder with at least one layer that has a smaller number of nodes, which functions as a bottleneck. After training the neural network using back-propagation, it is separated into two parts: the layers up to the bottleneck are used as an encoder, and the remaining layers are used as a decoder. In the simplest case, there is only one hidden layer (the bottleneck), and the layers in the network are fully connected. The compression capacity of autoencoders motivates their application in finance as a non-parametric, non-linear, analogue of the heavily used principal component analysis (PCA). It has been well known since the pioneering work of Baldi and Hornik (1989) that autoencoders are closely related to PCA. We follow Plaut (2018) and begin with a brief review of PCA and then show how exactly linear autoencoders enable PCA. Example 8.3

A Simple Autoencoder

For example, under a L2 -loss function, we wish to solve minimize ||FW,b (X) − Y ||2F W,B

subject to a regularization penalty on the weights and offsets. An autoencoder with two layers can be written as a feedforward network: Z (1) =f (1) (W (1) Y + b1) ), Yˆ =f (2) (W (2) Z (1) + b(2) ), where Z (1) is a low-dimensional representation of Y . We find the weights and biases so that the number of rows of W (1) equals the number of columns of W (2) and the number of rows is much smaller than the columns, which looks like the architecture in Fig. 8.7.

6 Autoencoders

267

Fig. 8.7 Autoencoders. Source: Van Veen, F. & Leijnen, S. (2019), “The Neural Network Zoo”, Retrieved from https://www. asimovinstitute.org/neuralnetwork-zoo

6.1 Linear Autoencoders In the case that no non-linear activation function is used, xi = W (1) yi + b(1) and yˆ i = W (2) xi + b(2) . If the cost function is the total squared difference between output and input, then training the autoencoder on the input data matrix Y solves min

W (1) ,b(1) ,W (2) ,b(2)

8

82 8 8 8Y − W(2) W (1) Y + b(1) 1TN + b(2) 1TN 8 . F

(8.85)

If we set the partial derivative with respect to b2 to zero and insert the solution into (8.85), then the problem becomes min

W (1) ,W (2)

8 82 8 8 8Y0 − W (2) W (1) Y0 8

F

Thus, for any b1 , the optimal b2 is such that the problem becomes independent of b1 and of y¯ . Therefore, we may focus only on the weights W (1) , W (2) . Linear autoencoders give orthogonal projections, even though the columns of the weight matrices are not orthogonal. To see this, set the gradients to zero, W (1) is the left Moore–Penrose pseudoinverse of W (2) (and W (2) is the right pseudoinverse of W (1) ):

−1 † W (1) = (W (2) ) = W (2)T W (2) (W (2) )T The minimization with respect to a single matrix is min

W (2) ∈Rn×m

8 82 8 8 8Y0 − W (2) (W (2) )† Y0 8

F

(8.86)

& '−1 (W (2) )T is the orthogonal The matrix W (2) (W (2) )† = W (2) (W (2) )T W (2) (2) projection operator onto the column space of W when its columns are not

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8 Advanced Neural Networks

necessarily orthonormal. This problem is very similar to (6.52), but without the orthonormality constraint. It can be shown that W (2) is a minimizer of Eq. 8.86 if and only if its column space is spanned by the first m loading vectors of Y. The linear autoencoder is said to apply PCA to the input data in the sense that its output is a projection of the data onto the low-dimensional principal subspace. However, unlike actual PCA, the coordinates of the output of the bottleneck are correlated and are not sorted in descending order of variance. The solutions for reduction to different dimensions are not nested: when reducing the data from dimension n to dimension m1 , the first m2 vectors (m2 < m1 ) are not an optimal solution to reduction from dimension n to m2 , which therefore requires training an entirely new autoencoder.

6.2 Equivalence of Linear Autoencoders and PCA Theorem The first m loading vectors of Y are the first m left singular vectors of the matrix W (2) which minimizes (8.86). A sketch of the proof now follows. We train the linear autoencoder on the original dataset Y and then compute the first m left singular vectors of W (2) ∈ Rn×m , where typically m 0 is the trading volume of the sold stock, Zt ∼ N (0, 1) is a standard Gaussian noise, and σ is the stock volatility. The problem faced by the broker is to find an optimal partitioning of the sell order into smaller blocks of shares that are to be executed sequentially, for example, each minute over the T = 10 min time period. This can also be formulated as a Markov Decision Process (MDP) problem (see Sect. 3 for further details) with a state variable Xt given by the number of shares outstanding (plus potentially other relevant variables such as limit order book data). For a one-step reward rt of selling at shares at step t, we could consider a risk-adjusted payoff given by the total expected payoff μ := at St penalized by the variance of the remaining inventory price at the next step t + 1: rt = μ − λVar [St+1 Xt+1 ]. We will return to the optimal stock execution problem later in this chapter, after we introduce the relevant mathematical constructs.

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9 Introduction to Reinforcement Learning

2.2 Value and Policy Functions In addition to an action chosen by the agent, its local reward depends on the state St of the environment. Different states of the environment can have different amount of attractiveness, or value, for the agent. Certain states may not exhibit any good options to receive high rewards. Furthermore, states of the environment in general change over a multi-step sequence of actions taken by the agent, and their future may be partly driven by their present state (and possibly actions of the agent). Therefore, reinforcement learning uses the concept of a value function as a numerical value of attractiveness of a state St for the agent, with a view on a multi-step character of an optimization problem faced by the agent. To relate it to the goal of reinforcement learning of maximization the cumulative reward over a period of time, a value function can be specified as a mean (expected) cumulative reward, that can be obtained by starting from this state, over the whole period (but as we will see below, there are other choices too). However, such a quantity would be under-specified as it stands, because rewards depend on actions at , in addition to their dependence on the state St . If we want to define the value function of a current (time-t) expected value of the cumulative reward obtained in T steps, we must know beforehand how the agent should act in any given possible state of the environment. This rule is specified by a policy function πt (St ) for how the agent should act at time t given the state St of the environment. A policy function may be a deterministic function of the state St , or alternatively it can be a probability distribution over a range of possible actions, specified by the current value of St . A value function V π (St ) is therefore a function of the current state St and a functional of the policy π .

2.3 Observable Versus Partially Observable Environments Finally, to complete the list of main concepts of reinforcement learning, we have to specify the notion of the state St , as well as define the law of its evolution. This process of an agent perceiving the environment and taking actions is extended over a certain period of time. During this period, the state of the environment changes. These changes might be determined by the previous history of the environment, and in addition, can be partially driven by some random factors, as well as an agent’s own actions. An immediate problem to be addressed is therefore how we model evolution of the environment. We start by describing autonomous evolution of the environment without any impact from the agent, and then generalize below to the case when such impact, i.e. a feedback loop, is present. In a most general form, the joint probability p0:T = p(s0 , . . . , sT −1 ) of a particular path (S0 = s0 , . . . , ST −1 = sT −1 ) can be written as follows:

2 Elements of Reinforcement Learning

p(s0 , s1 , . . . , sT −1 ) =

287 T −1

p(si |s0:i−1 ).

(9.3)

i=1

Note that this expression does not make any assumptions about the true datagenerating process—only the composition law of joint probabilities is used here. Unfortunately this expression is too general and useless in practice. This is because, for most problems of practical interest, the number of time steps encountered is in the tens or hundreds. Modeling path probabilities for sequences of states only relying on this general expression would lead to an exponential growth of the number of model parameters. We have to make some further assumptions in order to have a practically useful sequence model for the evolution of the environment. The simplest, and in many practical cases a reasonable “first order approximation” approach is to additionally assume Markovian dynamics where one assumes that conditional probabilities p(xi |x0:i−1 ) depend only on K most recent values, rather than on the whole history: p(st |s0:t−1 ) = p(st |st−K:i−1 ).

(9.4)

The most common case is K = 1 where probabilities of states at time t only depend on the previously observed values of the state. Following the common tradition in the literature, unless specifically mentioned, in what follows we refer to K = 1 Markov processes as simply “Markov processes.” This is also the basic setting for more general Markov processes with K > 1: such cases can be still viewed as Markov processes with K = 1 but with an extended definition of a time-t state St → Sˆt = (St , St−1 , . . . , St−K ). If we assume Markov dynamics for the evolution of the environment, this results in tractable formulations for sequence modeling. But in many practical cases, dynamics of a system cannot be as simple as a Markov process with a sufficiently low value of K, say 1 < K < 10. For example, financial markets often have longer memories than 10 time steps. For such systems, approximating essentially non-Markov dynamics by Markov dynamics with K ∼ 10 may not be satisfactory. A better and still tractable way to model a non-Markov environment is to use hidden variables zt . The dynamics is assumed to be jointly Markov in the pair (st , zt ) so that path probabilities factorize into the product of single-step probabilities: p(s0:T −1 , z0:T −1 ) = p(s0 , s0 )

T −1

p(zt |zt−1 )p(st |zt ).

(9.5)

t=1

Such processes with joint Markov dynamics in a pair (st , zt ) of an observable and unobservable components of a state are called hidden Markov models (HMM). Note that the dynamics of the marginal xt alone in HMMs would be in general nonMarkovian.

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9 Introduction to Reinforcement Learning

Remarkably, introducing hidden variables zt , we may construct models that both produce rich dynamics of observables, and have a substantially lower number of parameters than we would need with order K Markov processes. This means that such models might need considerably less data for training, and may behave better out-of-sample than Markov models. At the same time, models that are Markov in the pair (st , zt ) can be implemented in computationally efficient ways. Multiple examples in applications to speech and text recognition, robotics, finance demonstrated that HMMs are able to produce highly complex and sufficiently realistic sequences and time series. The key question in any type of HMM model is how to model a hidden state zt . Should it have a discrete or a continuous distribution? How many states are there? How should we specify the dynamics of hidden states, etc? While these are all practically important questions, here we want to focus on the conceptual aspect of modeling. First, introduction of hidden variables zt has a strong intuitive appeal. Traditionally, many important factors, e.g. political risk, are left outside of financial models, therefore they are “unknown” to the model. Treating such unknown risks as a de-correlated noise at each time step may be insufficient because such hidden risk factors usually include strong autocorrelations. This provides a second motivation to incorporate hidden processes in modeling of financial markets as not only a conventional tool to account for complex time dependencies of observable quantities xt , but also on their own, as a way to account for risk factors that we do not directly include in an observable state of the model. HMMs with their Markov dynamics in the pair (xt , zt ) provide a rather flexible set of non-Markov dynamics in observables xt . Even richer types of non-Markov dynamics can be obtained with recurrent extensions (such as RNN and LSTM neural networks) where hidden state probabilities depend on a long history of previous hidden states rather than just on a single last hidden state. While this implies that hidden variables might be very useful for financial machine learning, modeling decision-making of an agent in such a partially observable environment is more difficult than when an environment is fully observable. Therefore, in the rest of this chapter we will deal with models of decision-making within fully observable systems whose dynamics are assumed to be Markovian. The agent’s actions are added to the framework of Markov modeling, producing Markov Decision Process (MDP) based model. We will consider this topic next. Example 9.2

Portfolio Trading and Reinforcement Learning

The problem of multi-period portfolio management can be described as a problem of stochastic optimal control. Consider a portfolio of stocks and a single Treasury bond, where stocks are considered risky investments with random returns, while the Treasury bond is a risk-free investment with a fixed return determined by a risk-free discount rate. Let pn (t) with n = 1, . . . , N be investments at time t in N different stocks, and bt is the investment in the (continued)

3 Markov Decision Processes

Example 9.2

289

(continued)

bond. Let Xt be a vector of all other relevant portfolio-specific and marketwide dynamic variables that may impact an investment decision at time t. The vector Xt may, e.g., include market prices of all stocks in the portfolio, plus market indices such as SPY500 and various sector indices, macroeconomic factors such as the inflation rate, etc. The total state vector for such system is then st = (pt , bt , Xt ). The action at would be an (N + 1)-dimensional vector of capital allocations to all stocks and the bond at time t. If all components of vector Xt are observable, we can model its dynamics as Markov. Otherwise, if some components of vector Xt are non-observable, we can use an HMM formulation for the dynamics. The dynamics of some components of Xt can be partially affected by actions of the trader agent, for example, market prices of stock can be moved by large trades via market impact mechanisms. A model of the interaction of the trader agent and its environment (the “market”) can therefore include feedback loop effects. The objective of multi-step portfolio optimization is to maximize the expected cumulative reward. We can, e.g., consider one-step random rewards given by the one-step portfolio return r$ (st , at ) at time step t, penalized by the one-step variance of the return: R(st , at ) = r$ (st , at ) − λVar [r$ (st , at )], where λ is a risk-aversion rate. Taking the expectation of this random reward, we recover the classical Markowitz quadratic reward (utility) function for portfolio optimization. Therefore, reinforcement learning with random rewards R(st , at ) extends Markowitz single-period portfolio optimization to a samplebased, multi-period setting.

3 Markov Decision Processes Markov Decision Process models extend Markov models by adding new degrees of freedom describing controls. In reinforcement learning, control variables can describe agents’ actions. Controls are decided upon by the agent, and via the presence of a feedback loop, can modify the future evolution of the environment. When we embed the idea of controls with a feedback loop into the framework of Markov processes, we obtain Markov Decision Process (MDP) models. The MDP framework provides a stylized description of goal-directed learning from interaction. It describes the agent–environment interaction as message-passing of three signals: a signal of actions by the agent, a signal of the state of an environment, and a signal defining the agent’s reward, i.e. the goal. In mathematical terms, a Markov6Decision Process is defined 7 by a set of discrete time steps t0 , . . . , tn and a tuple S, A(s), p(s |s, a), R, γ with the following elements. First, we have a set of states S, so that each observed state St ∈ S. The

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9 Introduction to Reinforcement Learning

space S can be either discrete or continuous. If S is finite, we have a finite MDP, otherwise we have a MDP with a continuous state space. Second, a set of actions A(s) defines possible actions At ∈ A(s) that can be taken in a state St = s. Again, the set A(s) can be either discrete or continuous. In the former case, we have a MDP model with a discrete action space, and in the latter case we obtain a continuous-action MDP model. Next, a MDP is specified by transition probabilities p(s |s, a) = p(St = s |St−1 = s, at−1 = a) of a next state St given a previous state St−1 and an action At−1 taken in this state. Slightly more generally, we may specify a joint probability of a next state s and reward r ∈ R where R is a set of all possible rewards: p(s , r|s, a) = Pr St = s , Rt = r|St−1 = s, at−1 = a , p(s , r|s, a) = 1, ∀s ∈ S, a ∈ A.

(9.6)

s ∈S,r∈R

This joint probability specifies the state transition probabilities p(s |s, a) = Pr St = s |St−1 = s, at−1 = a = p(s , r|s, a),

(9.7)

r∈R

as well as the expected reward function r : S × A × S → R that gives the expected value of a random reward Rt received in the state St−1 = s upon taking action at−1 = a: r(s, a, s ) = E Rt |St−1 = s, At−1 = a, St = s p St = s , Rt = r|St−1 = s, at−1 = a = . r p [St = s |St−1 = s, at−1 = a]

(9.8)

r∈R

Finally, a MDP needs to specify a discount factor γ which is a number between 0 and 1. We need the discount factor γ to compute the total cumulative reward given by the sum of all single-step rewards, where each next term gets an extra power of γ in the sum: R(s0 , a0 ) + γ R(s1 , a1 ) + γ 2 R(s2 , a2 ) + . . . .

(9.9)

This means that as long as γ < 1, receiving a larger reward now and a smaller reward later is preferred to receiving a smaller reward now and a larger reward later. The discount factor γ just controls by how much the first scenario is preferable to the second one. This means that the discount factor for a MDP plays a similar role to a discount factor in finance, as it reflects the time value of rewards. Therefore, in financial application the discount factor γ can be identified with a continuously-

3 Markov Decision Processes

291

Fig. 9.1 The causality diagram for a Markov Decision Process

compounded interest rate. Note that for infinite-horizon MDPs, a value γ < 1 is required in order to have a finite total reward.2 A diagram describing a Markov Decision Process is shown in Fig. 9.1. The blue circles show the evolving state of the system St at discrete-time steps. These states are connected by arrows representing causality relations. We have only one arrow that enters each blue circle from a previous blue circle, emphasizing a Markov property of the dynamics: each next state depends only on the previous state, but not on the whole history of previous states. The green circles denote actions At taken by the agent. The upwards pointing arrow denote rewards Rt received by the agent upon taking actions At . The goal in a Markov Decision Process problem or in reinforcement learning is to maximize the expected total cumulative reward (9.9). This is achieved by a proper choice of a decision policy that specifies how the agent should act in each possible state.

3.1 Decision Policies Let us now consider how we can actually achieve the goals of maximizing the expected total rewards for Markov Decision Processes. Recall that the goal of reinforcement learning is to maximize the expected total reward from all actions performed in the future. Because this problem has to be solved now, but actions will be performed in the future, to solve this problem, we have to define a “policy”. A policy π(a|s) is a function that takes a current state St = s, and translates it into an action At = a. In other words, this function maps a state space onto an action space in the Markov Decision Process. If a system is currently in a state described by a vector St , then the next action At is given by a policy function π(St ). If the

2 An

alternative formulation for infinite-horizon MDPs is to consider maximization of an average reward rather than total reward. Such approach allows one to proceed without introducing a discount factor. We will not pursue reinforcement learning with average rewards in this book.

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9 Introduction to Reinforcement Learning

policy function is a conventional function of its argument St , then the output At will be a single number. For example, if the policy function is π(St ) = 0.5St , then for each possible value of St we will have one action to take. Such specification of a policy is called a deterministic policy. Another way to analyze policies in MDPs is to consider stochastic policies. In this case, a policy π(a|s) describes a probability distribution rather than a function. For example, suppose that there are two actions a0 and a1 , then the stochastic policy might be given by the logistic function π0 := π(a = a0 |s) = σ (θ T s) = 1 and π1 := π(a = a1 |s) = 1 − π0 . 1+exp (−θ T s) Now we take a look at differences between these two specifications in slightly more detail. First, consider deterministic policies. In this case, the action At to take is given by the value of a deterministic policy function π(St ) applied to a current state St . If the RL agent finds itself in the same state St of the system more than once, each time it will act exactly the same way. How it will act depends only on the current state St , but not on any previous history. This assumption is made to ensure consistency with the Markov property of the system dynamics, where probabilities to observe specific future states depend only on the current state, but not on any previous states. It can actually be proven that an optimal deterministic policy π always exists for a Markov Decision Process, so our task is simply to identify it among all possible deterministic policies. As long as that is the case, it might look like deterministic policies is all we ever need to solve Markov Decision Processes. But it turns out that the second class of policies, namely stochastic policies, are also often useful for reinforcement learning. For stochastic policies, a policy π(a|s) becomes a probability distribution over possible actions At = a. This distribution may depend on the current value of St = s as a parameter of such distribution. So, if we use a stochastic policy instead of a deterministic policy, when an agent visits the same state again, it might take a different action from an action it took the last time in this state. Note that the class of stochastic policies is wider than the class of deterministic policies, as the latter can always be thought as limits of stochastic policies where a distribution collapses to a Dirac delta-function. For example, if a stochastic policy is given by a Gaussian distribution with variance σ 2 , then a deterministic Dirac-like policy can be obtained in this setting by taking the limit σ 2 → 0. We might consider stochastic policies as a more general specification, but why would we want to consider such stochastic policies? It turns out that in certain cases, introducing stochastic policies might be an unnecessary complication. In particular, if we know transition probabilities in the MDP, then we can just consider only deterministic policies to find an optimal deterministic policy. For this, we just need to solve the Bellman equation that we introduce in the next section. But if we do not know the transition probabilities, we must either estimate them from data and use them to solve the Bellman equation, or we have to rely on samples, following the reinforcement learning approach. In this case, randomization

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of possible actions following some stochastic policy may provide some margin for exploration—for a better estimation of the model. A second case when a stochastic policy may be useful is when instead of a fully observed (Markov) process, we use a partially observed dynamic model of an environment which can be implemented as, e.g. a HMM process. Introducing additional control variables in this model would produce a Partially Observable Markov Decision Process, or POMDP for short. Stochastic policies may be optimal for POMDP processes, which is different from a fully observable MDP case where the best policy can always be found within the class of deterministic policies. While many problems of practical interest in finance would arguably lend themselves to a POMDP formulation, we leave such formulations outside the scope of this book where we rather focus on reinforcement learning for fully observed MDP settings.

3.2 Value Functions and Bellman Equations As mentioned above, a common auxiliary task of learning by acting is to evaluate a given state of the environment. Because it is defined by a sequence of actions taken by the agent, we want to relate it to local rewards obtained by the agent. One way to define the value function V π (s) for policy π is to specify it as an expected total reward that can be obtained starting from state St = s and following policy π : Vtπ (s)

=

Eπt

T −t−1 i=0

γ R (St+i , at+i , St+i+1 ) St = s , i

(9.10)

where R (St+i , at+i , St+i+1 ) is a random future reward at time t + i, T is the planning time horizon (an infinite-horizon case corresponds to T = ∞), and Eπt [ ·| St = s] means time-t averaging (conditional expectation) over all future states of the world given that future actions are selected according to policy π . The value function (9.10) can be used for two main settings for reinforcement learning. For episodic tasks, the problem of the agent learning naturally breaks into episodes which can be either of fixed or variable length T < ∞. For such tasks, using a discount factor γ for future rewards is not strictly necessary, and we could set there γ = 1 if so desired. On the other hand, for continuing tasks, the agent–environment interaction does not naturally break into episodes. Such tasks corresponds to the case T = ∞ in (9.10), which makes it necessary to keep γ < 1 to ensure convergence of an infinite series of rewards. The state-value function Vtπ (s) is therefore given by a conditional expectation of the total cumulative reward, also known in reinforcement learning as the random return Gt :

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Gt =

T −t−1

γ i R (St+i , at+i , St+i+1 ) .

(9.11)

i=0

For each possible state St = s, the state-value function V π (s) gives us the “value” of this state for the task of maximizing the total reward using policy π . The statevalue function Vtπ (s) is thus a function of s and a functional (i.e., a function of a function) of a decision policy π . For time-homogeneous problems, the time index can be omitted, Vtπ (s) → V π (s). Similarly, we can specify a value of simultaneously being in state s and take action a as a first action, and following policy π for all the following actions. This defines the action-value function Qπ (s, a): Qπt (s, a) = Eπt

T −t−1 i=0

γ i R (St+i , at+i , St+i+1 ) St = s, at = a .

(9.12)

Note that in comparison to the state-value function Vtπ (s), the state-value function Qπ (s, a) depends on an additional argument a which is an action taken at time step t. While we may consider arbitrary inputs a ∈ A to the action-value function Qπ (s, a), it is of a particular interest to consider a special case when the first action a is also drawn from policy π(a|s). In this case, we obtain a relation between the state-value function and the action-value function: π(a|s)Qπt (s, a). (9.13) Vtπ (s) = Eπ Qπt (s, a) = a∈A

(Note that while we assume a finite MDP formulation in this section, the same formulas applied to continuous-state MDPs model provided by replace sums by integrals). Both the state-value function Vtπ (s) and action-value function Qπt (s, a) are given by conditional expectations of the return Gt from states. Therefore, they could be estimated directly from data if we have observations of total returns Gt obtained in different states. The simplest case arises for a finite MDP model. Let us assume that we have K discrete states, and for each state, we have data referring observed sequences of states, actions, and rewards obtained while starting in each one of the K states. Then values functions in different states can be estimated empirically using these observed (or simulated) sequences. Such methods are referred to in reinforcement learning literature as Monte Carlo methods. Another useful way to analyze value functions is to rely on their particular structure defined as an expectation of cumulative future rewards. As we will see next, this can be used to derive certain equations for the state-value function (9.10) and action-value function (9.12).

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Consider first the state-value function (9.10). We can separate the first term from the sum, and write the latter as the first term plus a similar sum but starting with the second term:

T −t−1 π π π i Vt (s) = Et [R (s, at , St+1 )] + γ Et γ R (St+i , at+i , St+i+1 ) St = s . i=1

Note that the first term in the right-hand side of this equation is just the expected reward from the current step. The second term is the same as the expectation of the value function at the next time step t + 1 in some other state s , multiplied by the discount factor γ . This gives & ' π (s ). Vtπ (s) = Eπt Rt s, a, s + γ Eπt Vt+1

(9.14)

Equation (9.14) then gives an expression for a value of the state as a sum of immediate expected reward and a discounted expectation of the expected next state value. Note further that the conditional time-t expectations Eπt [·] in this expression involve conditioning on the current state St = s. Equation (9.14) thus presents a simple recursive scheme that enables computation of the value function at time t in terms of its future values at time t + 1 by going backward in time, starting with t = T − 1. This relation is known as the “Bellman equation for the value function”. Later on we will introduce a few more equations that are also called Bellman equations. It was proposed in 1950s by Richard Bellman in the context of his pioneering work on dynamic programming. Note that this is a linear equation, because the second expectation term is a linear functional of π (s ). In particular, for a finite MDP with N distinct states, the Bellman equation Vt+1 produces a set of N linear equations defining the value function at time t for each state in terms of expected immediate rewards, and transition probabilities to states at π (s ) that enter the expectation in its time t + 1 and next-period value functions Vt+1 right-hand side. If transition probabilities are known, we can easily solve this linear system using methods of linear algebra. Finally, note that we could repeat all steps leading to the Bellman equation (9.14), but starting with the action-value function rather than the state-value function. This produces the Bellman equation for the action-value function: & ' π Qπt (s, a) = Eπt Rt s, a, s + γ Eπt Vt+1 (s ) .

(9.15)

Similarly to (9.14), this is a linear equation that can be solved backward in time using linear algebra for a finite MDP model.

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•

> Learning with a Finite vs Infinite Horizon While an MDP problem is formulated in the same way for both cases of a finite T < ∞ or infinite T → ∞ time horizon, computationally they produce different algorithms. For infinite-horizon MDP problems with rewards that do not explicitly depend on time, a state- and action-value function should also not depend explicitly on time: Vt (st ) → V (st ) and similarly for Gt (st , at ) → G(st , at ), which expresses time-invariance of such problem. For time-invariant problems, the Bellman equation (9.15) becomes a fixedpoint equation for the same function Q(s, t) rather than a recursive relation between different functions Qt (st , at ) and Qt+1 (st+1 , at+1 ). While many of existing textbooks and other resources are often focused on presenting and pursuing algorithms for MDP and RL for a time-independent setting, including analyses of convergence, a proper question to ask is which one of the two types of MDP problem is more relevant for financial applications? We tend to suggest that finite-horizon MDP problems are more common in finance, as most of goals in quantitative finance focus on performance over some pre-specified time horizon such as one day, one month, one quarter, etc. For example, an annual performance time horizon of T = 1Y for a mutual fund or an ETF sets up a natural time horizon for planning in the MDP formulation. On the other hand, a given fixed time horizon may consist of a large number of smaller time steps. For example, a daily fund performance can result from multiple trades executed on a minute scale during the day. Alternatively, multi-period optimization of a long-term retirement portfolio with a time horizon of 30 years can be viewed at the scale of monthly or quarterly steps. For such cases, it may be reasonable to approximate an initially time-dependent problem with a fixed but large number of time steps by a time-independent infinite-horizon problem. The latter is obtained as an approximation of the original problem when the number of time steps goes to infinity. Therefore, an infinite-horizon MDP formulation can serve as a useful approximation for problem involving long sequences.

3.3 Optimal Policy and Bellman Optimality Next, we introduce an optimal value function Vt% . The optimal value function for a state is simply the highest value function for this state among all possible policies. So, the optimal value function is attained for some optimal policy that we will call π% . The important point here is that an optimal policy π% is optimal for all states of the system. This means that Vt% should be larger or equal than Vtπ with any other

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policy π , and for any state S, i.e. π% > π if Vt% := Vtπ% (s) ≥ V π (s), ∀s ∈ S. We may therefore express V% as follows: Vt% (s) := Vtπ% (s) = max Vtπ (s), ∀s ∈ S. π

(9.16)

Equivalently, the optimal policy π% can be determined in terms of the action-value function: Q%t (s, a) := Qπt % (s, a) = max Qπt (s, a), ∀s ∈ S. π

(9.17)

This function gives the expected reward from taking action a in state s, and following an optimal policy thereafter. The optimal action-value function can therefore be represented by the following Bellman equation that can be obtained from Eq. (9.15): % Q%t (s, a) = E%t Rt (s, a, s ) + γ E%t Vt+1 (s ) .

(9.18)

Note that this equation might be inconvenient to work with in practice, as it involves two optimal functions Q%t (s, a) and Vt% (s), rather than values of the same function at different time steps, as in the Bellman equation (9.14). However, we can obtain a Bellman equation for the optimal action-value function Q%t (s, a) in terms of this function only, if we use the following relation between two optimal value functions: Vt% (s) = max Q%t (s, a). a

(9.19)

We can now substitute Eq. (9.19) evaluated at time t + 1 into Eq. (9.18). This produces the following equation: Q%t (s, a)

=

E%t

% Rt (s, a, s ) + γ max Qt+1 (s , a ) . a

(9.20)

Now, unlike Eq. (9.18), this equation relates the optimal action-value function to its values at a later time moment. This equation is called the Bellman optimality equation for the action-value function, and it plays a key role in both dynamic programming and reinforcement learning. This expresses the famous Bellman’s principle of optimality, which states that optimal cumulative rewards should be obtained by taking an optimal action now, and following an optimal policy later. Note that unlike Eq. (9.18), the Bellman optimality equation (9.20) is a nonlinear equation, due to the max operation inside of the expectation. Respectively, the Bellman optimality equation is harder to solve than the linear Bellman equation (9.14) that holds for an arbitrary fixed policy π . The Bellman optimality equation is usually solved numerically. We will discuss some classical methods of solving it in the next sections.

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The Bellman equation for the optimal state-value function Vt% (s) can be obtained using Eqs. (9.18) and (9.19): % (s ) . Vt% (s) = max E%t Rt (s, a, s ) + γ Vt+1 a

(9.21)

Like Eq. (9.20), the Bellman optimality equation (9.21) for the state-value function is a non-linear equation due to the presence of the max operator. If the optimal state-value or action-value function is already known, then finding the optimal action is simple, and essentially amounts to “greedy” onestep maximization. The term “greedy” is used in computer science to describe algorithms that are based only on intermediate (single-step) considerations but not on considerations of longer-term implications. If we already know the optimal statevalue function Vt% (s), this implies that all actions at all time steps implied in this value function are already optimal. This still does not determine on its own what action should be chosen at a current time step. However, because we know that actions at the subsequent steps are already optimal, to find the best next action in this setting we need only perform a greedy one-step search that just takes into account the immediate successor states for the current state. In other words, when the optimal state-value function Vt% (s) is used, a one-step-ahead search for optimal action produces long-term optimal actions. With the optimal action-value function Q%t (s, a), the search of an optimal action at the current step is even simpler, and amounts to simply maximizing Q%t (s, a) with respect to a. This does not require using any information about possible successor states and their values. This means that all relevant information of the dynamics is already encoded in Q%t (s, a). The Bellman optimality equations (9.20) and (9.21) are central objects for decision-making under the formalism of MDP models. The methods of dynamic programming focus on exact or numerical solutions of these equations. Many (though not all) methods of reinforcement learning are also based on approximate solutions to the Bellman optimality equations (9.20) and (9.21). As we will see in more details later on, the main difference of these methods from traditional dynamic programming methods is that they rely on empirically observed transitions rather than on a theoretical model of transitions.

•

> Existence of Bellman Equations: Infinite-Horizon vs Finite Horizon Cases For time-homogeneous (infinite horizon) problems, a value function does not explicitly depend on time, and the Bellman equation (9.14) can be written in this case in a compact form T πV π = V π,

(9.22) (continued)

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where T π : RS → RS stands for the Bellman operator & π ' T V (s) = r(s, π(s)) + γ p(s , s, π(s))V (s ), ∀s ∈ S.

(9.23)

s ∈S

Therefore, for time-stationary MDP problems the Bellman equation becomes a fixed-point equation. If 0 < γ < 1, then T π is a maximum-norm contraction and the fixed-point equation (9.22) has a unique solution, see, e.g., Bertsekas (2012). Similarly, the Bellman optimality equation (9.21) can be written in a timestationary case as T %V % = V %,

(9.24)

where T % : RS → RS is for the Bellman optimality operator + & π ' p(s , s, a)V (s ) , ∀s ∈ S. T V (s) = max r(s, a) + γ a∈A

(9.25)

s ∈S

Again, if 0 < γ < 1, then T % is a maximum-norm contraction and the fixedpoint equation (9.22) has a unique solution, see Bertsekas (2012). For finite-horizon MDP problem the Bellman equations (9.14) and (9.21) become recursive relations between different functions Vtπ (st ) and π (s Vt+1 t+1 ). The existence of a solution for this case can be established by mapping the finite-horizon MDP problem onto a stochastic shortest-path (SSP) problem. SSP problems have a certain terminal (absorbing) state, and the problem of an agent is to incur a minimum expected total cost on a path to the terminal state. For details on existence of the Bellman equation for the SSP, see Vol II of Bertsekas (2012). In a finite-horizon MDP, time t can be thought of as a stage of the process, such that after N stages the system enters the absorbing state with probability one. The finite-horizon problem is therefore mapped onto a SSP problem with an augmented state s˜t = (st , t).

4 Dynamic Programming Methods Dynamic programming (DP), pioneered by Bellman (1957), is a collection of algorithms that can be used to solve problems of finding optimal policies when an environment is Markov and fully observable, so that we can use the formalism of Markov Decision Processes. Dynamic programming approaches work for finite MDP models with typically a low number of states, and moreover assume that the

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model of the environment is perfectly known. Both cases are rarely encountered in problems of practical interest where a model of the environment is typically not known beforehand, and a state space is often either discrete and high-dimensional, or continuous, or continuous with multiple dimensions. Methods of DP which find exact solutions for finite MDP become infeasible for such situations. Still, while they are normally not very useful for practical problems, methods of DP have fundamental importance for understanding other methods that do work in “real-world” applications and are applied in reinforcement learning, as well as in other related approaches such as approximate dynamic programming. For example, we may use DP methods to validate RL algorithms applied to simulated data with known probability transition matrix. Here we want to analyze the most popular DP algorithms. A common feature in all these algorithms is that they all rely on the notion of a state-value function (or action-value function, or both) as a condensed representation of quality of both policies and states. Respectively, extensions of such algorithms to (potentially high-dimensional) sample-based approached performed by RL methods are generically called the value function based RL. Some RL methods do not rely on the notion of a value function, and operate only with policies. In this book, we will focus mostly on the value function based RL but we shall occasionally see examples of an alternative approach based on “policy iteration.” In addition, all approaches considered in this section imply that a state-action space is discrete. While usually not explicitly specified, the dimensionality of a state space is assumed to be sufficiently low, so that the resulting computational algorithm would be feasible in practice. DP approaches are also used for systems with a continuous state by discretizing a range of values. For a multi-dimensional continuous state space, we could discretize each individual component, and then produce a discretized uni-dimensional representation by taking a direct product of individual grids, and indexing all resulting states. However, this approach is straightforward and feasible when the number of continuous dimensions is sufficiently low, e.g. do not exceed 3 or 4. For higher dimensional continuous problems a naive discretization by forming cross-products of individual grids produces an exponential explosion of a number of discretized steps, and quickly becomes infeasible in practice.

4.1 Policy Evaluation As we mentioned above, the state-value function Vtπ (s) gives the value of the current state St = s provided the agent follows policy π in choosing its actions. For a time-stationary MDP, which will be considered in this section, the timeindependent version of the Bellman equation (9.14) reads V π (s) = Eπt R (s, at , St+1 ) + γ V π (s ) ,

(9.26)

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while the time-independent version of the Bellman optimality equation (9.21) is V % (s) = max E%t R(s, a, s ) + γ V % (s ) . a

(9.27)

Thus, for time-stationary problems, the problem of finding the state-value function V π (s) for a given policy π amounts to solving a set of |S| linear equations, where |S| stands for the dimensionality of the state space. As solving such a system directly (in one step) involves matrix inversion, such a solution can be costly when the dimensionality |S| of the state-space grows. As an alternative, the Bellman equation (9.26) can be used to set up a recursive approach to solve it. While this produces a multi-step numerical algorithm to find the state-value function rather than an explicit one-step formulas obtained with the linear algebra approach, in practice it often works better than the latter. The idea is simply to view the Bellman equation (9.26) as a stationary point at convergence as k → ∞ of an iterative map indexed by steps k = 0, 1, . . ., which is obtained by applying the Bellman operator T π [V ] := Eπt R (s, at , St+1 ) + γ V (s ) ,

(9.28)

(π )

to the previous iteration Vk (s) of the value function. Here the lower index k enumerates iterations, and replaces the time index t which is omitted because we work in this section with time-homogeneous MDPs. This produces the following update rule: π Vkπ (s) = Eπt R (s, at , St+1 ) + γ Vk−1 (s ) .

(9.29)

The informal way to understand this relation is to think about a recursion as a sequential process. The sequential nature of the process can be mapped onto some notion of time. Therefore, if we formally replace T − t → k in the original time-dependent Bellman equation (9.14), it produces Eq. (9.29). The relation (9.29) is often referred to as the Bellman iteration. The Bellman iteration is proven to converge under some technical conditions on the Bellman operator (9.28). In particular, rewards need to be bounded to guarantee convergence of the map. For any given policy π , policy evaluation algorithm amounts to a repeated application of the Bellman iteration (9.29) starting with some initial guess, e.g. (π V0 (s) = 0. The iteration continues until convergence at a given tolerance level is achieved, or alternatively it can run for a pre-specified number of steps. Each iteration involves only linear operations, therefore it can be performed very fast. Recall here that the model of the environment is assumed to be perfectly known in the DP approach, therefore all expectations are linear and fast to compute. The method is scalable to high-dimensional discrete state spaces. While a policy evaluation algorithm can be quite fast in evaluating a single policy π , the ultimate goal of both dynamic programming and reinforcement learning approaches is to find the optimal policy π% . This opens the door to a plethora of different approaches. In one class of approaches, we consider a set of candidate

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policies {π } and we find the policy π% by selecting between them. These methods are called “policy iteration” algorithms, and they use the policy evaluation algorithms as their integral part. We will consider such algorithms next.

4.2 Policy Iteration Policy iteration is a classical algorithm for finite MDP models. Again, we consider here only stationary problems, therefore we again replace the time index by an iteration index VTπ−t (s) → Vk% (s). We can also omit the index π here as the purpose of this algorithm is to find the optimal π = π% . To produce a policy iteration algorithm, we need two features: a way to evaluate a given policy and a way to improve a given policy. Both can be considered subalgorithms in an overall algorithm. Note that the first component in this scheme is already available, and is given by the policy evaluation method just presented. Therefore, to have a complete algorithms, our only remaining task is to supplement this with a method to improve a given policy. To this end, consider the Bellman equation for the action-value function Qπ (s, a) = Et R(s, a, s ) + γ V π (s ) = p(s , r|s, a) R(s, a, s ) + γ V π (s ) . s ,r

(9.30) The current action a is a control variable here. If we follow policy π in choosing a, then the action taken is a = π(s), and the action value is Qπ (s, π(s)) = V π (s). This means that by taking different actions a ∼ π (with another policy π ) rather than prescribed by policy π , we can produce higher values Qπ (s, π (s)). It can be shown that this implies that the new policy also improves the state-value function, i.e. V π (s) ≥ V π (s) for all states s ∈ S (the latter statement is called the policy improvement theorem, see, e.g., Sutton and Barto (2018) or Szepesvari (2010) for details.) Now imagine we want to choose action a in Eq. (9.30) so that to maximize Qπ (s, a). Maximizing a → a% means that we find a value a% such that Qπ (s, a% ) ≥ Q(s, a) for any a = a% . This can be equivalently thought of as a greedy policy π that is given by π except for the state s, where it should be such that a% = π (s). If some greedy policy π = π can be found by maximizing Qπ (s, a), or equivalently the right-hand side of Eq. (9.30), then it will satisfy the policy improvement theorem, and then can be used to find the optimal policy via policy iteration. Therefore, an inexpensive search for a better greedy policy π by a local optimization over possible next actions a ∈ A is guaranteed to produce a sequence of policies that are either better than the previous ones, or at worst keep them unchanged. This observation underlies the policy iteration algorithm which proceeds as follows. We start with some initial policy π (0) . Often, a purely random initialization is used. After that, we repeat the following set of two calculations, for a fixed number of steps or until convergence:

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– Policy evaluation: For a given policy π (k−1) , compute the value function V (k−1) by solving the Bellman equation (9.26) – Policy improvement: Calculate new policy π (k) = arg max

a∈A

! " p(s |s, a) R(s, a, s ) + γ V (k−1) (s ) .

(9.31)

s

In words, at each iteration step we first compute the value function using the previous policy, and then update the policy using the current value function. The algorithm is guaranteed to converge for a finite state MDP with bounded rewards. Note that if the dimensionality of the state space is large, multiple runs of policy evaluation can be quite costly, because it would involve high-dimensional systems of linear equations. But many practical problems of optimal control involve large discrete state-action spaces, or continuous state-action space. In these settings, methods of DP introduced by Bellman (1957), and algorithms like policy iteration (or value iteration, to be presented next), do not work anymore. Reinforcement learning methods were developed in particular as a practical answer to such challenges.

4.3 Value Iteration Value iteration is another classical algorithm for finite and time-stationary MDP models. Unlike the policy iteration method, it bypasses the policy improvement stage, and uses a recursive procedure to directly find the optimal state-value function V % (s). The value iteration method works by applying the Bellman optimality equation as an update rule in an iterative scheme. In more detail, we start with initialization of the value function at some initial values V (s) = V (0) (s) for all states, with some choice of function V (0) (s), e.g. V (0) (s) = 0. Then we continue iterating the evaluation of the value function using the Bellman optimality equation as the definition of the update rule. That is, for each iteration k = 1, 2, . . ., we use the result of the previous iteration to compute the right-hand side of the equation: ! " " ! p(s , r|s, a) r +γ V (k−1) . V (k) (s)= maxE%t R(s, a, s ) + γ V (k−1) (s ) = max a

a

s ,r

(9.32) This can be thought of as combining the two steps of policy improvement and policy evaluation into one update step. Note that the new value iteration update rule (9.32) is similar to the policy evaluation update (9.29) except that it also involves taking a maximum over all possible actions a ∈ A. Now, there are a few ways to update the value function in such a value iteration. One approach is to complete re-computing the value function for all states s ∈ S,

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and then simultaneously update the value function over all states, V (k−1) (s) → V (k) (s). This is referred to as synchronous updating. The other approach is to update the value function V (k−1) (s) on the fly, as it is re-computed in the current iteration k. This is called asynchronous updating. Asynchronous updates are often used for problems with large state-action spaces. When only a relatively small number of states matter for an optimal solution, updating all states after a complete sweep, as is done with synchronous updates, might be inefficient for high-dimensional state-action spaces. For either way of updating, it can be proven that the algorithm converges to the optimal value function V % (s). After V % (s) is found, the optimal policy π% can be found using the same formula as before. As one can see, the basic algorithm is very simple, and works well, as long your state-action space is discrete and has a small number of states. However, similarly to policy iteration, the value iteration algorithm quickly becomes unfeasible in highdimensional discrete or continuous state spaces, due to exponentially large memory requirements. This is known as the curse of dimensionality in the DP literature.3 Given that the time needed for DP solutions to be found is polynomial in the number of states and actions, this may also produce prohibitively long computing times.4 For low-dimensional continuous state-action spaces, the standard approach enabling applications of DP is to discretize variables. This method can be applied only if the state dimensionality is very low, typically not exceeding three or four. For higher dimensions, a simple enumeration of all possible states leads to an exponential growth of the number of discretized states, making the classical DP approaches unfeasible for such problems due to memory and speed limitations. On the other hand, as will be discussed in details below, the RL approach relies on samples, which are always discrete-valued even for continuous distributions. When a sampling-based approach of RL is joined with some reasonable choices for a low-dimensional bases in such continuous spaces (i.e., using some methods of function approximation), RL is capable of working with continuously valued multidimensional states and action. Recall that DP methods aim for a numerically exact calculation of optimal value function at all points of a discrete state space. However, what is often needed in high-dimensional problems is an approximate way of computing the value functions using simultaneously a lower, and often much lower, number of parameters than the original dimensionality of the state space. Such methods are called approximate dynamic programming, and can be applied in situations when the model of the world is known (or independently estimated beforehand from data), but the dimensionality of a (discretized) state space is too large to apply the standard value or policy 3 While

high dimensionality is a curse for DP approaches as it makes them infeasible for highdimensional problems, with some other approaches this may rather bring simplifications, in which case the “curse of dimensionality” is replaced by the “blessing of dimensionality.” 4 Computing times that are polynomial in the number of states and actions are obtained for worst-case scenarios in DP. In practical applications of DP, convergence is sometimes faster than constraints given by worst-case scenarios.

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iteration methods. On the other hand, the reinforcement learning approach works directly with samples from data. When it is combined with a proper method of function approximation to handle a high-dimensional state space, it provides a sample-based RL approach to optimal control—which is the topic we will pursue next. Example 9.3

Financial Cliff Walking

Consider an over-simplified model of household finance. Let St be the amount of money the household has in a bank account at time t. We assume for 6 7N −1 simplicity that St can only take values in a discrete set S (i) i=0 . The account has to be maintained for T time steps, after which it should be closed, so T is the planning horizon. The zero level S (0) = 0 is a bankruptcy level—it has to be avoided, as reaching it means inability to pay on household’s liabilities. At each step, the agent can deposit to the account S (i) → S (i+1) (action a+ ), withdraw from the account S (i) → S (i−1) (action a− ), or keep the same amount S (i) → S (i) (action a0 ). The initial amount in the account is zero. For any step before the final step T , if the agent moves to the zero level S0 = 0, it receives a negative reward of −100, and the episode terminates. Otherwise, the agent continues for all T steps. Any action not leading to the zero level gets a negative reward of −1. At the terminal time, if the final state is ST > 0, the reward is -1, but if the account goes back to zero exactly at time T , i.e. ST = 0, the last action gets a positive reward of +10. The learning task is to maximize the total reward over T time steps (Fig. 9.2). The RL agent has to learn the optimal depository policy online, by trying different actions during a training episode. Note that while this is a time-dependent problem, we can map it onto a stationary problem with an episodic task and a target state, such as the original cliff walking problem in Sutton and Barto (2018).

Balance

Time

Optimal path

S

Bankruptcy

G

Fig. 9.2 The financial cliff walking problem is closely based on the famous cliff walking problem by Sutton and Barto (2018). Given a bank account with an initial zero balance (“start”), the objective is to deposit and deplete the account by a unit of currency so that the account ends with a zero balance at the final time step (“goal”). Premature depletion is labeled as a bankruptcy and terminates the game. Reaching the final time with a surplus amount in the account results in a penalty. At each time step, the agent may choose from a number of actions if the account is not zero: deposit (“U”), withdraw (“D”), or do nothing (“Z”). Transactions costs are imposed so that the optimal policy is to hold the balance at unity

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5 Reinforcement Learning Methods Reinforcement learning methods aim at the same goal of solving MDP models as do the DP methods. The main differences are in how the problems of data processing and computational design are approached. This section provides a brief overview of some of the most popular reinforcement learning methods for solving MDP problems. Three main classes of approaches to reinforcement learning are Monte Carlo methods, policy search methods, and value-based RL. The first two methods do not rely on Bellman equations, and thus do not have direct links to the Bellman equation introduced in this chapter. While we present a brief overview of Monte Carlo and policy search methods, most of the material presented in the later chapters of this book uses value-based RL. In what follows in later sections, we will often refer to value-based RL as simply “RL”. As we just mentioned, these RL approaches use the Bellman optimality equations (9.20) and (9.21); however, they proceed differently. With DP approaches, one attempts to solve these equations exactly. This is only feasible if a perfect model of the world is known and the dimensionality of the state-action space is sufficiently low. As we remarked earlier, both assumptions do not hold in most problems of practical interest. Reinforcement learning approaches do not assume that a perfect model of the world is known. Instead, they simply rely on actual data that are viewed as samples from a true data-generating distribution. The problem of estimating this distribution can be bypassed altogether with reinforcement learning. In particular, model-free reinforcement learning operates directly with samples of data, and relies only on samples when optimizing its policy. This is not to say, of course, that models of the future are useless for reinforcement learning. Model-based reinforcement learning approaches build an internal model of the world as a part of their ultimate goal of policy optimization. We will discuss model-based reinforcement learning later in this book, but in this section, we will generally restrict consideration to model-free RL. Related to the first principle of relying on the data is the second key difference of reinforcement learning methods from DP methods. Because data is always noisy, reinforcement learning cannot aim at an exact solution as the standard DP methods, but rather aim at some good approximate, rather than exact, solutions. Clearly, this does not prevent an exploration of the behavior of RL solutions when the number of data points becomes infinite. If we knew an exact model of the world, we could reconstruct it exactly in this limit. Therefore, theoretically sound (rather than purely empirically driven) RL algorithms should demonstrate convergence to known solutions in this asymptotic limit. In particular, if we deal with a sufficiently low-dimensional system, such solutions can be independently calculated using the standard DP methods. This can be used for testing and benchmarking RL algorithms, as will be discussed in more details in later sections of this book.

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Finally, the last key difference of reinforcement learning methods from DP is that they do not seek the best solution, they simple seek a “sufficiently good” solution. The main motivation for such a paradigm change is the “curse of dimensionality” that was mentioned above. DP methods for finite MDPs operate with tabular representations of value functions, rewards, and transition probabilities. Memory requirements and speed constraints make this approach infeasible for high-dimensional discrete or continuous state-action spaces. Therefore, when working with such problems, reinforcement learning approaches rely on function approximation for quantities of interest such as value functions or action policies. Reinforcement learning algorithms with function approximation will be discussed later in this section, after we introduce tabulated versions of these algorithms for finite MDPs with sufficiently low dimensionality of discrete-valued state and action spaces. The purpose of this section is to introduce some of the most popular RL algorithms for both finite and continuous-state MDP problems. We will start with methods developed for finite MDPs, and then later show how they can be extended to continuous state-action spaces using function approximation approaches.

•

? Multiple Choice Question 1

Select all the following correct statements: a. Unlike DP, RL needs to know rewards and transition probability functions. b. Unlike DP, RL does not need to know reward and transition probability functions, as it relies on samples.

(n) (n) (n) c. The information set Ft for RL includes a triplet Xt , at , Xt+1 for each step.

(n) (n) (n) (n) d. The information set Ft for RL includes a tuplet Xt , at , Rt , Xt+1 for each step.

5.1 Monte Carlo Methods Monte Carlo methods, as other methods of reinforcement learning, do not assume complete knowledge of the environment, nor do they rely on any model of the environment. Instead, Monte Carlo methods rely on experience, that is samples of states, actions, and rewards. When working with real data, this amounts to learning without any prior knowledge of the environment. Experience can also be simulated. In this case, Monte Carlo methods provide a simulation-based approach to solving MDP problems. Recall that the DP approach requires knowledge of exact transition probabilities to perform iteration steps in policy iteration or value iteration algorithms. With reinforcement learning Monte Carlo methods, only samples from these distributions are needed, but not their explicit form.

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Monte Carlo methods are normally restricted to episodic task with a finite planning horizon T < ∞. Rather than relying on Bellman equations, they operate directly with the definition of the action-value function Qπt (s, a)=Eπt

T −1 i=0

γ

t−i

R (St+i , at+i , St+i+1 ) St = s, at = a =Eπt [ Gt | St =s, at =a] , (9.33)

where Gt is the total return, see Eq. (9.11). If we have access to data consistent of (n) N set of T -step trajectories each producing return Gt , then we could estimate the action-value function at the state-action values (s, a) using the empirical mean: Qπt (s, a)

N " 1 ! (n) Qt St = s, at = a . ( N

(9.34)

n=1

Note that for each trajectory, a complete T -step trajectory should be observed, so that its return can be observed and used to update the action-value function. It is worth clarifying the meaning of index π in this relation. With the Monte Carlo estimation of Eq. (9.34), π should be understood as a policy that was applied when collecting the data. This means that this Monte Carlo method is an on-policy algorithm. On-policy algorithms are only able to learn an optimal policy from samples if these samples themselves are produced using the optimal policy. Conversely, off-policy algorithms are able to learn an optimal policy from data generated using other, sub-optimal policies. Off-policy Monte Carlo methods will not be addressed here, and the interested reader is referred to Sutton and Barto (2018) for more details on this topic. As indicated by Eq. (9.33), the action-value process should be calculated separately for each combination of a state and action in a finite MDP assumed here. The number of such combinations will be |S| · |A|. Respectively, for each combination of s and a from this set, we should only select those trajectories that encounter such a combination, and only include returns from these trajectories in the sum in Eq. (9.34). For every combination of (s, a), the empirical estimate (9.34) asymptotically converges to the exact answer in the limit N → ∞. Also note that these estimates are independent for different values of (s, a). This could be useful, as it enables a trivial parallelization of the calculation. On the other hand, independence of estimates for different pairs (s, a) means that this algorithm does not bootstrap, i.e. it does not use previous or related evaluations to estimate the action-value function at node (s, a). Such a method may miss some regularities observed or expected in true solutions (e.g., smoothness of the value function with respect to its arguments), and therefore may produce some spurious jumps in estimated statevalue functions produced due to noise in the data. Beyond empirical estimation of the action-value function as in Eq. (9.34) or the state-value function, Monte Carlo methods can also be used to find the optimal control, provided the Monte Carlo RL agent has access to a real-world or simulated environment. To this end, the agent should produce trajectories using different trial

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policies. For each policy π , a number of N trajectories are sampled using this policy. The action-value function is estimated using an empirical mean as in Eq. (9.34). After this, one follows with the policy improvement step which coincides with the greedy update of the policy iteration method: π (s) = arg maxa Qπ (s, a). The new policy is used to sample a new set of trajectories, and the process runs until convergence or for a fixed number of steps. Note that generating new trajectories corresponding to newly improved policies may not always be feasible. For example, an agent may only have access to one fixed set of trajectories obtained using a certain fixed policy. In such cases, one may resort to importance sampling techniques which use trajectories obtained under different policies to estimate the return under a given policy. This is achieved by re-weighting observed trajectories by likelihood ratio factors obtained as ratios of probabilities of observing given reward under the trial policy π and the policy π used in the data collection stage. Instead of updating the action-value function (or the state-value function) simultaneously after all N trajectories are sampled, which is a batch mode evaluation, we could convert the problem into an online learning problem where updates occur after observing each individual trajectory according to the following rule: Q(s, a) ← Q(s, a) + α [Gt (s, a) − Q(s, a)] ,

(9.35)

where 0 < α < 1 is a step-size parameter usually referred to as the “learning rate.” It can be shown that such iterative update converges to true empirical and theoretical averages in the limit N → ∞. Yet this update is not entirely real time, as it requires finishing each T -step trajectory, until its total return Gt can be used in the update (9.35). This may be inefficient, especially if multiple trajectory generations and evaluations are required as a part of policy optimization. As we will show below, there are other methods of learning that are free of this drawback.

5.2 Policy-Based Learning In value-based RL, the optimal policy is obtained from an optimal value function, and thus is not modeled separately. Policy-based reinforcement learning takes a different approach, and directly models policy. Unlike the value-based RL where we considered deterministic policies, policy-based RL operates with stochastic policies πθ (a|s) that define probability distributions over a set of possible actions a ∈ A, where θ defines parameters of this distribution. Recall that deterministic policies can be considered special cases of stochastic policies where a distribution of possible actions degenerates into a Dirac deltafunction concentrated in a single action prescribed by the policy: πθ (a|s) = δ(a − a% (s, θ )) where a% (s, θ ) is a fixed map from states to actions, parameterized by θ . With either deterministic or stochastic policies, learning is performed by tuning the free parameters θ to maximize the total expected reward.

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Policy-based methods are based on a simple relation commonly known as the “log-likelihood trick” which is obtained by computing the derivative of an expectation J (θ ) = Eπθ (a) [G(a)]. Here function G(a) can be arbitrary, but to connect to reinforcement learning, we will generally mean that G(a) stands for the expectation of the random return (9.11), which we write here as G(a) to emphasize its dependence on the actions taken. The gradient of the expectation with respect to parameters θ can be computed as follows: (

(

∇θ πθ (a) πθ (a)da =Eπθ (a) G(a)∇θ log πθ (a) . πθ (a) (9.36) This shows that the gradient of J with respect to θ is the expected value of the function G(a)∇θ log πθ (a). Therefore, if we can sample from the distribution πθ (a), we can compute this function and have an unbiased estimate of the gradient of G(a) by sampling. This is the reason the relation (9.36) is called the “log-likelihood trick”: it allows one to estimate the gradient of the functional J (θ ) by sampling or simulation. The log-likelihood trick underlies the simplest policy search algorithm called REINFORCE. The algorithm starts with some initial values of parameters θ0 and the iteration counter is k = 0. Using a size-step hyperparameter α, the update of θk amounts to first sampling ak ∼ pπk (a), and then updating the vector of parameters using the incremental version of Eq. (9.36) ∇θ J (θ )=

G(a)∇θ πθ (a)dz=

G(a)

θk+1 = θk + αk G(ak )∇θ log πθk (ak ).

(9.37)

Here α is a learning rate parameter defining the speed of updates along the negative of the gradient of G(a). The algorithm continues until convergence, or for a fixed number of steps. As one can see, this algorithm is very simple to implement as long as sampling from distribution πθ (a) is easy. On the other hand, Eq. (9.37) is a version of stochastic gradient descent which can be noisy and thus produce high variance estimators. The REINFORCE algorithm (9.37) is a pure policy search method that does not use any value function. A more sophisticated version of learning can be obtained where we simultaneously model both the policy and the action-value functions. Such methods are called actor-critic methods , where “actor” is an algorithm that generates a policy from a family πθ (a|x), and “critic” evaluates the results of applying the policy, expressing it in terms of a state-value or action-value function. Following such terminology, the REINFORCE algorithm could be considered an “actor-only” algorithm, while SARSA or Q-learning, to be presented below, could be viewed as “critic-only” methods. One advantage of policy-based algorithms is that they admit very flexible parameterizations of action policies, which can also work for continuous action spaces. One popular and quite general type of action policies is the so-called softmax in actions policy

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eh(s,a,θ) . h(s,a ,θ) a e

πθ (a|s) =

(9.38)

Functions h(s, a, θ ) in this expression would be interpreted as action preferences. They could be taken linear functions in parameters θ , for example h(s, a, θ ) = θ T U (s, a),

(9.39)

where U (s, a) would be a vector of features constructed on the product space S × A. Models of this kind are known as linear architecture models. Alternatively, preference functions h(s, a, θ ) could be modeled non-parametrically using neural networks (or some other universal approximators such as decision trees). Parameters θ in this case would be weights of such neural network. Indeed many implementations of actor-critic algorithms involve using two separate neural networks that serve as general function approximations for the action policy and value functions, respectively. More on actor-critic algorithms can be found in Sutton and Barto (2018), Szepesvari (2010).

5.3 Temporal Difference Learning We have seen that Monte Carlo methods must wait until the end of each episode to determine the increment of the action-value function update (9.35). Temporal difference (TD) methods perform updates differently, by waiting only until the next time step, and incrementing the value function at each time step. TD methods can be used for both policy evaluation and policy improvements. Here we focus on how they can be used to evaluate a given policy π by computing a state-value function Vtπ . How it can be done can be seen from the Bellman equation (9.14) which we repeat here for convenience: π Vtπ (s) = Eπt Rt (s, a, s ) + γ Vt+1 (s ) .

(9.40)

As we discussed above, this equation can be converted into an update equation, if the state-value function from a previous iteration is used to evaluate the right-hand side to define the update rule. This idea was used in value iteration and policy iteration algorithms of DP. TD methods use the same idea, and add to this an estimation of the expectation entering Eq. (9.40) from a single observation—which is the observation obtained at the next time step. Without relying on a theoretical model of the world which, similarly to the DP approach, would calculate the expectation in Eq. (9.40) exactly, TD methods rely on a simplest possible estimation of this expectation, essentially by computing an empirical mean from a single observation!

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Clearly, relying on a single observation to estimate an empirical mean can lead to very volatile updates, but this is the price one should be prepared to pay for a truly online method. On the other hand, it is extremely fast. Even if it might bring only a marginal improvement (on average) for maximization of a value function, it might produce a workable and efficient algorithm, because such updates may be repeated many times and at a low cost the during steps of policy improvement. A TD method, when applied to the state-value function Vtπ (s), takes a mismatch between the right-hand side of Eq. (9.40) (estimated with a single observation) and its left-hand side as a measure of an error δt , also called the TD error: δt = Rt (s, a, s ) + γ Vt+1 (s ) − Vt (s).

(9.41)

Note as we deal here with updating the state-value function for a fixed policy π , we omitted explicit upper indices π in this relation. This error defines the rule of update of the state-value function at node s: Vt (s) ← V (s) + α Rt (s, a, s ) + γ Vt+1 (s ) − Vt (s) ,

(9.42)

where α is a learning rate. Note that the learning rate should not be a constant, but rather can vary with the number of iterations. In fact, as we will discuss shortly, a certain decay schedule for the learning rate α = αt should be implemented to guarantee convergence where the number of updates goes to infinity. Note that the TD error δt is not actually available at time t as it depends on the next-step value s , and is therefore only available at time t + 1. The update (9.42) relies only on the information from the next step, and is therefore often referred to as the one-step TD update, also known as the TD(0) rule. It is helpful to compare this with Eq. (9.35) that was obtained for a Monte Carlo (MC) method. Note the Eq. (9.35) requires observing a whole episode in order to compute the full trajectory return Gt . Rewards observed sequentially do not produce updates of a value function until a trajectory is completed. Therefore, the MC method cannot be used in an online setting. On the other hand, the TD update rule (9.42) enables updates of the state-value function after each individual observation, and therefore can be used as an online algorithm that bootstraps by combining a reward signal observed in a current step with an estimation of a next-period value function, where both values are estimated from a sample. TD methods thus combine the bootstrapping properties of DP with a sample-based approach of Monte Carlo methods. That is, updates in TD methods are based on samples, while in DP they are based on computing expectations of next-period value functions using a specific model of the environment. For any fixed policy π , the TD(0) rule (9.42) can be proved to converge to a true state-value function if the learning rate α slowly decreases with the number of iterations. Such proofs of convergence hold for both finite MDPs and for MDPs with continuous

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state-action spaces—with the latter case only established with a linear function approximation, but not with more general non-linear function approximations.5 The ability of TD learning to produce updates after each observation turns out to be very important in many practical applications. Some RL tasks involve long episodes, and some RL problems such as continuous learning do not have any unambiguous definition of finite-length episodes. Whether an episodic learning appears natural or ad hoc, delaying learning until the end of each episode can slow it down and produce inefficient algorithms. Because each update of model parameters requires re-running of all episodes, Monte Carlo methods become progressively more inefficient in comparison to TD methods with increased model complexity. There exist several versions of TD learning. In particular, instead of applying it to learning a state-value function Vt (s), we could use a similar approach to update the action-value function Qt (s, a). Furthermore, for both types of TD learning, instead of one-step updates such as the TD(0) rule (9.42), one could use multi-step updates, leading to more general TD(λ) methods and n-step TD methods. We refer the reader to Sutton and Barto (2018) for a discussion on these algorithms, while focusing in the next section on one-step TD learning methods for an action-value function.

5.4 SARSA and Q-Learning We now arrive at arguably the most important material in this chapter. In applying TD methods to learn an action-value function Q(s, a) instead of a state-value function V (s), one should differentiate between on-policy and off-policy algorithms. Recall that on-policy algorithms assume that policy used to produce a dataset used for learning is an optimal policy, and the task is therefore is to learn the optimal policy function from the data. In contrast, off-policy algorithms assume that the policy used in a particular dataset may not necessarily be an optimal policy, but can be sub-optimal or even purely random. The purpose of off-policy algorithms is to find an optimal policy when data is collected under a different policy. This task is in general more difficult than the first case of on-policy learning which can be viewed as a direct inference problem of fitting a function (a policy function, in this case) to observed data. For both on-policy and off-policy learning with TD methods, the starting point is the Bellman optimality equation (9.20) that we repeat here Q%t (s, a) = E%t Rt (s, a, s ) + γ max Q%t+1 (s , a ) . a

5 We

(9.43)

will discuss function approximations below, after we present TD algorithms in a tabulated setting that is appropriate for finite MDPs with a sufficiently low number of possible states and actions.

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The idea of TD methods for the action-value function is the same as before, to estimate the right-hand side of Eq. (9.43) from observations, and then use a mismatch between the right- and left-hand sides of this equation to define the rule of an update. However, details of such procedure depend on whether we use on-policy or off-policy learning. Consider first the case of on-policy learning. If we know that data was collected under an optimal policy, the max operator in Eq. (9.43) becomes redundant, as observed actions should in this case correspond to maximum of a value function. Similar to the TD method for the state-value function, we replace the expectation in (9.43) by its estimation based on a single observation. The update in this case becomes Qt (s, a) ← Qt (s, a) + α Rt (s, a, s ) + γ Qt+1 (s , a ) − Qt (s, a) .

(9.44)

This on-policy algorithm is known as SARSA, to emphasize that it uses a quintuple (s, a, r, s , a ) to make an update. The TD error for this case is δt = Rt (s, a, s ) + γ Qt+1 (s , a ) − Qt (s, a).

(9.45)

Convergence of the SARSA algorithm depends on a policy used to generate data. If the policy converges to a greedy policy in the limit of an infinite number of steps, SARSA converges to the true policy and action-value functions in the limit when each state-action pair is visited an infinite number of times.

•

> SARSA vs Q-Learning – SARSA is an On-Policy method, which means that it computes the Q-value according to a certain policy and then the agent follows that policy. – Q-learning is an Off-Policy method. It consists of computing the Q-value according to a greedy policy, but the agent does not necessarily follow the greedy policy.

Now consider the case of off-policy learning. In this case, the data available for learning is collected using some sub-optimal, or possibly even purely random policy. Can we still learn from such data? The answer to this question is in the affirmative - we simply replace the expectation in (9.43) by its estimate obtained from a single observation, as in SARSA, but keeping this time the max operator over next-step actions a : Qt (s, a) ← Qt (s, a) + α Rt (s, a, s ) + γ max Qt+1 (s , a ) − Qt (s, a) . a

(9.46)

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This is known as Q-learning. It was proposed by Watkins in 1989, and since then has become one of the most popular approaches in reinforcement learning. Q-learning is provably convergent for finite MDPs when the learning rate α slowly decays with the number of iterations, in the limit when each state-action pair is visited an infinite number of times. Extensions of a tabulated-form Q-learning (9.46) for finite MDPs to systems with a continuous state space will be presented in the following sections. Q-learning is thus a TD(0) learning applied to an action-value function. Note the key difference between SARSA and Q-learning in an online setting when an agent has to choose actions during learning. In SARSA, we use the same policy (e.g., an ε-greedy policy, see Exercise 9.8) to generate both the current action a and the next action a . In contrast to that, in Q-learning the next action a is a greedy action that maximizes the action-value function Qt+1 (s , a ) at the next time moment. It is exactly the choice of a greedy next action a that makes Q-learning an off-policy algorithm that can learn an optimal policy from different and sub-optimal execution policies. The reason Q-learning works as an off-policy method is that the TD(0) rule (9.46) does not depend on the policy used to obtain data for training. Such dependence enters the TD rule only indirectly, via an assumption that this policy should be such that each state-action pair is encountered in the data many times—in fact, an infinite number of times asymptotically, when the number of observations goes to infinity. The TD rule (9.46) does not try to answer the question how the observed values of such pairs are computed. Instead, it directly uses these observed values to make updates in values of the action-value function. It is its ability to learn from off-policy data that makes Q-learning particularly attractive for many tasks in reinforcement learning. In particular, in batch reinforcement learning, an agent should learn from some data previously produced by some other agent. Assuming that the agent that produced the historical data was acting optimally may in many cases be too stringent or unrealistic an assumption. Moreover, in certain cases the previous agent might have been acting optimally, but an environment could change due to some trend (drift) effects. Even though a policy used in the data could be optimal for previous periods, due to the drift, this becomes off-policy learning. In short, it appears there are more examples of offpolicy learning in real-world applications than of on-policy learning. The ability to use off-policy data does not come without a price which is related to the presence of the max operator in Eq. (9.46). This operator in fact provides a mechanism for comparison between different policies during learning. In particular, Eq. (9.46) implies that one cannot learn an optimal policy from a single observed transition (s, a) → (r, s , a ) and nothing else, as the chosen next action a may not necessarily be an optimal action that maximizes Q%t+1 (s , a ), as required in the Q-learning update rule (9.46). This suggests that online Q-learning could maintain a tabulated representation of the action-value function Q(s, a) for all previously visited pairs (s, a), and use it in order to estimate the maxa Q%t+1 (s , a ) term using both the past data and a newly observed transition. Such a method could be viewed as an incremental version of batch learning where a batch dataset is continuously updated by

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adding new observations, and possibly remove observations that are too old and possibly correspond to very sub-optimal policies. This approach is called experience replay in the reinforcement learning literature. The same procedure of adding one observation (or a few of them) at the time can also be used in a pure batch-mode Qlearning as a computational method that allows one to make updates as processing trajectories stored in the datafile. The difference between online Q-learning with experience replay and a pure batch-mode Q-learning therefore amounts to different rules of updating the batch file. In batch-mode Q-learning, it stays the same during learning, but could also be built in increments of one or a few observations to speed up the learning process. In online Q-learning, the experience replay buffer is continuously updated by adding new observations, and removing distant ones to keep the buffer size fixed. Example 9.4

Financial Cliff Walking with SARSA and Q-Learning

The “financial cliff walking” example introduced earlier in this chapter can serve as a simple test case for SARSA and Q-learning for a finite MDP. We assume N = 4 values of possible funds in the account, and assume T = 12 time steps. All combinations of state and time can then be represented as a two-dimensional grid of size N × T = 4 × 12. A time-dependent action-value function Qt (st , at ) with three possible actions at = {a+ , a− , a0 } can then be stored as a rank-three tensor of dimension 4 × 12 × 3. To facilitate exploration required in online applications of RL, we can use a ε-greedy policy. The ε-greedy policy is a simple stochastic policy where the agent takes an action that maximizes the action-value function with probability 1 − ε, and takes a purely random action with probability ε. The ε-greedy policy is used to produce both actions a, a in the SARSA update (9.44), while with Q-learning it is only used to pick the action at the current step. A comparison of the optimal policies is given in Table 9.1. For sufficiently small α and under tapering of (see Fig. 9.3, both methods are shown by Fig. 9.4 to converge to the same cumulative reward. This example is implemented in the financial cliff walking with Q-learning notebook. See Appendix “Python Notebooks” for further details.

5.5 Stochastic Approximations and Batch-Mode Q-learning A more systematic view of TD methods is given by their interpretation as stochastic approximations to solve Bellman equations. As we will show in this section, such a view both helps to better understand the meaning of TD update rules presented above, as well as to extend them to learning using batches of observations in each step, instead of taking all individual observations one by one.

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Table 9.1 The optimal policy for the financial cliff walking problem using (top) SARSA (S) and (below) Q-learning (Q). The row indices denote the balance and the column indices denote the time period. Note that the two optimal policies are almost identical. Starting with a zero balance, both optimal policies will almost surely result in the agent following the same shortest path, with a balance of 1, until the final time period S 3 2 1 0 Q 3 2 1 0

0 Z Z Z U 0 Z Z Z U

1 Z Z Z Z 1 Z Z Z Z

2 Z Z Z Z 2 Z Z Z Z

3 Z Z Z Z 3 Z Z Z Z

4 Z Z Z Z 4 Z Z Z Z

5 Z Z Z Z 5 Z Z Z Z

250

500

750

6 Z Z Z Z 6 Z Z Z Z

7 Z Z Z Z 7 Z Z Z Z

8 Z Z Z Z 8 Z Z Z Z

9 Z D Z Z 9 Z D Z Z

10 Z Z D Z 10 Z D D Z

1500

1750

2000

11 Z Z Z G 11 Z Z Z G

0.10

Epsilon

0.08

0.06

0.04

0.02

0.00 1000 1250 Episode

Fig. 9.3 This figure illustrates how is tapered in the financial cliff walking problem with increasing episodes so that Q-learning and SARSA converge to the same optimal policy and cumulative reward as shown in Table 9.1 and Fig. 9.4

When the model on an environment is unknown, we try to approximately solve the Bellman optimality equation (9.20) by replacing expectations entering this equation by their empirical averages. Stochastic approximations such as the Robbins–Monro algorithm (Robbins and Monro 1951) take this idea one step further, and estimate the mean without directly summing the samples. We can illustrate the idea behind this method using a simple example of estimation of a mean value K1 K k=1 xk of a sequence of observations xk with k = 1, . . . , K. Instead of waiting for all K observations, we can add them one by one, and iteratively update the running estimation of the mean xˆk , where k is the number of iteration, or the number of data points in a dataset:

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Sum of rewards during episode

–20

–40

–60

–80 Sarsa Q-Learning

–100 0

250

500

750

1000 1250 Episodes

1500

1750

2000

Fig. 9.4 Q-learning and SARSA are observed to converge to almost the same optimal policy and cumulative reward in the financial cliff walking problem under the -tapering schedule shown in Fig. 9.3. Note that the cumulative rewards are averaged over twenty simulations

xˆk+1 = (1 − αk )xˆk + αk xk ,

(9.47)

and where αk < 1 denotes the step size (learning rate) at step k, that should satisfy the following conditions: lim

K→∞

K k=1

αk = ∞ ,

lim

K→∞

K

(αk )2 < ∞.

(9.48)

k=1

Robbins and Monro have shown that under these constraints, an iterative method of computing the mean (9.47) converges to a true mean with probability one (Robbins and Monro 1951). In general, the optimal choice of a (step-dependent) learning rate αk is not universal but specific to a problem, and may require some experimentation. Q-learning presented in Eq. (9.46) can now be understood as the Robbins–Monro stochastic approximation (9.47) to estimate the unknown expectation in Eq. (9.43) & ' as a current estimate Q(k) t (s, a) corrected by a current observation Rt s, a, s + & ' γ maxa ∈A Q(k) t+1 s , a : & ' (k) & ' Qt(k+1) (s, a) = (1 − αk )Q(k) R (s, a) + α Q , a . s, a, s + γ max s k t t t+1 a ∈A

(9.49) The single-observation Q-update in Eq. (9.49) corresponds to a pure online version of the Robbins–Monro algorithm. Alternatively, stochastic approximations can be employed in an off-line manner, by using a chunk of data, instead of a single data point, to iteratively update model parameters. Such approaches are useful when working with large datasets, and are frequently used in machine learning, e.g. for a mini-batch stochastic gradient descent method, as a way to more efficiently train a

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model by feeding it mini-batches of data. In addition, batch versions of stochastic approximation methods are widely used when doing reinforcement learning in continuous state-action spaces, which is a topic we discuss next.

•

? Multiple Choice Question 2

Select all the following correct statements: a. Q-learning is obtained by using the Robbins–Monro stochastic approximation to estimate the max(·) term in the Bellman optimality equation. b. Q-learning is obtained by using the Robbins–Monro stochastic approximation to estimate the unknown expectation in the Bellman optimality equation. c. The optimal Q-function in Q-learning is obtained when an optimal learning rate αk = αk where α ( 1/137 is used for learning. d. The optimal Q-function is learned in Q-learning iteratively, where each step (Qiteration) implements one iteration of the Robbins–Monro algorithm.

Example 9.5

Optimal Stock Execution with SARSA and Q-Learning

A setting that is very similar to the “financial cliff walking” example introduced earlier in this chapter can serve to develop a toy MDP model for optimal stock execution. Assume that the broker has to sell N blocks of shares with n shares in each block, e.g. we can have N = 10, n = 1000. The state of the inventory at time t is then given by the variable Xt taking values in a set X with N = 10 states X(n) , so that the start point at t = 0 is X0 = X(N −1) and the target state is XT = X(0) = 0. In each step, the agent has four possible actions at = a (i) that measure the number of blocks of shares sold at time t where a (0) = 0 stands for no action, and a (i) = i with i = 1, . . . , 3 is the number of blocks sold. The update equation is Xt+1 = (Xt − at )+ .

(9.50)

Trades influence the stock price dynamics through a linear market impact St+1 = St e(1−νat ) + σ St Zt ,

(9.51)

where ν is a market friction parameter. To map onto a finite MDP problem, a range of possible stock prices S can be discretized to M values, e.g. M = 12. The state space of the problem is given by a direct product of states X × S of dimension N × M = 10 · 12 = 120. The dimension of the extended space including the time is then 120 · 10 = 1200. (continued)

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Example 9.5 (continued) The payoff of selling at blocks of shares when the stock price is St is nat St . A risk-adjusted payoff adds a penalty on variance of the remaining inventory price at the next step t + 1: rt = nat St − λnVar [St+1 Xt+1 ]. All combinations of state and time can then be represented as a three-dimensional grid of size N × M × T = 10 · 12 · 10. A time-dependent action-value function Qt (st , at ) with four possible actions at = {a0 , a1 , a2 , a3 } can then be stored as a rankfour tensor of dimension 10 × 12 × 10 × 4. We can now apply SARSA or Q-learning to learn optimal stock execution in such a simplified setting. For exploration needed for online learning, one can use a ε-greedy policy. At each time step, a time-dependent optimal policy is therefore found with 10 × 12 (for inventory and stock price level) states and four possible actions at = {a0 , a1 , a2 , a3 } can be viewed as a 10 × 12 matrix as shown, for the second time step, in Table 9.2. This example is implemented in the market impact problem with Q-learning notebook. See Appendix “Python Notebooks” for further details (Fig. 9.5).

Fig. 9.5 The optimal execution problem: how to break up large market orders into smaller orders with lower market impact? In the finite MDP formulation, the state space is the inventory, shown by the number of blocks, stock price, and time. In this illustration, the agent decides whether to sell {0, 1, 2, 3} blocks at each time step. The problem is whether to retain inventory, thus increasing market risk but reduces the market impact, or quickly sell inventory to reduce exposure but increase the market impact

t =2 Inventory 0 1 2 3 4 5 6 7 8 9

Price level 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 3 1 0 3 2 0 3 2 0 0 0 0 0 0

4 0 0 0 0 0 0 3 2 2 0

5 0 0 0 0 0 0 0 0 1 0

6 0 0 0 0 0 0 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 0 0

9 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 0 0 0 0

t =2 Inventory Price level 1 2 3 0 0 0 1 0 0 2 0 0 3 0 2 4 0 2 5 0 2 6 0 1 7 0 3 8 0 0 9 0 0 4 0 0 0 0 0 0 0 2 0 0

5 0 0 0 0 0 0 1 2 2 0

6 0 0 0 0 0 0 0 0 3 0

7 0 0 0 0 0 0 0 0 0 0

8 0 0 0 0 0 0 0 0 3 0

9 0 0 0 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0 0 0 0

11 0 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 0 0 0 0

Table 9.2 The optimal policy, at time step t = 2, for the trade execution problem using (left) SARSA and (right) Q-learning. The rows denote the inventory level and the columns denote the stock price level. Each element denotes an action to sell {0, 1, 2, 3} blocks of shares

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322

Example 9.6

9 Introduction to Reinforcement Learning

Electronic Market Making with SARSA and Q-Learning

We can build on the previous two examples by considering the problem of high-frequency market making. Unlike the previous example, we shall learn a time-independent optimal policy. Assume that a market maker seeks to capture the bid–ask spread by placing one lot best bid and ask limit orders. They are required to strictly keep their inventory between −1 and 1. The problem is when to optimally bid to buy (“b”), bid to sell (“s”), or hold (“h”), each time there is a limit order book update. For example, sometimes it may be more advantageous to quote a bid to close out a short position if it will almost surely yield an instantaneous net reward, other times it may be better to wait and capture a larger spread. In this toy example, the agent uses the liquidity imbalance in the top of the order book as a proxy for price movement and, hence, fill probabilities. The example does not use market orders, knowledge of queue positions, cancelations, and limit order placement at different levels of the ladder. These are left to later material and exercises. A simple illustration of the market making problem is shown in Fig. 9.6. At each non-uniform time update, t, the market feed provides best prices and depths {pta , ptb , qta , qtb }. The state space is the product of the inventory, Xt ∈ qa {−1, 0, 1}, and gridded liquidity ratio Rˆ t = a t b N ∈ [0, 1], where N is qt +qt

the number of grid points and qta and qtb are the depths of the best ask and bid. Rˆ t → 0 is the regime where the mid-price will go up and an ask is filled. Conversely for Rˆ t → 1. The dimension of the state space is chosen to be 3 · 5 = 15. A bid is filled with probability t := Rˆ t and an ask is filled with probability 1 − t . The rewards are chosen to be the expected total P&L. If a bid is filled to close out a short holding, then the expected reward rt = −t (pt + c), where pt is the difference between the exit and entry price and c is the transaction cost. For example, if the agent entered a short position at time s < t with a filled ask at psa = 100 and closed out the position with a filled bid at ptb = 99, then pt = 1. The agent is penalized for quoting an ask or bid when the position is already short or long, respectively. As with previous examples, we apply SARSA or Q-learning to optimize market making. For exploration needed for online learning, one can use a εgreedy policy. A comparison of the optimal policies is given in Table 9.3. For sufficiently large number of iterations in each episode and under tapering of , both methods are observed to converge to the same cumulative reward in Fig. 9.7. This example is implemented in the electronic market making with Q-learning notebook. See Appendix “Python Notebooks” for further details.

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Cumulated PnL: 12.40 ~ Position: short ~ Entry Price: 2243.00

Probability of fill 1.0

Prices

2240

0.8 0.6

2240 0.4 0.2

2240

0.0 0

10

20

30 Iteration

40

50

bid

60

ask

12.5 Total PnL

10.0 7.5 5.0 2.5 0.0 0

10

20

30 40 Iteration

50

60

Fig. 9.6 The market making problem requires the placement of bid and ask quotes to maximize P&L while maintaining a position within limits. For each limit order book update, the agent must anticipate which quotes shall be filled to capture the bid–ask spread. Transaction costs, less than a tick, are imposed to penalize trading. This has the net effect of rewarding trades which capture at least a tick. A simple model is introduced to determine the fill probabilities and the state space is the product of the position and gridded fill probabilities Table 9.3 The optimal policy for the market making problem using (top) SARSA (S) and (below) Q-learning (Q). The row indices denote the position and the column indices denote the predicted ask fill probability buckets. Note that the two optimal policies are almost identical S Flat Short Long Q Flat Short Long

0–0.1 b b h 0–0.1 b b s

0.1–0.2 b b s 0.1–0.2 b b s

0.2–0.3 b b s 0.2–0.3 b b s

0.3–0.4 b b s 0.3–0.4 b b s

0.4–0.5 b b s 0.4–0.5 b b s

0.5–0.6 b b s 0.5–0.6 b b s

0.6–0.7 s b s 0.6–0.7 s b s

0.7–0.8 s b s 0.7–0.8 b b s

0.8–0.9 s b s 0.8–0.9 b b s

0.9–1.0 s h s 0.9–1.0 s b s

5.6 Q-learning in a Continuous Space: Function Approximation Our previous presentation of reinforcement learning algorithms assumed a setting of a finite MDP model with discrete state and action spaces. For this case, all combinations of state-action pairs are enumerable. Therefore, the action-value and state-value functions can be maintained in a tabulated form, while TD methods such as Q-learning (9.49) can be used to compute the action-value function values at these nodes in an iterative manner. While such methods are simple and can be proven to converge in a tabulated setting, they also quickly run into their limitations for many interesting real-world

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15500

SARSA Q-Learning

Sum of rewards during episode

15000 14500 14000 13500 13000 12500 12000 11500 0

20

40

60

80

100

120

140

Episodes

Fig. 9.7 Q-learning and SARSA are observed to converge to almost the same optimal policy and cumulative reward in the market making problem

problems. The latter are often high-dimensional in discrete or continuous stateaction spaces. If we use a straightforward discretization of each continuous state and/or action variable, and then enumerate all possible combinations of states and actions, we end up with an exponentially large number of state-action pairs. Even storing such data can pose major challenges with memory requirements, not to speak about an exponential slow-down in a cost of computations, where all such pairs essentially become parameters of a very high-dimensional optimization problem. This calls for approaches based on function approximation methods where functions in (high-dimensional) discrete or continuous spaces are represented in terms of a relatively small number of freely adjustable parameters. To motivate linear function approximations, let us start with a finite MDP with a set of nodes {sn }M n=1 where M is the number of nodes. The state-value function V (s) in this case is determined by a set of node values Vn for each node of the state grid. We can present this set using an “index-free” notation: V (s) =

M

Vn δs,sn ,

(9.52)

1 if s = sn 0 otherwise.

(9.53)

n=1

where δs,sn is the Kronecker symbol: 2 δs,sn

=

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Eq. (9.52) can be viewed as an approach to conveniently and simultaneously represent all values Vn of the value function on the grid in the form of a map V (s) that points to a given node value Vn for any particular choice s = sn . Now we can write the same equation (9.52) in a more suggestive form as an expansion over a set of basis functions V (s) =

M

Vn δs,sn =

n=1

M

Vn φn (s),

(9.54)

n=1

with “one-hot” basis functions φn (s): 2 φn (s) = δs,sn =

1 if s = sn 0 otherwise.

(9.55)

The latter form helps to understand how this setting can now be generalized for a continuous state space. In a discrete-state representation (9.54), we use “one-hot” (Dirac-like) basis functions φn (s) = δs,sn . We can now envision a transition to a continuous time limit as a process of adding new points to the grid, while at the same time keeping the size M of the sum in Eq. (9.54) by aggregating (taking partial sums) within some neighborhoods of a set of M node points. Each term in such a sum would be given by the product of an average mass of nodes Vn and a smoothedout version of the original Dirac-like basis function of a finite MDP. Such partial aggregation of neighboring points produces a tractable approximation to the actual value function defined in a continuous limit. A function value at any point of a continuous space is now mapped onto a M-dimensional approximation. The quality of such finite-dimensional function approximation is determined by the number of terms in the expansion, as well as the functional form of the basis functions. A smoothed-out localized basis could be constructed using, e.g., B-splines or Gaussian kernels, while for a multi-dimensional continuous-state case one could use multivariate B-splines or radial basis functions (RBFs). A set of one-dimensional B-spline basis functions is illustrated in Fig. 9.8. As one can see, B-splines produce well localized basis functions that differ from zero only on a limited segment of a total support region. Other alternatives, for example, polynomials or trigonometric functions, can also be considered for basis functions; however, they would be functions of a global, rather than local, variation. Similar expansions can be considered for action-value functions Q(s, a). Let us assume that we have a set of basis functions ψk (s, a) with k = 0, 1, . . . , K defined on the direct product S × A. Respectively, we could approximate an action-value function as follows: Q(s, a) =

K−1 k=0

θk ψk (s, a).

(9.56)

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9 Introduction to Reinforcement Learning

1.0

0.8

0.6

0.4

0.2 0.0 4.0

4.2

4.4

4.6

4.8

5.0

5.2

Fig. 9.8 B-spline basis functions

Here coefficients θk of such expansion can be viewed as free parameters. Respectively, we can find the values of these parameters which provide the best fit to the Bellman optimality equation. For a fixed and finite set of K basis functions ψk (s, a), the problem of functional optimization of the action-value function Q(s, a) is reduced by Eq. (9.56) to a much simpler problem of K-dimensional numeric optimization over parameters θk , irrespective of the actual dimensionality of the state space. Note that having only a finite (and not too large) value of K, we can at best hope only for an approximate match between the optimal value function obtained in this way with a “true” optimal value function. The latter could in principle be obtained with the same basis function expansion (9.56), provided the set {ψk (s, a)} is complete, by taking the limit K → ∞. Equation (9.56) thus provides an example of function approximation, where a function of interest is represented as expansion over a set of K basis functions ψk (s, a), with coefficients θk being adjustable parameters. Such linear function representations are usually referred to as linear architectures in the machine learning literature. Functions of interest such as value functions and/or policy functions are represented and computed in these linear architecture methods as linear functions of parameters θk and basis functions ψk (s, a). The main advantage of linear architecture approaches is their relative robustness and computational simplicity. The possible amount of variation in functions being approximated is essentially determined by a possible amount of variation in basis functions, and thus can be explicitly controlled. Furthermore, because Eq. (9.56) is linear in θ , this can produce analytic solutions if a loss function is quadratic, or unique and easily computed numerical solutions when a loss function is nonquadratic but convex. Reinforcement learning methods with linear architectures are provably convergent.

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On the other hand, the linear architecture approach is not free of drawbacks. The main one is that it gives no guidance on how to choose a good set of basis functions. For a bounded one-dimensional continuous state, it is not difficult to derive a good set of basis functions. For example, we could use a trigonometric basis, or splines, or even a polynomial basis. However, with multiple continuous dimensions, to find a good set of basis functions is non-trivial. This goes beyond reinforcement learning, and is generally known in machine learning as a feature construction (or extraction) problem. One possible approach to deal with such cases is to use non-linear architectures that use generic function approximation tools such as trees or neural networks, in order to have flexible representations for functions of interest that do not rely on any pre-defined set of basis functions. In particular, deep reinforcement learning is obtained when one uses a deep neural network to approximate value functions or action policy (or both) in reinforcement learning tasks. We will discuss deep reinforcement learning below, after we present an off-line version of Q-learning that for both discrete and continuous state spaces while operating with mini-batches of data, instead of updating each observation.

5.7 Batch-Mode Q-Learning Assume that we have a set of basis functions ψk (s, a) with k = 0, 1, . . . , K defined on the direct product S × A, and the action-value function is represented by the linear expansion (9.56). Such a representation applies to both finite and continuous MDP problems—as we have seen above, a finite MDP case can be considered a special case of the linear architecture (9.56) with Dirac delta-functions taken as basis functions ψk (s, a). Therefore, using the linear specification (9.56), we can provide a unified description of algorithms of Q-learning for both cases of finite and continuous MDPs. Solving the Bellman optimality equation (9.20) now amounts to finding parameters θk . Clearly, if we want to find all K > 1 such parameters, observing just one data point in each iteration would be insufficient to determine them, or update their previous estimation in a unique and well-specified way. To this end, we need to tackle at least K observations (and, to avoid high variance estimations, a multiple of this number) to produce such estimate. In other words, we need to work in the setting of batch-mode, or off-line, reinforcement learning. With batch reinforcement learning, an agent does not have access to an environment, but rather can only operate with some historical data collected by observing actions of another agent over a period of time. Based on the law of large numbers, one can expect that whenever batch reinforcement learning can be used for training a reinforcement learning algorithm, it can provide estimators with lower variance than those obtained with a pure online learning. To obtain a batch version of Q-learning, the one-step Bellman optimality equation (9.20) is interpreted as regression of the form

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9 Introduction to Reinforcement Learning

& ' & ' K−1 Rt s, a, s + γ max Q%t+1 s , a = θk ψk (s, a) + εt , a ∈A

(9.57)

k=0

where εt is a random noise at time t with mean zero. Parameters & θk 'now appear as regression coefficients of the dependent variable R s, a, s + t & ' γ maxa ∈A Q%t+1 s , a on the regressors given by the basis functions ψk (s, a). Equations (9.57) and (9.20) are equivalent in expectations, as taking the expectation of both sides of (9.57), we recover (9.20) with function approximation (9.56) used for the optimal Q-function Q%t (s, a). A batch data file consists of tuples (s, a, r, s ) = (st , at , rt , st+1 ) for t = 0, . . . , T − 1. Each tuple record (s, a, r, s ) contains the current state s = st , action taken a = at , reward received next-step & r = ' rt , and the next state s . If we know the % action-value function Qt+1 s , a as a function of state s and action a (via either an explicit formula or a numerical algorithm), the tuple record (s, a, r, s ) could be used to view them as pairs (s, y) of a supervised regression with& the independent ' variable s = st and dependent variable y := r + γ maxa ∈A Q%t+1 s , a . Note that there is a nuance here related to taking the max maxa ∈A over all actions in the next time step. We will return to this point momentarily, but for now let us assume that this operation can be performed in one way or another, so that each tuple (s, a, r, s ) can indeed be used as an observation for the regression (9.57). Assume that for each time step t, we have samples from N trajectories.6 Using a conventional squared loss function, coefficients θk can be found by solving the standard least-square optimization problem: Lt (θ ) =

N k=1

,

& ' & ' K−1 θk ψk (s, a) Rt sk ak , sk + γ max Q%t+1 s , a − a ∈A

-2 .

k=0

(9.58) This is known as the Fitted Q Iteration (FQI) method. We will discuss an application of this method in the next chapter when we present reinforcement learning for option pricing. & ' Let us now address the challenge of computing the term maxa ∈A Q%t+1 s , a that appears in the regression (9.57) when we are only given samples of transitions in form of tuples (s, a, r, s ). One simple way would be to replace the theoretical maximum by an empirical maximum observed in the dataset. This would amount to using the same dataset to estimate both the optimal action and the optimal Qfunction. & ' It turns out that such a procedure leads to an overestimation of maxa Q%t+1 s, a in the Bellman optimality equation (9.20), due to Jensen’s inequality and convexity of the max(·) function: E [max f (x)] ≤ max E [f (x)], where f (x) is an arbitrary function. Indeed, by replacing expected maximum of Q(s , a ) by an empirical maximum, we replace the expected maximum E maxa Q(s , a ) by 6 These

may be Monte Carlo trajectories or trajectories obtained from real-world data.

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329

maxa E Q(s , a ) , and then further use a sample-based estimation of the inner expectation in the last expression. Due to Jensen’s inequality, such replacement generally leads to overestimation of the action-value function. When repeated multiple times during iterations over time steps or during optimization over parameters, this overestimation can lead to distorted and sometimes even diverging value functions. This is known in the reinforcement learning literature as the overestimation bias problem. There are two possible approaches to address a potential overestimation bias with Q-learning. One of them is to use two different datasets to train the actionvalue function and the optimal policy. But this is not directly implementable with Q-learning where policy is determined by the action-value function Q(s, a). Instead of optimizing both the action-value function and the policy on different subsets of data, a method known as Double Q-learning (van Hasselt 2010) introduces two action-value functions QA (s, a) and QB (s, a). At each iteration, when presented with a new mini-batch of data, the Double Q-learning algorithm randomly choses between an update of QA (s, a) and update of QB (s, a). If one chooses to update QA (s, a), the optimal action is determined by finding a maximum a% of QA (s , a ), and then QA (s, a) is updated using the TD error of r + γ QB (s , a% ) − QA (s, a). If, on the other hand, one chooses to update QB (s, a), then the optimal action a% is calculated by maximizing QB (s , a ), and then updating QB (s, a) using the TD error of r + γ QA (s , a% ) − QB (s, a). As was shown by van Hasselt (2010), actionvalue functions QA (s, a) and QB (s, a) converge in Double Q-learning to the same limit when the number of observation grows. The method avoids the overestimation bias problem of a naive sample-based Q-learning, though it can at times lead to underestimation of the action-value function. Double Q-learning is often used with model-free Q-learning using neural networks to represent an action-value function Qθ (s, a) where θ is a set of parameters of the neural network. Another possibility arises if the action-value function Q (s, a) has a specific parametric form, so that the maximum over the next-step action a can be performed analytically or numerically. In particular, for linear architectures (9.56) the maximum can be computed once the form of basis functions ψk (s, a) and coefficients θk are fixed. With such an independent calculation of the maximum in the Bellman optimality equation, splitting a dataset into two separate datasets for learning the action-value function and optimal policy, as is done with Double Q-learning, can be avoided. As we will see in later chapters, such scenarios can be implemented for some problems in quantitative trading, including in particular option pricing.

•

> Bellman Equations and Non-expansion Operators As we saw above, a non-analytic term in the Bellman optimality equation involving a max over all actions at the next step poses certain computational challenges. It turns out that this term can be relaxed using differentiable parameterized operators constructed in such a way that the “hard” max operator is recovered in a certain parametric limit. It turns out that operators (continued)

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of certain type, called non-expansion operators, can replace the max operator in a Bellman equation without loosing the existence of a solution, such as a fixed-point solution for a time-stationary MDP problem. Let h be a real-valued function over a finite set I , and let * be a summary operator that maps values of function h onto a real number. The maximum operator maxi∈I h(i) and the minimum operators mini∈I h(i) are examples of summary operators. A summary operator * is called a non-expansion if it satisfies two properties min h(i) ≤ *h(i) ≤ max h(i)

(9.59)

*h(i) − *h (i) ≤ max h(i) − h (i) ,

(9.60)

i∈I

i∈I

and i∈I

where h is another real-valued function over the same set. Some examples of non-expansion include the mean and max operator, as well as epsilon-greedy operator epsε (X) = εmean (X) + (1 − ε) max (X). As was shown by Littman and Szepasvari (1996), value iteration for the action-value function ˆ , a) ˆ a) ← r(s, a) + γ p(s, a, s ) max Q(s (9.61) Q(s, a ∈A

s ∈S

can be replaced by a generalized value iteration

ˆ a) ← r(s, a) + γ Q(s,

ˆ , a) p(s, a, s ) *a ∈A Q(s

(9.62)

s ∈S

which converges to a unique fixed point if operator * is a non-expansion with respect to the infinity norm: ˆ ˆ a) − Q ˆ (s, a) ˆ (s, a) ≤ max Q(s, *Q(s, a) − *Q a

ˆ Qˆ , s. for any Q,

•

? Multiple Choice Question 3

Select all the following correct statements: a. Fitted Q Iteration is a method to accelerate on-line Q-learning.

(9.63)

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331

b. Similar to the DP approach, Fitted Q Iteration looks at only one trajectory at each update, so that Q-iteration fits better when extra noise from other trajectories is removed. c. Fitted Q Iteration works only for discrete state-action spaces. d. Fitted Q Iteration works only for continuous state-action spaces. e. Fitted Q Iteration works for both discrete and continuous state-action spaces.

•

> Online Learning with MDPs Recall that in Chap. 1, we presented the Multi-Armed Bandit (MAB) as an example of online sequential learning. Similar to the MAB formulation, for a more general setting of online learning with Markov Decision Processes,7 a training algorithm A aims at minimization of the regret of A defined as follows: A % RA T = RT − T ρ ,

(9.64)

T −1 where RTA = t=0 Rt+1 is the total reward received up to time T while following A, and ρ % stands for the optimal long-run average reward: ρ % = max ρ π = max π

π

μπ (s)R(s, π(s)),

(9.65)

s∈S

where μπ (s) is a stationary distribution of states induced by policy π . The problem of minimization of regret is clearly equivalent to maximization of the total reward. For a discussion of algorithms for online learning with the MDP, see Szepesvari (2010). Online learning with MDP can be of interest in the financial context for certain tasks that may require real-time adjustments of policy following new data received, such as intraday trading. As we mentioned earlier, a combination of off-line and online learning using experience replay is often found to produce better and more stable behavior than pure online learning.

5.8 Least Squares Policy Iteration Recall that for every MDP, there exists an optimal policy, π ∗ , which maximizes the expected, discounted return of every state. As discussed earlier in this chapter, policy iteration is a method of discovering this policy by iterating through a sequence of

7 See

Sect. 3 for further details of MDPs.

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monotonically improving policies. Each iteration consists of two phases: (i) Value determination computes the state-action values for a policy, π by solving the above system; (ii) Policy improvement defines the next policy π . These steps are repeated until convergence. The Least Squares Policy Iteration (LSPI) (Lagoudakis and Parr 2003) is a model-free off-policy method that can efficiently use (and re-use at each iteration) sample experiences collected using any policy . The LSPI method can be understood as a sample-based and model-free approximate policy iteration method that uses a linear architecture where an action-value function Qt (xt , at ) is sought as a linear expansion in a set of basis functions. The LSPI approach proceeds as follows. Assume we have a set of K basis functions { k (x, a)}K k=1 . While particular choices for such basis functions will be presented below, in this section their specific form does not matter, as long as the set of basis function is expressive enough so that the true optimal action-value function approximately lies in a span of these basis functions. Provided such a set is fixed, we use a linear function approximation for the action-value function: Qπt (xt , at ) = Wt (xt , at ) =

K

Wtk k (xt , at ).

(9.66)

k=1

Note that a dependence on a policy π enters this expression through a dependence of coefficients Wik on π . The LSPI method can be thought of a process of finding an optimal policy via adjustments of weights Wik . The policy π is a greedy policy that maximizes the action-value function: at% (xt ) = πt (xt ) = argmax Qt (xt , at ) .

(9.67)

a

The LSPI algorithm continues iterating between computing coefficients Wtk (and hence the action-value function Qt (xt , at )) and computing the policy given the action-value function using Eq. (9.74). This is done for each time step, proceeding backward in time for t = T − 1, . . . , 0. To find coefficients Wt for a given time step, we first note that a linear Bellman equation (9.18) for a fixed policy π can be expressed in a form that only involve the action-value function, by noting that for an arbitrary policy π , we have Vtπ (xt ) = Qπt (xt , π(xt )).

(9.68)

Using this in the Bellman equation (9.18), we write it in the following form: Qπt (xt , at ) = Rt (xt , at ) + γ Et Qπt+1 (Xt+1 , π(Xt+1 )) xt , at . Similar to Eq. (9.57), we can interpret Eq. (9.69) as regression of the form

(9.69)

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Rt (xt , at , xt+1 ) + γ Qπt+1 (xt+1 , π(xt+1 )) = Wt (xt , at ) + εt ,

(9.70)

where Rt (xt , at , xt+1 ) is a random reward, and εt is a random

noise at time t with (k) (k) (k) (k) mean zero. Assume we have access to sample transitions Xt , at , Rt , Xt+1 with k = 1, . . . , N for each t = T − 1, . . . , 0. For a given policy π , coefficients Wt can then found by solving the following least squares optimization problem, which is similar to Eq. (9.58) above: N

(k) (k) (k) Rt Xt , at , Xt+1 Lt (Wt ) = k=1

+γ Qπt+1

2 (k) (k) (k) (k) Xt+1 , π Xt+1 − Wt Xt , at .

(9.71)

For a solution of this equation, see Exercise 9.14. Note that for an MDP with a finite state-action space, finding an optimal policy using (9.67) is straightforward, and achieved by enumeration of possible actions for each state. When the action space is continuous, it takes more effort. Consider, for example, the case where both the state and action spaces are one-dimensional continuous spaces. To use Eq. (9.67) in such setting, we discretize the range of values of xt to a set of discrete values xn . We can first compute the optimal action for these values, and then use splines to interpolate for the rest of values of xt . For a given set of coefficients Wtk , the policy πt (xt ) is then represented by a splineinterpolated function. Example 9.7

LSPI for Optimal Allocation

Recall from Chap. 1 the problem of an investor who starts with an initial wealth W0 = 1 at time t = 0 and, at each period t = 0, . . . , T − 1 allocates a fraction ut = ut (St ) of the total portfolio value to the risky asset, and the remaining fraction 1 − ut is invested in a risk-free bank account that pays a risk-free interest rate rf = 0. If the wealth process is self-financing and the one-step rewards Rt for t = 0, . . . , T − 1 are the risk-adjusted portfolio returns Rt = rt − λVar [rt |St ] = ut φt − λu2t Var [φt |St ]

(9.72)

then the optimal investment problem for T − 1 steps is given by

T

V π (s) = max E Rt St = s = max E ut φt − λu2t Var [φt |St ] St = s . ut ut t=0 t=0 (9.73)

T

(continued)

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9 Introduction to Reinforcement Learning

Example 9.7

(continued)

where we allow for short selling in the ETF (i.e., ut < 0) and borrowing of cash ut > 1. We apply the LSPI algorithm to N = 2000 simulated stock (i) T prices over T = 10 periods: {{St }N i=1 }t=1 . At each time period, we construct K a basis { k (s, a)}k=1 over the state-action space using K = 256 B-spline basis (i) N functions, where s ∈ [min({St(i) }N i=1 ), max({St }i=1 )] and a ∈ [−1, 1]. Note that in this particular simple problem, the actions are independent of the state space and hence the basis construction over state-action space is not really needed. However, our motive here is to show that we can obtain E[φt ] using a more general an estimate close to the exact solution, u∗t = 2λVar[φ t] methodology. aT −1 is initialized with uniform random samples at time step t = T − 1. In subsequent time steps, at , t ∈ {T − 1, T − 2, . . . , 0}, we initialize with ∗ . LSPI updates the policy iteratively the previous optimal action, at = at+1 k−1 k π → π until convergence of the value function. The Q-function is maximized over a gridded state-action space, h , with 200 stock values and 20 action values, to give the optimal action atk (s) = πtk (s) = argmax Qπt

k−1

a

(s, a) , (s, a) ∈ h .

(9.74)

For each time step, LSPI updates the policy π k until the following stopping k k−1 criterion is satisfied ||V π − V π ||2 ≤ τ where τ = 1 × 10−6 . The optimal allocation using the LSPI algorithm (red) is compared against the exact solution (blue) in Fig. 9.9. The implementation of the LSPI and its application to this optimal allocation problem is given in the Python notebook ML_in_Finance_LSPI_Markowitz.ipynb.

120

–0.30

110 action

–0.32

St

100

–0.34

90 –0.36

80

–0.38

70 0

2

4 6 Time Steps (a)

8

10

2

4 time (b)

6

8

Fig. 9.9 Stock prices are simulated using an Euler scheme over a one year horizon. At each of ten periods, the optimal allocation is estimated using the LSPI algorithm (red) and compared against the exact solution (blue). (a) Stock price simulation. (b) Optimal allocation

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5.9 Deep Reinforcement Learning A good choice of basis functions that could be used in linear architectures (9.56) may be a hard problem for many practical applications, especially if the dimensionality of data grows, or if data become highly non-linear, or both. This problem is also known as the feature engineering problem, and it is common for all types of machine learning, rather than being specific to reinforcement learning. Learning representative features is an interesting and actively researched topic in the machine learning literature, and various supervised and unsupervised algorithms have been suggested to address such tasks. Instead of pursuing hand-engineered or algorithm-driven features defined in general as parameter-based functional transforms of original data, we can resort to universal function approximation methods such as trees or neural networks considered as parameterized “black-box” algorithms. In particular, deep reinforcement learning approaches rely on multi-level neural networks to represent value functions and/or policy functions. For example, if an action-value function Q(s, a) is represented by a multilayer neural network, one way of thinking about it would be in terms of the linear architecture specification (9.56) where parameters θk represent the weight of a last linear layer of a neural network, while the previous layer generates certain “black-box”-type basis functions ψk (s, a) that can be parameterized in terms of their own parameters θ . This approach may be advantageous when an action-value function is highly non-linear and no clear-cut choice of a good set of basis function can be immediately suggested. In particular, functions of high variations appear in analysis of images, videos, and video games. For such applications, using deep neural networks as function approximation is very useful. A strong push to this area of research was initiated by Google’s DeepMind’s work on using deep Q-learning for playing Atari video games.

5.9.1

Preliminaries

Since we cannot reasonably learn and store a Q value for each state-action tuple when the state space is continuous, we will represent our Q values as a function q(s, ˆ a, w) where w are parameters of the function (typically, a neural network’s weights and bias). In this approximation setting, our update rule becomes * ) & ' w = w + α r + γ max qˆ s , a , w − qˆ (s, a, w) ∇w q(s, ˆ a, w). a ∈A

(9.75)

In other words, we seek to minimize L(w) = E

s,a,r,s

2 & ' r + γ max qˆ s , a , w − q(s, ˆ a, w) . a ∈A

(9.76)

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9 Introduction to Reinforcement Learning

5.9.2

Target Network

DeepMind (Mnih et al. 2015) maintain two sets of parameters, w (to compute ˆ , a )) s.t. our update rule becomes q(s, ˆ a)) and w− (target network, to compute q(s * ) ' & ˆ a, w). w = w + α r + γ max qˆ s , a , w− − qˆ (s, a, w) ∇w q(s, a ∈A

(9.77)

The target network’s parameters are updated with the Q-network’s parameters occasionally and are kept fixed between individual updates. Note that when computing the update, we do not compute gradients with respect to w− (these are considered fixed weights).

5.9.3

Replay Memory

As we play, we store our transitions (s, a, r, s ) in a buffer. Old examples are deleted as we store new transitions. To update our parameters, we sample a mini-batch from the buffer and perform a stochastic gradient descent update. -Greedy Exploration Strategy During training, we use an -greedy heuristic strategy. DeepMind (Mnih et al. 2015) decrease from 1 to 0.1 during the first million steps. At test time, the agent chooses a random action with probability = 0.05. π(a|s) =

1 − ,

a = argmaxa Qt (s, a)

/|A|, a = argmaxa Qt (s, a)

There are several points to note: a. w updates every learning_freq steps by using a mini-batch of experiences sampled from the replay buffer. b. DeepMind’s deep Q-network takes as input the state s and outputs a vector of size = number of actions. In our environment, we have |A| actions, thus q(s, ˆ w) ∈ R|A| . c. The input of the deep Q-network can be based on both the current and history of observations of the environment. The practice of using Deep Q-learning on finance problems is problematic. The sheer number of parameters that need to be tuned and configured renders the approach much more complex than Q-learning, LSPI, and of course deep learning in a supervised learning setting. However, deep Q-learning is one of the few approaches which scales to high-dimensional discrete and/or continuous state and action spaces.

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337

6 Summary This chapter has introduced the reader to reinforcement learning, with some toy examples of how it is useful for solving problems in finance. The emphasis of the chapter is on understanding the various algorithms and RL approaches. The reader should check the following learning objectives: – Gain familiarity with Markov Decision Processes; – Understand the Bellman equation and classical methods of dynamic programming; – Gain familiarity with the ideas of reinforcement learning and other approximate methods of solving MDPs; – Understand the difference between off-policy and on-policy learning algorithms; and – Gain insight into how RL is applied to optimization problems in asset management and trading. The next chapter will present much more in depth examples of how RL is applied in more realistic financial models.

7 Exercises Exercise 9.1 Consider an MDP with a reward function rt = r(st , at ). Let Qπ (s, a) be an actionvalue function for policy π for this MDP, and π% (a|s) = arg maxπ Qπ (s, a) be an optimal greedy policy. Assume we define a new reward function as an affine transformation of the previous reward: r˜ (t) = wrt + b with constant parameters b and w > 0. How does the new optimal policy π˜ % relate to the old optimal policy π% ? Exercise 9.2 With True/False questions, give a short explanation to support your answer. – True/False: Value iteration always find the optimal policy, when run to convergence. [3] – True/False: Q-learning is an on-policy learning (value iteration) algorithm and estimates updates to the action-value function, Q(s, a) using actions taken under the current policy π . [5] – For Q-learning to converge we need to correctly manage the exploration vs. exploitation tradeoff. What property needs to hold for the exploration strategy? [4] – True/False: Q-learning with linear function approximation will always converge to the optimal policy. [2]

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9 Introduction to Reinforcement Learning

Table 9.4 The reward function depends on fund wealth w and time

w 2 1

t0 0 0

t1 0 −10

Exercise 9.3* Consider the following toy cash buffer problem. An investor owns a stock, initially valued at St0 = 1, and must ensure that their wealth (stock + cash) is not less than a certain threshold K at time t = t1 . Let Wt = St + Ct denote their at time t, where Ct is the total cash in the portfolio. If the wealth Wt1 < K = 2 then the investor is penalized with a -10 reward. The investor chooses to inject either 0 or 1 amounts of cash with a respective penalty of 0 or −1 (which is not deducted from the fund). Dynamics The stock price follows a discrete Markov chain with P (St+1 = s | St = s) = 0.5, i.e. with probability 0.5 the stock remains the same price over the time interval. P (St+1 = s + 1 | St = s) = P (St+1 = s − 1 | St = s) = 0.25. If the wealth moves off the grid it simply bounces to the nearest value in the grid at that time. The states are grid squares, identified by their row and column number (row first). The investor always starts in state (1,0) (i.e., the initial wealth Wt0 = 1 at time t0 = 0—there is no cash in the fund) and both states in the last column (i.e., at time t = t1 = 1) are terminal (Table 9.4). Using the Bellman equation (with generic state notation), give the first round of value iteration updates for each state by completing the table below. You may ignore the time value of money, i.e. set γ = 1. Vi+1 (s) = max( a

T (s, a, s )(R(s, a, s ) + γ Vi (s )))

s

(w,t) V0 (w) V1 (w)

(1,0) 0 ?

(2,0) 0 NA

Exercise 9.4* Consider the following toy cash buffer problem. An investor owns a stock, initially valued at St0 = 1, and must ensure that their wealth (stock + cash) does not fall below a threshold K = 1 at time t = t1 . The investor can choose to either sell the stock or inject more cash, but not both. In the former case, the sale of the stock at time t results in an immediate cash update st (you may ignore transactions costs). If the investor chooses to inject a cash amount ct ∈ {0, 1}, there is a corresponding penalty of −ct (which is not taken from the fund).

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Let Wt = St + Ct denote their wealth at time t, where Ct is the total cash in the portfolio. Dynamics The stock price follows a discrete Markov chain with P (St+1 = s | St = s) = 0.5, i.e. with probability 0.5 the stock remains the same price over the time interval. P (St+1 = s + 1 | St = s) = P (St+1 = s − 1 | St = s) = 0.25. If the wealth moves off the grid it simply bounces to the nearest value in the grid at that time. The states are grid squares, identified by their row and column number (row first). The investor always starts in state (1,0) (i.e., the initial wealth Wt0 = 1 at time t0 = 0—there is no cash in the fund) and both states in the last column (i.e., at time t = t1 = 1) are terminal. Using the Bellman equation (with generic state notation), give the first round of value iteration updates for each state by completing the table below. You may ignore the time value of money, i.e. set γ = 1. Vi+1 (s) = max( a

T (s, a, s )(R(s, a, s ) + γ Vi (s )))

s

(w,t) V0 (w) V1 (w)

(0,0) 0 ?

(1,0) 0 ?

Exercise 9.5* Deterministic policies such as the greedy policy pi% (a|s) = arg maxπ Qπ (s, a) are invariant with respect to a shift of the action-value function by an arbitrary ˜ π (s, a) where function of a state f (s): π% (a|s) = arg maxπ Qπ (s, a) = arg maxπ Q π π ˜ (s, a) = Q (s, a) − f (s). Show that this implies that the optimal policy is also Q invariant with respect to the following transformation of an original reward function r(st , at , st+1 ): r˜ (st , at , st+1 ) = r(st , at , st+1 ) + γf (st+1 ) − f (st ). This transformation of a reward function is known as reward shaping (Ng, Russell 1999). It has been used in reinforcement learning to accelerate learning in certain settings. In the context of inverse reinforcement learning, reward shaping invariance has far-reaching implications, as we will discuss later in the book. Exercise 9.6** Define the occupancy measure ρπ : S × A → R by the relation Table 9.5 The reward function depends on fund wealth w and time

w 1 0

t0 0 0

t1 0 −10

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9 Introduction to Reinforcement Learning

ρπ (s, a) = π(a|s)

∞

γ t Pr (st = s|π ) ,

t=0

where Pr (st = s|π ) is the probability density of the state s = st at time t following policy π . The occupancy measure ρπ (s, a) can be interpreted as an unnormalized density of state-action pairs. It can be used, e.g., to specify the value function as an expectation value of the reward: V =< r(s, a) >ρ . a. Compute the policy in terms of the occupancy measure ρπ . b. Compute a normalized occupancy measure ρ˜π (s, a). How different the policy will be if we used the normalized measure ρ˜π (s, a) instead of the unnormalized measure ρπ ? Exercise 9.7** Theoretical models for reinforcement learning typically assume that rewards rt := r(st , at , st+1 ) are bounded: rmin ≤ rt ≤ rmax with some fixed values rmin , rmax . On the other hand, some models of rewards used by practitioners may produce (numerically) unbounded rewards. For example, with linear architectures, a popular choice of a reward function is a linear expansion rt = K k=1 θk k (st , at ) over a set of K basis functions k . Even if one chooses a set of bounded basis functions, this expression may become unbounded via a choice of coefficients θt . a. Use the policy invariance under linear transforms of rewards (see Exercise 9.1) to equivalently formulate the same problem with rewards that are bounded to the unit interval [0, 1], so they can be interpreted as probabilities. b. How could you modify a linear unbounded specification of reward rθ (s, a, s ) = k=1K θk k (s, a, s ) to a bounded reward function with values in a unit interval [0, 1]? Exercise 9.8 Consider an MDP with a finite number of states and actions in a real-time setting where the agent learns to act optimally using the ε-greedy policy. The ε-greedy policy amounts to taking an action a% = argmaxa Q(s, a ) in each state s with probability 1−ε, and taking a random action with probability ε. Will SARSA and Qlearning converge to the same solution under such policy, using a constant value of ε? What will be different in the answer if ε decays with the epoch, e.g. as εt ∼ 1/t? Exercise 9.9 Consider the following single-step random cost (negative reward) C (st , at , st+1 ) = ηat + (K − st+1 − at )+ , where η and K are some parameters. You can use such a cost function to develop an MDP model for an agent learning to invest. For example, st can be the current assets in a portfolio of equities at time t, at be an additional cash added to or subtracted from the portfolio at time t, and st+1 be the portfolio value at the end of time interval

7 Exercises

341

[t, t + 1). The second term is an option-like cost of a total portfolio (equity and cash) shortfall by time t + 1 from a target value K. Parameter η controls the relative importance of paying costs now as opposed to delaying payment. a. What is the corresponding expected cost for this problem, if the expectation is taken w.r.t. to the stock prices and at is treated as deterministic? b. Is the expected cost a convex or concave function of the action at ? c. Can you find an optimal one-step action at% that minimizes the one-step expected cost? Hint: For Part (i), you can use the following property: d d [y − x]+ = [(y − x)H (y − x)] , dx dx where H (x) is the Heaviside function. Exercise 9.10 Exercise 9.9 presented a simple single-period cost function that can be used in the setting of model-free reinforcement learning. We can now formulate a model based formulation for such an option-like reward. To this end, we use the following specification of the random end-of-period portfolio state: st+1 = (1 + rt )st rt = G(Ft ) + εt . In words, the initial portfolio value st + at in the beginning of the interval [t, t + 1) grows with a random return rt given by a function G(Ft ) of factors Ft corrupted by noise ε with E [ε] = 0 and E ε2 = σ 2 . a. Obtain the form of expected cost for this specification in Exercise 9.9. b. Obtain the optimal single-step action for this case. c. Compute the sensitivity of the optimal action with & respect' to the i-th factor Fit assuming the sigmoid link function G(Ft ) = σ i ωi Fit and a Gaussian noise εt . Exercise 9.11 Assuming a discrete set of actions at ∈ A of dimension K show that deterministic policy optimization by greedy policy of Q-learning Q(st , at% ) = maxat ∈A Q(st , at ) can be equivalently expressed as maximization over a set probability distributions π(at ) with probabilities πk for at = Ak , k = 1, . . . K (this relation is known as Fenchel duality): max Q(st , at ) = max

at ∈A

{π }k

K k=1

πk Q (st , Ak ) s.t. 0 ≤ πi ≤ 1,

K k=1

πk = 1.

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9 Introduction to Reinforcement Learning

Exercise 9.12** The reformulation of a deterministic policy search in terms of search over probability distributions given in Exercise 9.11 is a mathematical identity where the end result is still a deterministic policy. We can convert it to a probabilistic policy search if we modify the objective function max Q(st , at ) = max

at ∈A

{π }k

K

πk Q (st , Ak ) s.t. 0 ≤ πi ≤ 1,

k=1

K

πk = 1

k=1

by adding to it a KL divergence of the policy π with some reference (“prior”) policy ω: G% (st , at ) = max π

K

πk Q (st , Ak ) −

k=1

K πk 1 πi log , β ωk k=1

where β is a regularization parameter controlling the relative importance of the two terms that enforce, respectively, maximization of the action-value function and a preference for a previous reference policy ω with probabilities ωk . When parameter β < ∞ is finite, this produces a stochastic rather than deterministic optimal policy π % (a|s). Find the optimal policy π % (a|s) from the entropy-regularized functional G(st , at ) (Hint: use the method of Lagrange multipliers to enforce the normalization constraint k πk = 1). Exercise 9.13** Regularization by KL-divergence with a reference distribution ω introduced in the previous exercise can be extended to a multi-period setting. This produces maximum entropy reinforcement learning which augments the standard RL reward by an additional entropy penalty term in the form of KL divergence. The optimal value function in MaxEnt RL is ∞

* ) π(at |st ) 1 % t F (s) = max E γ r(st , at , st+1 ) − log (9.78) s0 = s , π β π0 (at |st ) t=0

where E [·] stands for an average under a stationary distribution ρπ (a) = s μπ (s)π(a|s) where μπ (s) is a stationary distribution over states induced by the policy π , and π0 is some reference policy. Show that the optimal policy for this entropy-regularized MDP problem has the following form: π % (a|s) =

π π 1 π0 (at |st )eβGt (st ,at ) , Zt ≡ π0 (at |st )eβGt (st ,at ) , Zt a t

(9.79)

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343

π (s where Gπt (st , at ) = Eπ [r(st , at , st+1 )] + γ st+1 p(st+1 |st , at )Ft+1 t+1 ). Check that the limit β → ∞ reproduces the standard deterministic policy, that is limβ→∞ V % (s) = maxπ V π (s), while in the opposite limit β → 0 we obtain a random and uniform policy. We will return to entropy-regularized value-based RL and stochastic policies such as (9.79) (which are sometimes referred to as Boltzmann policies) in later chapters of this book. Exercise 9.14* Show that the solution for the coefficients Wtk in the LSPI method (see Eq. (9.71)) is W%t = S−1 t Mt , where St is a matrix and Mt is a vector with the following elements: (t) Snm =

N

n Xt(k) , at(k) m Xt(k) , at(k)

k=1

Mn(t) =

N

(k) (k) (k) (k) (k) (k) (k) Rt Xt , at , Xt+1 + γ Qπt+1 Xt+1 , π Xt+1 . n Xt , at

k=1

Exercise 9.15** Consider the Boltzmann weighted average of a function h(i) defined on a binary set I = {1, 2}: Boltzβ h(i) =

h(i)

i∈I

eβh(i) βh(i) i∈I e

a. Verify that this operator smoothly interpolates between the max and the mean of h(i) which are obtained in the limits β → ∞ and β → 0, respectively. b. By taking β = 1, h(1) = 100, h(2) = 1, h (1) = 1, h (2) = 0, show that Boltzβ is not a non-expansion. c. (Programming) Using operators that are not non-expansions can lead to a loss of a solution in a generalized Bellman equation. To illustrate such phenomenon, we use the following simple example.Consider the MDP problem on the set I = {1, 2} with two actions a and b and the following specification: p(1|1, a) = 0.66, p(2|1, a) = 0.34, r(1, a) = 0.122 and p(1|1, b) = 0.99, p(1|1, b) = 0.01, r(1, b) = 0.033. The second state is absorbing with p(1|2) = 0, p(2|2) = 1. The discount factor is γ = 0.98. Assume we use the Boltzmann policy ˆ

eβ Q(s,a) π(a|s) = . ˆ β Q(s,a) ae

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9 Introduction to Reinforcement Learning

Show that the SARSA algorithm " ! ˆ a) , ˆ a) ← Q(s, ˆ a) + α r(s, a) + γ Q(s ˆ , a ) − Q(s, Q(s, where a, a are drawn from the Boltzmann policy with β = 16.55 and α = 0.1, ˆ 1 , a) that do not achieve stable ˆ 1 , a) and Q(s leads to oscillating solutions for Q(s states with an increased number of iterations. Exercise 9.16** An alternative continuous approximation to the intractable max operator in the Bellman optimality equation is given by the mellowmax function (Asadi and Littman 2016) , n 1 ωxi 1 mmω (X) = log e ω n i=1

a. Show that the mellowmax function recovers the max function in the limit ω → ∞. b. Show that mellowmax is a non-expansion.

Appendix Answers to Multiple Choice Questions Question 1 Answer: 2, 4. Question 2 Answer: 2, 4 Question 3 Answer: 5

Python Notebooks The notebooks provided in the accompanying source code repository accompany many of the examples in this chapter, including Q-learning and SARSA for the financial cliff walking problem, the market impact problem, and electronic market making. The repository also includes an example implementation of the LSPI algorithm for optimal allocation in a Markowitz portfolio. Further details of the notebooks are included in the README.md file.

References

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References Asadi, K., & Littman, M. L. (2016). An alternative softmax operator for reinforcement learning. Proceedings of ICML. Bellman, R. E. (1957). Dynamic programming. Princeton, NJ: Princeton University Press. Bertsekas, D. (2012). Dynamic programming and optimal control (vol. I and II), 4th edn. Athena Scientific. Lagoudakis, M. G., & Parr, R. (2003). Least-squares policy iteration. Journal of Machine Learning Research, 4, 1107–1149. Littman, M. L., & Szepasvari, S. (1996). A generalized reinforcement-learning model: convergence and applications. In Machine Learning, Proceedings of the Thirteenth International Conference (ICML ’96), Bari, Italy. Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., et al. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540), 529–533. Robbins, H., & Monro, S. (1951). A stochastic approximation method. Ann. Math. Statistics, 22, 400–407. Sutton, R. S., & Barto, A. G. (2018). Reinforcement learning: An introduction, 2nd edn. MIT. Szepesvari, S. (2010). Algorithms for reinforcement learning. Morgan & Claypool. Thompson, W. R. (1935). On a criterion for the rejection of observations and the distribution of the ratio of deviation to sample standard deviation. Ann. Math. Statist., 6(4), 214–219. Thompson, W. R. (1993). On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika, 25(3), 285–94. van Hasselt, H. (2010). Double Q-learning. Advances in Neural Information Processing Systems. http://papers.nips.cc/paper/3964-double-q-learning.pdf.

Chapter 10

Applications of Reinforcement Learning

This chapter considers real-world applications of reinforcement learning in finance, as well as further advances in the theory presented in the previous chapter. We start with one of the most common problems of quantitative finance, which is the problem of optimal portfolio trading in discrete time. Many practical problems of trading or risk management amount to different forms of dynamic portfolio optimization, with different optimization criteria, portfolio composition, and constraints. This chapter introduces a reinforcement learning approach to option pricing that generalizes the classical Black–Scholes model to a data-driven approach using Q-learning. It then presents a probabilistic extension of Q-learning called G-learning and shows how it can be used for dynamic portfolio optimization. For certain specifications of reward functions, G-learning is semi-analytically tractable and amounts to a probabilistic version of linear quadratic regulators (LQR). Detailed analyses of such cases are presented, and show their solutions with examples from problems of dynamic portfolio optimization and wealth management.

1 Introduction In this chapter, we consider real-world applications of reinforcement learning in finance. We start with one of the most common problems of quantitative finance, which is the problem of optimal portfolio trading. Many practical problems of trading or risk management amount to different forms of dynamic portfolio optimization, with different optimization criteria, portfolio composition, and constraints. For example, the problem of optimal stock execution can be viewed as a problem of optimal dynamic management of a portfolio of stocks of the same company, with the objective being minimization of slippage costs of selling the stock. A more traditional example of dynamic portfolio optimization is given by asset managers and mutual or pension funds who usually manage investment © Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_10

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portfolios over long time horizons (months or years). Intra-day trading which is more typical of hedge funds can also be thought of as a dynamic portfolio optimization problem with a different portfolio choice, time step, constraints, and so on. In addition to different time horizons and objective functions, details of a portfolio optimization problem determine choices for features and hence for a state description. For example, management of long-horizon investment portfolios typically involves macroeconomic factors but not the limit order book data, while for intra-day trading it is the opposite case. Dynamic portfolio management is a problem of stochastic optimal control where control variables are represented by changes in positions in different assets in a portfolio made by a portfolio manager, and state variables describe the current composition of the portfolio, prices of its assets, and possibly other relevant features including market indices, bid–ask spreads, etc. If we consider a large market player whose trade can move the market, actions of such trader may produce a feedback loop effect. The latter is referred to in the financial literature as a “market impact effect.” All the above elements of dynamic portfolio management make it suitable for applying methods of dynamic programming and reinforcement learning. While the previous chapter introduced the main concepts and methods of reinforcement learning, here we want to take a more detailed look at practical applications for portfolio management problems. When viewed as problems of optimal control to be addressed using reinforcement learning, such problems typically have a very high-dimensional state-action space. Indeed, even if we constrain ourselves by actively traded US stocks, we get around three thousands stocks. If we add to this other assets such as futures, exchangetraded funds, government and corporate bonds, etc., we may end up with state spaces of dimensions of many thousands. Even in a more specialized case of an equity fund whose objective is to beat a given benchmark portfolio, the investment universe may be tens or hundreds of stocks. This means that such applications of reinforcement learning in finance have to handle (very) high-dimensional and typically continuous (or approximately continuous) state-action spaces. Clearly, such high-dimensional RL problems are far more complex than simple low-dimensional examples typically used to test reinforcement learning methods, such as inverted pendulum or cliff walking problems described in the Sutton–Barto book, or the “financial cliff walking” problem presented in the previous chapter. Modern RL methods applied to problems outside of finance typically operate with action space dimensions measured in tens but not hundreds or thousands, and typically have a far larger signal-to-noise ratio than financial problems. Low signal-to-noise ratios and potentially very high dimensionality are therefore two marked differences of applications of reinforcement learning in finance as opposed to applications to video games and robotics. As high-dimensional optimal control problems are harder than low-dimensional ones, we first want to explore applications of reinforcement learning with lowdimensional portfolio optimization problems. Such an approach both sets the

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grounds for more complex high-dimensional applications and can be of independent interest when applied to practically interesting problems falling in this general class of dynamic portfolio optimization. The first problem that we address in this chapter is exactly of this kind: it is both low-dimensional and of practical interest on its own, rather than being a toy example for a multi-asset portfolio optimization. Namely, we will consider the classical problem of option pricing, in a formulation that closely resembles the framework of the celebrated Black–Scholes–Merton (BSM) model , also known as the Black– Scholes (BS) model, one of the cornerstones of modern quantitative finance (Black and Scholes 1973; Merton 1974). Chapter Objectives This chapter will present a few practical cases of using reinforcement learning in finance: – – – –

RL for option pricing and optimal hedging (QLBS); G-learning for stock portfolios and linear quadratic regulators; RL for optimal consumption using G-learning; and RL for portfolio optimization using G-learning.

The chapter is accompanied by two notebooks implementing the QLBS model for option pricing and hedging, and G-learning for wealth management. See Appendix “Python Notebooks” for further details.

2 The QLBS Model for Option Pricing The BSM model was initially developed for the so-called plain vanilla European call and put options. A European call option is a contract that allows a buyer to obtain a given stock at some future time T for a fixed price K. If ST is the stock price at time T , then the payoff to the option buyer at time T is (ST − K)+ . Similarly, a buyer of a European put option has a right to sell the stock at time T for a fixed price K, with a terminal payoff of (K − ST )+ . European call and put options are among the simplest and most popular types of financial derivatives whose value is derived from (or driven by) the underlying stock (or more generally, the underlying asset). The core idea of the BSM model is that options can be priced using the relative value approach to asset pricing, which prices assets in terms of other tradable assets. The relative pricing method for options is known as dynamic option replication. It is based on the observation that an option payoff depends only on the price of a stock at expiry of the option. Therefore, if we neglect other sources of uncertainty such as stochastic volatility, the option value at arbitrary times before the expiry should only depend on the stock value. This makes it possible to mimic the option using a simple portfolio made of the underlying stock and cash, which is called the hedge

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portfolio. The hedge portfolio is dynamically managed by continuously rebalancing its wealth between the stock and cash. Moreover, this is done in a self-financing way, meaning that there are no cash inflows/outflows in the portfolio except at the time of inception. The objective of dynamic replication is to mimic the option using the hedge portfolio as closely as possible. In the continuous-time setting of the original BSM model, it turns out that such dynamic replication can be made exact by a continuous rebalancing of the hedge portfolio between the stock and cash, such that the amount of stock coincides with the option price sensitivity with respect to the stock price. This makes the total portfolio made of the option and the hedge portfolio instantaneously risk-free, or equivalently it makes the option instantaneously perfectly replicable in terms of the stock and cash. Risk of mis-hedging between the option and its underlying is instantaneously eliminated; therefore, the full portfolio involving the option and its hedge should earn a risk-free rate. The option price in this limit does not depend on risk preferences of investors. Such analysis performed in the continuous-time setting gives rise to the celebrated Black–Scholes partial differential equation (PDE) for option prices, whose solution produces option prices as deterministic functions of current stock prices. The Black–Scholes PDE can be derived using analysis of the hedge portfolio in discrete time with time steps t, and then taking the continuous-time limit t → 0, see, e.g., Wilmott (1998). It can be shown that the resulting continuous-time BSM model does not amount to a problem of sequential decision making and does not in general reduce to any sort of optimization problem. However, in option markets, rebalancing of option replication (hedge) portfolio occurs at finite frequency, e.g. daily. A frequent rebalancing can be costly due to transaction costs which are altogether neglected in the classical BSM model. When transaction costs are added, a formal continuous-time limit may not even exist as it leads to formally infinite option prices due to an infinite number of portfolio rebalancing acts. With a finite rebalancing frequency, perfect replication is no longer feasible, and the replicating portfolio will in general be different from the option value according to the amount of hedge slippage. The latter depends on the stock price evolution between consecutive rebalancing acts for the portfolio. Respectively, in the absence of perfect replication, a hedged option position carries some mis-hedging risk which the option buyer or seller should be compensated for. This means that once we revert from the idealized setting of continuous-time finance to a realistic setting of discretetime finance, option pricing becomes dependent on investors’ risk preferences. If we take a view of an option seller agent in such a discrete-time setting, its objective should be to minimize some measures of slippage risk, also referred to as a “risk-adjusted cost of hedging” the option, by dynamic option replication. When viewed over the lifetime of an option, this setting can be considered a sequential decision-making process of minimization of slippage cost (or equivalently maximization of rewards determined as negative costs). While such a discrete-time approach converges to the Black–Scholes formulation in the limit of vanishing time steps, it offers both a more realistic setting, and allows one to focus on the

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key objective of option trading and pricing, which is risk minimization by hedging in a sequential decision-making process. This makes option pricing amenable to methods of reinforcement learning, and indeed, as we will see below, option pricing and hedging in discrete time amounts to reinforcement learning. Casting option pricing as a reinforcement learning task offers a few interesting insights. First, if we select a specific model for the stock price dynamics, we can use model-based reinforcement learning as a powerful sample-based (Monte Carlo) computational approach. The latter may be advantageous to other numerical methods such as finite differences for computing option prices and hedge ratios, especially when dimensionality of the state space goes beyond three or four. Second, we may rely on model-free reinforcement learning methods such as Q-learning, and bypass the need to build a model of stock price dynamics altogether. RL provides a framework for model-free learning of option prices and hedges.1 While we only consider the simplest setting for a reinforcement learning approach to pricing and hedging of European vanilla options (e.g., put or call options), the approach can be extended in a straightforward manner to more complex instruments including options on multiple assets, early exercises, option portfolios, market frictions, etc. The model presented in this chapter is referred to as the QLBS model, in recognition of the fact that it combines the Q-learning method of Watkins (1989); Watkins and Dayan (1992) with the method of dynamic option replication of the (time-discretized) Black–Scholes model. As Q-learning is a model-free method, this means that the QLBS model is also model-free. More accurately, it is distributionfree: option prices in this approach depend on the chosen utility function, but do not rely on any model for the stock price distribution, and instead use only samples from this distribution. The QLBS model may also be of interest as a financial model which relates to the literature on hedging and pricing in incomplete markets (Föllmer and Schweizer 1989; Schweizer 1995; Cerný and Kallsen 2007; Potters et al. 2001; Petrelli et al. 2010; Grau 2007). Unlike many previous models of this sort, QLBS ensures a full consistency of hedging and pricing at each time step, all within an efficient and datadriven Q-learning algorithm. Additionally, it extends the discrete-time BSM model. Extending Markowitz portfolio theory (Markowitz 1959) to a multi-period setting, Sect. 3 incorporates a drift in a risk/return analysis of the option’s hedge portfolio. This extension allows one to consider both hedging and speculation with options in a consistent way within the same model, which is a challenge for the standard BSM model or its “phenomenological” generalizations, see, e.g., Wilmott (1998). Following this approach, it turns out that all results of the classical BSM model (Black and Scholes 1973; Merton 1974) can be obtained as a continuous-time limit t → 0 of a multi-period version of the Markowitz portfolio theory (Markowitz 1959), if the dynamics of stock prices are log-normal, and the investment portfolio 1 Here

we use the notion of model-free learning in the same context as it is normally used in the machine learning literature, namely as a method that does not rely on an explicit model of feature dynamics. Option prices and hedge ratios in the framework presented in this section depend on a model of rewards, and in this sense are model-dependent.

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is self-replicating. However, this limit is degenerate: all fluctuations of the “true” option price asymptotically decay in this limit, resulting in a deterministic option price which is independent of risk preferences of an investor. However, as long as the time step t is kept finite, both risk of option mis-hedging and dependence of the option price on investor risk preferences persist. To the extent that option pricing in discrete time amounts to either DP (a.k.a. model-based RL), if a model is known, or RL if a model is unknown, we may say that the classical continuous-time BSM model corresponds to the continuous-time limit of model-based reinforcement learning. In such a limit, all data requirements are reduced to just two numbers—the current stock price and volatility.

3 Discrete-Time Black–Scholes–Merton Model We start with a discrete-time version of the BSM model. As is well known, the problem of option hedging and pricing in this formulation amounts to a sequential risk minimization. The main open question is how to define risk in an option. In this part, we follow a local risk minimization approach pioneered in the work of Föllmer and Schweizer (1989), Schweizer (1995), Cerný and Kallsen (2007). A similar method was developed by physicists Potters et al. (2001), see also the work by Petrelli et al. (2010). We use a version of this approach suggested in Grau (2007). In this approach, we take the view of a seller of a European option (e.g., a put option) with maturity T and the terminal payoff of HT (ST ) at maturity, that depends on a final stock price ST at that time. To hedge the option, the seller uses proceeds of the sale to set up a replicating (hedge) portfolio $t composed of the stock St and a risk-free bank deposit Bt . The value of hedge portfolio at any time t ≤ T is $t = ut St + Bt ,

(10.1)

where ut is a position in the stock at time t, taken to hedge risk in the option.

3.1 Hedge Portfolio Evaluation As usual, the replicating portfolio tries to exactly match the option price in all possible future states of the world. If we start at maturity T when the option position is closed, the hedge ut should be closed at the same time, thus we set uT = 0 and therefore $T = BT = HT (ST ),

(10.2)

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353

which sets a terminal condition for BT that should hold in all future states of the world at time T .2 To find an amount needed to be held in the bank account at previous times t < T , we impose the self-financing constraint which requires that all future changes in the hedge portfolio should be funded from an initially set bank account, without any cash infusions or withdrawals over the lifetime of the option. This implies the following relation that ensures conservation of the portfolio value by a re-hedge at time t + 1: ut St+1 + ert Bt = ut+1 St+1 + Bt+1 .

(10.3)

This relation can be expressed recursively in order to calculate the amount of cash in the bank account to hedge the option at any time t < T using its value at the next time step: Bt = e−rt [Bt+1 + (ut+1 − ut ) St+1 ] , t = T − 1, . . . , 0.

(10.4)

Substituting this into Eq. (10.1) produces a recursive relation for $t in terms of its values at later times, which can therefore be solved backward in time, starting from t = T with the terminal condition (10.2), and continued through to the current time t = 0: $t = e−rt [$t+1 − ut St ] , St = St+1 − ert St , t = T − 1, . . . , 0. (10.5) Note that Eqs. (10.4) and (10.5) imply that both Bt and $t are not measurable at any t < T , as they depend on the future. Respectively, their values today B0 and $0 will be random quantities with some distributions. For any given hedging strategy {ut }Tt=0 , these distributions can be estimated using Monte Carlo simulation, which first simulates N paths of the underlying S1 → S2 → . . . → SN , and then evaluates $t going backward on each path. Note that because the choice of a hedge strategy does not affect the evolution of the underlying, such simulation of forward paths should only be performed once, and then re-used for future evaluations of the hedge portfolio under difference hedge strategy scenarios. Alternatively, the distribution of the hedge portfolio value $0 can be estimated using real historical data for stock prices, together with a pre-determined hedging strategy {ut }Tt=0 and a terminal condition (10.2). To summarize, the forward pass of Monte Carlo simulation is done by simulating the process S1 → S2 → . . . → SN , while the backward pass is performed using the recursion (10.5) which takes a prescribed hedge strategy, {ut }Tt=0 , and backpropagates uncertainty in the future into uncertainty today, via the self-financing constraint (10.3) (Grau 2007) which serves as a “time machine for risk.” transaction costs are neglected, taking uT = 0 simply means converting all stock into cash. For more details on the choice uT = 0, see Grau (2007).

2 When

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As a result of such “back-propagation of uncertainty” from the future to the current time t, the option replicating portfolio $t at time t is a random quantity with a certain distribution. The option price acceptable to the option seller would then be determined by risk preferences of the option seller. The option price can, for example, be taken to be the mean of the distribution of $t , plus some premium for risk. Clearly, the option price can be determined only after the seller decides on a hedging strategy, {ut }Tt=0 , to be used in the future, which would be applied in the same way (as a mapping) for any future value, {$t }Tt=0 . The choice of an optimal hedge strategy, {ut }Tt=0 , will therefore be discussed next.

3.2 Optimal Hedging Strategy Unlike the recursive calculation of the hedge portfolio value (10.5) which is performed path-wise, optimal hedges are computed using a cross-sectional analysis that operates simultaneously over all paths. This is because we need to learn a hedging strategy, {ut }Tt=0 , which would apply to all states that might be encountered in the future, but each given path only produces one value St at time t. Therefore, to compute an optimal hedge, ut (St ), for a given time step t, we need cross-sectional information on all concurrent paths. As with the portfolio value calculation, the optimal hedges, {ut }Tt=0 , are computed backward in time, starting from t = T . However, because we cannot know the future when we compute a hedge, for each time t, any calculation of an optimal hedge, ut , can only condition on the information Ft available at time t. This calculation is similar to the American Monte Carlo method of Longstaff and Schwartz (2001).

•

> Longstaff–Schwartz American Monte Carlo Option Pricing

While the objective of the American Monte Carlo method of Longstaff and Schwartz (2001) is altogether different from the problem addressed in this chapter (a risk-neutral valuation of an American option vs a real-measure discrete-time hedging/pricing of a European option), the mathematical setting is similar. Both problems look for an optimal strategy, and their solution requires a backward recursion in combination with a forward simulation. Here we provide a brief outline of their method. The main idea of the LSM approach of Longstaff and Schwartz (2001) is to treat the backward-looking stage of the security evaluation as a regression problem formulated in a forward-looking manner which is more suited for (continued)

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355

a Monte Carlo (MC) setting. The starting point is the (backward-looking) Bellman equation, the most fundamental equation of the stochastic optimal control (otherwise known as stochastic optimization). For an American option on a financial underlying, the control variable is binary: “exercise” or “not exercise.” The Bellman equation for this particular case produces the continuation value, Ct (St ), at time t as a function of the current underlying value St : Ct (St ) = E e−rt max (ht+t (St+t ), Ct+t (St+t )) Ft .

(10.6)

Here hτ (Sτ ) is the option payoff at time τ . For example, for an American put option, hτ (Sτ ) = (K − Sτ )+ . Note that for American options, the continuation value should be estimated as a function Ct (x) of the value x = Xt , as long we want to know whether it is larger or smaller than the intrinsic value, H (Xt ), for a particular realization Xt = x of the process Xt at time t. The problem is, of course, that each Monte Carlo path has exactly one value of Xt at time t. One way to estimate a function Ct (St ) is to use all paths, i.e. use the cross-sectional information. To this end, the one-step Bellman equation (10.6) is interpreted as a regression of the form max (ht+t (St+t ), Ct+t (St+t )) = ert Ct (St ) + εt (St ),

(10.7)

where εt (St ) is a random noise at time t with mean zero, which may in general depend on the underlying value St at that time. Clearly (10.7) and (10.6) are equivalent in expectations, as taking the expectation of both sides of (10.7), we recover (10.6). Next the unknown function Ct (St ) is expanded in a set of basis functions: Ct (x) = an (t)φn (x), (10.8) n

for some particular choice of the basis {φn (x)}, and the coefficients an (t) are then calculated using the least squares regression of max (ht+t (St+t ), Ct+t (St+t )) on the value St of the underlying at time t across all Monte Carlo paths.

The optimal hedge, u% (St ), in this model is obtained from the requirement that the variance of $t across all simulated paths at time t is minimized when conditioned on the currently available cross-sectional information Ft , i.e.

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u%t (St ) = argmin V ar [$t |Ft ] u

= argmin V ar [$t+1 − ut St |Ft ] , u

t = T − 1, . . . , 0.

(10.9)

Note the first expression in Eq. (10.9) implies that all uncertainty in $t is due to uncertainty regarding the amount Bt needed to be held in the bank account at time t in order to meet future obligations at the option maturity T . This means that an optimal hedge should minimize the cost of a hedge capital for the option position at each time step t. The optimal hedge can be found analytically by setting the derivative of (10.9) to zero. This gives u%t (St ) =

Cov ( $t+1 , St | Ft ) , t = T − 1, . . . , 0. V ar ( St | Ft )

(10.10)

This expression involves one-step expectations of quantities at time t+1, conditional on time t. How they can be computed depends on whether we deal with a continuous or a discrete state space. If the state space is discrete, then such one-step conditional expectations are simply finite sums involving transition probabilities of an MDP model. If, on the other hand, we work in a continuous-state setting, these conditional expectations can be calculated in a Monte Carlo setting by using expansions in basis functions, similar to the LSMC method of Longstaff and Schwartz (2001), or realmeasure MC methods of Grau (2007), Petrelli et al. (2010), Potters et al. (2001). In our exposition below, we use a general notation as in Eq. (10.10) to denote similar conditional expectations where Ft denotes cross-sectional information set at time t, which lets us keep the formalism general enough to handle both cases of a continuous and a discrete state spaces, and discuss simplifications that arise in a special case of a discrete-state formulation separately, whenever appropriate.

3.3 Option Pricing in Discrete Time We start with the notion of a fair option price Cˆ t defined as a time-t expected value of the hedge portfolio $t : Cˆ t = Et [ $t | Ft ] .

(10.11)

Using Eq. (10.5) and the tower law of conditional expectations, we obtain Cˆ t = Et e−rt $t+1 Ft − ut (St ) Et [ St | Ft ] = Et e−rt Et+1 [ $t+1 | Ft+1 ] Ft − ut (St ) Et [ St | Ft ] (10.12) " ! = Et e−rt Cˆ t+1 Ft − ut (St ) Et [ St | Ft ] , t = T − 1, . . . , 0.

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Note that we can similarly use the tower law of conditional expectations to express the optimal hedge in terms of Cˆ t+1 instead of $t+1 :

Cov Cˆ t+1 , St Ft | F , S Cov $ ( ) t+1 t t % ut (St ) = = . V ar ( St | Ft ) V ar ( St | Ft )

(10.13)

If we now substitute (10.13) into (10.12) and re-arrange terms, we can put the recursive relation for Cˆ t in the following form: " ! ˆ Cˆ t = e−rt EQ Cˆ t+1 Ft , t = T − 1, . . . , 0,

(10.14)

ˆ is a signed measure with transition probabilities where Q (St − Et [St ]) Et [St ] q˜ ( St+1 | St ) = p ( St+1 | St ) 1 − , V ar ( St | Ft )

(10.15)

and where p ( St+1 | St ) are transition probabilities under the physical measure P. Note that for sufficiently large moves of St , this expression may become negative. ˆ is not a genuine probability measure, but rather only a signed This means that Q measure (a signed measure, unlike a regular measure, can take both positive and negative values). A potential for a negative fair option price, Cˆ t , is a well-known property of quadratic risk minimization schemes (Cerný and Kallsen 2007; Föllmer and Schweizer 1989; Grau 2007; Potters et al. 2001; Schweizer 1995). However, we note that the “fair” (expected) option price (10.11) is not a price a seller of the option should charge. The actual fair risk-adjusted price is given by Eq. (10.16) below, which can always be made non-negative by a proper level of risk-aversion λ which is defined by the seller’s risk preferences.3 The reason why the fair option price is not yet the price that the option seller should charge for the option is that she is exposed to the risk of exhausting the bank account Bt at some time in the future, after any fixed amount Bˆ 0 = E0 [B0 ] is paid into the bank account at time t = 0 upon selling the option. If necessary, the option seller would need to add cash to the hedge portfolio and she has to be compensated for such a risk. One possible specification of a risk premium, that the dealer has to add on top of the fair option price, is her own optimal ask price to add to the cumulative expected discounted variance of the hedge portfolio along all time steps t = 0, . . . , N , with a risk-aversion parameter λ:

3 If

it is desired to have non-negative option prices for arbitrary levels of risk-aversion, the method developed below can be generalized by using non-quadratic utility functions instead of the quadratic Markowitz utility. This would incur a moderate computational overhead of numerically solving a convex optimization problem at each time step, instead of a quadratic optimization that is solved semi-analytically.

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(ask)

C0

(S, u) = E0 $0 + λ

T t=0

−rt e V ar [ $t | Ft ] S0 = S, u0 = u.

(10.16)

In order to proceed further, we first note that the problem of minimization of a fair (to the dealer) option price (10.16) can be equivalently expressed as the problem of (ask) , where maximization of its negative Vt = −Ct Vt (St ) = Et

−$t − λ

T

e

−r(t −t)

t =t

Example 10.8

V ar [ $t | Ft ] Ft .

(10.17)

Option pricing with non-quadratic utility functions

The fact that the “fair” option price, Cˆ t , can become negative when price fluctuations are large is attributed to the non-monotonicity of the Markowitz quadratic utility. Non-monotonicity violates the Von Neumann–Morgenstern conditions U (a) ≥ 0, U (a) ≤ 0 on a utility function U (a) of a rational investor. While this problem with the quadratic utility function can be resolved by adding a risk premium to the fair option price, this may require that the risk-aversion parameter, λ, exceeds some minimal value. We can obtain nonnegative option prices with arbitrary values of risk-aversion if, instead of a quadratic utility, we use utility functions that satisfy the Von NeumannMorgenstern conditions. In particular, one popular choice is given by the exponential utility function U (X) = − exp(−γ X),

(10.18)

where γ is a risk-aversion parameter whose meaning is similar to parameter ρ in the quadratic utility. As shown in Halperin (2018), the hedges and prices corresponding to the quadratic risk minimization scheme can be obtained with the exponential utility in the limit of a small risk-aversion γ → 0, alongside calculable corrections via an expansion in powers of γ . Note that while the idea of adding an option price premium proportional to the variance of the hedge portfolio as done in Eq. (10.16) was initially suggested on the intuitive grounds by Potters et al. (2001), a utility-based approach presented in Halperin (2018) actually derives it as a quadratic approximation to a utility-based option price, which also establishes an approximate relation between a risk-aversion parameter λ of the quadratic risk optimization and a parameter γ of the exponential utility U (X) = − exp(−γ X): λ(

1 γ. 2

(10.19)

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3.4 Hedging and Pricing in the BS Limit The framework presented above provides a smooth transition to the strict BS limit t → 0. In this limit, the BSM model dynamics under the physical measure P is described by a continuous-time geometric Brownian motion with a drift, μ, and volatility, σ : dSt = μdt + σ dWt , St

(10.20)

where Wt is a standard Brownian motion. Consider first the optimal hedge strategy (10.13) in the BS limit t → 0. Using the first-order Taylor expansion ∂Ct Cˆ t+1 = Ct + St + O (t) ∂St

(10.21)

∂Ct , ∂St

(10.22)

in (10.13), we obtain % uBS t (St ) = lim ut (St ) = t→0

which is the correct optimal hedge in the continuous-time BSM model. To find the continuous-time limit of the option price, we first compute the limit of the second term in Eq. (10.12): ∂Ct dt. t→0 dt→0 dt→0 ∂St (10.23) To evaluate the first term in Eq. (10.12), we use the second-order Taylor expansion: lim ut (St ) Et [ St | Ft ] = lim uBS t St (μ − r)dt = lim (μ − r)St

∂Ct ∂Ct 1 ∂ 2 Ct dt + dSt + Cˆ t+1 = Ct + (dSt )2 + . . . ∂t ∂St 2 ∂St2

(10.24)

∂Ct ∂Ct dt + St (μdt + σ dWt ) ∂t ∂St

1 ∂ 2 Ct 2 2 2 + σ S dW + 2μσ dW dt + O dt 2 . t t t 2 ∂St2

= Ct +

Substituting Eqs. (10.23) and (10.24) into Eq. (10.12), using E [dWt ] = 0 and E dWt2 = dt, and simplifying, we find that the stock drift μ under the physical measure P drops out from the problem, and Eq. (10.12) becomes the celebrated Black–Scholes equation in the limit dt → 0: ∂Ct ∂Ct ∂ 2 Ct 1 + σ 2 St2 − rCt = 0. + rSt ∂t ∂St 2 ∂St2

(10.25)

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Therefore, if the stock price is log-normal, both our hedging and pricing formulae become the original formulae of the Black–Scholes–Merton model in the strict limit t → 0.

•

? Multiple Choice Question 1

Select all the following correct statements: a. In the Black–Scholes limit t → 0, the optimal hedge ut is equal to the BS delta ∂ 2 Ct . ∂St2 b. In the Black–Scholes limit t → 0, the optimal hedge ut is equal to the BS delta ∂Ct ∂St . c. The risk-aversion parameter λ drops from the problem of option pricing and hedging in the limit t → 0. d. For finite t, the optimal hedge ut depends on λ.

4 The QLBS Model We shall now re-formulate and generalize the discrete-time BSM presented in Sect. 3 using the framework of Markov Decision Processes (MDPs). The key idea is that the risk-based pricing and hedging in discrete time can be understood as an MDP problem with the value function to be maximized determined by Eq. (10.17). Recall that we defined this value function as the negative of a risk-adjusted option price for the option seller. The availability of an MDP formulation for option pricing is beneficial in multiple ways. First, it generalizes the BSM by providing a consistent option pricing and hedging method which can take the expected return of the option into decision making, and thus can be used by both types of market players that trade options: hedgers and speculators. Previous incomplete-market models for option pricing either do not ensure consistency of hedging and pricing, or do not allow for incorporation of stock returns into analysis, or both.4 Therefore, the MDP formulation improves the original discrete-time BSM model by making it more generally applicable. Second, the MDP formulation can be used to formulate new computational approaches to option pricing and hedging. Particular methods are chosen depending on assumptions on a data-generating stock price process. If we assume it is known, so that both transition probabilities and a reward function are known, the option

4 The

standard continuous-time BSM model is equivalent to using a risk-neutral pricing measure for option valuation. This approach only enables pure risk-based option hedging, which might be suitable for a hedger but not for an option speculator.

4 The QLBS Model

361

pricing problem can be solved by solving a Bellman optimality equation using dynamic programming or approximate dynamic programming. For the simplest onestock model formulation, we will show how it can be solved using a combination of Monte Carlo simulation of the underlying process and a recursive semi-analytical procedure that only involves matrix linear algebra (OLS linear regression) for a numerical implementation. Similar methods based on approximate dynamic programming can also be applied to more complex multi-dimensional extensions of the model. On the other hand, we might know only the general structure of an MDP model, but not its specifications such as transition probability and reward function. In this case, we should solve a backward recursion for the Bellman optimality equation relying only on samples of data. This is the setting of reinforcement learning. It turns out that a Bellman optimality equation for our MDP model, without knowing model dynamics by relying only on data, can be easily solved (also semi-analytically, due to a quadratic reward function) using Q-learning or its modifications. A particular choice between different versions of Q-learning is determined by how the state space is modeled. One can discretize the state and action spaces and work with Markov chain approximation to continuous stock price dynamics, see, e.g., Duan and Simonato (2001). If such a finite-state approximation to dynamics converges to the actual continuous-state dynamics, optimal option prices and hedge ratios computed with this approach also converge to their continuous-state limits. If the log-normal model is indeed the data generation process, prices and hedge ratios convergence to the classical BSM limits, once one further takes the limit t → 0, λ → 0 in resulting expressions. Another possibility is to keep the state space continuous and work with approximate methods to represent a Q-function. In particular, if linear architectures are used, we can use the Fitted Q-iteration (FQI) method (Ernst et al. 2005). For nonlinear architectures, neural Q-iteration methods use neural networks to represent a Q-function. Our presentation below is mostly focused on the continuous-state FQI method for the basic single-stock option setting that uses a linear architecture and a fixed set of basis functions. However, all formulas presented below can be easily adjusted to a finite-state formulation by using “one-hot” basis functions.

4.1 State Variables As stock price dynamics typically involve a deterministic drift term, we can consider a change of state variable such that new, time-transformed variables would be stationary, i.e. non-drifting. For a given stock price process St , we can achieve this by defining a new variable Xt by the following relation: * ) σ2 t + log St . Xt = − μ − 2

(10.26)

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10 Applications of Reinforcement Learning

The advantage of this representation can be clearly seen in a special case when St is a geometric Brownian motion (GBM). For this case, we obtain )

σ2 dXt = − μ − 2

* dt + d log St = σ dWt .

(10.27)

Therefore, when the true dynamics of St is log-normal, Xt is a standard Brownian motion, scaled by volatility, σ . If we know the value of Xt in a given MC scenario, the corresponding value of St is given by the formula St = e

2 Xt + μ− σ2 t

.

(10.28)

Note that as long as {Xt }Tt=0 is a martingale, i.e. E [dXt ] = 0, ∀t, on average it should not run too far away from an initial value X0 during the lifetime of an option. The state variable, Xt , is time-uniform, unlike the stock price, St , which has a drift term. But the relation (10.28) can always be used in order to map non-stationary dynamics of St onto stationary dynamics of Xt . The martingale property of Xt is also helpful for numerical lattice approximations, as it implies that a lattice should not be too large to capture possible future variations of the stock price. The change of variables (10.26) and its reverse (10.28) can also be applied when the stock price dynamics are not GBM. Of course, the new state variable Xt will not in general be a martingale in this case; however, it is intrinsically useful for separating non-stationarity of the optimization task from non-stationarity of state variables.

4.2 Bellman Equations We start by re-stating the risk minimization procedure outlined above in Sect. 3.2 in the language of MDP problems. In particular, time-dependent state variables, St , are expressed in terms of time-homogeneous variables Xt using Eq. (10.28). In addition, we will use the notation at = at (Xt ) to denote actions expressed as functions of time-homogeneous variables Xt . Actions, ut = ut (St ), in terms of stock prices are then obtained by the substitution ) * * ) σ2 t , ut (St ) = at (Xt (St )) = at log St − μ − 2

(10.29)

where we have used Eq. (10.26). To differentiate between the actual hedging decisions, at (xt ), where xt is a particular realization of a random state Xt at time t, and a hedging strategy that applies for any state Xt , we introduce the notion of a time-dependent policy, π (t, Xt ). We consider deterministic policies of the form:

4 The QLBS Model

363

π : {0, . . . , T − 1} × X → A,

(10.30)

which is a deterministic policy that maps the time t and the current state, Xt = xt , to the action at ∈ A: at = π(t, xt ).

(10.31)

We start with the value maximization problem of Eq. (10.17), which we rewrite here in terms of a new state variable Xt , and with an upper index to denote its dependence on the policy π : Vtπ (Xt ) = Et

−$t (Xt ) − λ

T

e

−r(t −t)

t =t

⎡

V ar [ $t (Xt )| Ft ] Ft

= Et ⎣ −$t (Xt ) − λ V ar [$t ] − λ

T t =t+1

e

−r(t −t)

(10.32)

⎤ V ar [ $t (Xt )| Ft ] Ft ⎦ .

The last term in this expression, which involves a sum from t = t + 1 to t = T , can be expressed in terms of Vt+1 using the definition of the value function with a shifted time argument gives: ⎡ −λEt+1 ⎣

T

⎤ e

−r(t −t)

V ar [ $t | Ft ]⎦ = γ (Vt+1 + Et+1 [$t+1 ]) , γ := e−rt .

t =t+1

(10.33) Note that parameter γ , introduced in the last relation, is a discrete-time discount factor which in our framework is fixed in terms of a continuous-time risk-free interest rate r of the original BSM model. Substituting this into (10.32), re-arranging terms and using the portfolio process Eq. (10.5), we obtain the Bellman equation for the QLBS model:

π Vtπ (Xt ) = Eπt R(Xt , at , Xt+1 ) + γ Vt+1 (Xt+1 ) ,

(10.34)

where the one-step time-dependent random reward is defined as follows5 : Rt (Xt , at , Xt+1 ) = γ at St (Xt , Xt+1 ) − λ V ar [ $t | Ft ] , t = 0, . . . , T − 1 (10.35)

2 ˆ 2t+1 − 2at Sˆt $ ˆ t+1 + at2 Sˆt , = γ at St (Xt , Xt+1 ) − λγ 2 Et $

5 Note

that with our definition of the value function Eq. (10.32), it is not equal to a discounted sum of future rewards.

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10 Applications of Reinforcement Learning

ˆ t+1 := $t+1 − $ ¯ t+1 , and where we used Eq. (10.5) in the second line, and $ ¯ t+1 is the sample mean of all values of $t+1 , and similarly for Sˆt . For where $ t = T , we have RT = −λ V ar [$T ] , where $T is determined by the terminal condition (10.2). Note that Eq. (10.35) implies that the expected reward, Rt , at time step t is quadratic in the action variable at : Et [Rt (Xt , at , Xt+1 )] = γ at Et [St ] (10.36)

2 ˆ t+1 + at2 Sˆt ˆ 2t+1 − 2at Sˆt $ . − λγ 2 Et $ This expected reward has the same mathematical structure as a risk-adjusted return of a single-period Markowitz portfolio model for a special case of a portfolio made of cash and a single stock. The first term gives the expected return from such portfolio, while the second term penalizes for its quadratic risk. Note further that when λ → 0, the expected reward is linear in at , so it does not have a maximum. As the one-step reward in our formulation incorporates variance of the hedge portfolio as a risk penalty, this approach belongs to a class of risk-sensitive reinforcement learning. With our method, risk is incorporated to a traditional riskneutral RL framework (which only aims at maximization of expected rewards) by modifying the one-step reward function. A similar construction of a risk-sensitive MDP by adding one-step variance penalties to a finite-horizon risk-neutral MDP problem was suggested in a different context by Gosavi (2015). The action-value function, or Q-function, is defined by an expectation of the same expression as in Eq. (10.32), but conditioned on both the current state Xt and the initial action a = at , while following a policy π afterwards: (10.37) Qπt (x, a) = Et [ −$t (Xt )| Xt = x, at = a]

T − λ Eπt e−r(t −t) V ar [ $t (Xt )| Ft ] Xt = x, at = a . t =t

The optimal policy πt% (·|Xt ) is defined as the policy which maximizes the value function Vtπ (Xt ), or alternatively and equivalently, maximizes the action-value function Qπt (Xt , at ): πt% (Xt ) = argmaxπ Vtπ (Xt ) = argmaxat ∈A Q%t (Xt , at ).

(10.38)

The optimal value function satisfies the Bellman optimality equation Vt% (Xt ) = Eπt

%

% Rt (Xt , ut = πt% (Xt ), Xt+1 ) + γ Vt+1 (Xt+1 ) .

(10.39)

4 The QLBS Model

365

The Bellman optimality equation for the action-value function reads for t = 0, . . . , T − 1 Q%t (x, a) = Et Rt (Xt , at , Xt+1 ) + γ max Q%t+1 (Xt+1 , at+1 ) Xt = x, at = a , at+1 ∈A

(10.40) with a terminal condition at t = T Q%T (XT , aT = 0) = −$T (XT ) − λ V ar [$T (XT )] ,

(10.41)

and where $T is determined by Eq. (10.2). Recall that V ar [·] here means variance with respect to all Monte Carlo paths that terminate in a given state.

4.3 Optimal Policy Substituting the expected reward (10.36) into the Bellman optimality equation (10.40), we obtain & ' % + at St (10.42) Q%t (Xt , at ) = γ Et Q%t+1 Xt+1 , at+1

2 ˆ 2t+1 − 2at $ ˆ t+1 Sˆt + at2 Sˆt , t = 0, . . . , T − 1. − λγ 2 Et $ & ' % Note that the first term Et Q%t+1 Xt+1 , at+1 depends on the current action only through the conditional probability p (Xt+1 |Xt at ). However, the next-state probability depends on the current action, at , only when there is a feedback loop of trading in the option’s underlying stock on the stock price. In the present framework, we follow the standard assumptions of the Black–Scholes model which assumes an option buyer or seller does not produce any market impact. & ' % Neglecting the feedback effect, the expectation Et Q%t+1 Xt+1 , at+1 does not depend on at . Therefore, with this approximation, the action-value function Q%t (Xt , at ) is quadratic in the action variable at .

•

> The Black–Scholes Limit

Note that in the limit of zero risk-aversion λ → 0, this equation becomes & ' % + at St . Q%t (Xt , at ) = γ Et Q%t+1 Xt+1 , at+1

(10.43) (continued)

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10 Applications of Reinforcement Learning

As in this limit Q%t (Xt , at ) = −$ (Xt , at ), using the fair option price definition (10.11), we obtain ! " Cˆ t = γ Et Cˆ t+1 − at St .

(10.44)

This equation coincides with Eq. (10.12), showing that the recursive formula (10.42) correctly rolls back the BS fair option price Cˆ t = Et [$t ], which corresponds to first taking the limit λ → 0, and then taking the limit t → 0 of the QLBS price (while using the BS delta for at in Eq. (10.44), see below).

As Q%t (Xt , at ) is a quadratic function of at , the optimal action (i.e., the hedge) at% (St ) that maximizes Q%t (Xt , at ) is computed analytically: ! " ˆ t+1 + 1 St Et Sˆt $ 2γ λ at% (Xt ) = .

2 Et Sˆt

(10.45)

If we now take the limit of this expression as t → 0 by using Taylor expansions around time t as in Sect. 3.4, we obtain (see also Problem 1): lim at% =

t→0

∂ Cˆ t μ−r 1 + . ∂St 2λσ 2 St

(10.46)

Note that if we set μ = r, or alternatively if we take the limit λ → ∞, it becomes identical to the BS delta, while the finite-t delta in Eq. (10.45) coincides in these cases with a local risk-minimization delta given by Eq. (10.10). Both these facts have related interpretations. The quadratic hedging that approximates option delta (see Sect. 3.4) only accounts for risk of a hedge portfolio, while here we extend it by adding a drift term Et [$t ] to the objective function, see Eq. (10.17), in the style of Markowitz risk-adjusted portfolio return analysis (Markowitz 1959). This produces a linear first term in the quadratic expected reward (10.36). Resulting hedges are therefore different from hedges obtained by only minimizing risk. Clearly, a pure risk-focused quadratic hedge corresponds to either taking the limit of infinite riskaversion rate in a Markowitz-like risk-return analysis, or setting μ = r in the above formula, to achieve the same effect. Both factors appearing in Eq. (10.46) show these two possible ways to obtain pure risk-minimizing hedges from our more general hedges. Such hedges can be applied when an option is considered for investment/speculation, rather than only as a hedge instrument.

4 The QLBS Model

367

To summarize, the local risk-minimization hedge and fair price formulae of Sect. 3 are recovered from Eqs. (10.45) and (10.42), respectively, if we first set μ = r in Eq. (10.45), and then set λ = 0 in Eq. (10.42). After that, the continuoustime BS formulae for these expressions are reproduced in the final limit t → 0 in these resulting expressions, as discussed in Sect. 3. Note that the order of taking the limits is to start with the hedge ratio (10.46), set there μ = r, then substitute this into the price equation (10.42), and take the limit λ → 0 there, leading to Eq. (10.44). The latter relation yields the Black–Scholes equation in the limit t → 0 as shown in Eq. (10.25). This order of taking the BS limit is consistent with the principle of hedging first and pricing second, which is implemented in the QLBS model, as well as consistent with market practices of working with illiquid options. Substituting Eq. (10.45) back into Eq. (10.42), we obtain an explicit recursive formula for the optimal action-value function for t = 0, . . . , T − 1:

2 & ' % ˆ 2t+1 + λγ at% (Xt ) 2 Sˆt Q%t (Xt , at% ) = γ Et Q%t+1 (Xt+1 , at+1 , ) − λγ $ (10.47) where at% (Xt ) is defined in Eq. (10.45). Note that this relation does not have the right risk-neutral limit when we set λ → 0 in it. The reason is that setting λ → 0 in Eq. (10.47) is equivalent to setting λ → 0 in Eq. (10.45), but, as we just discussed, this would not be the right way to reproduce the BS option price equation (10.25). The correct procedure of taking the limit λ → 0 in the recursion for the Qfunction is given by Eq. (10.43) which implies that action at used there is obtained as explained above by setting μ = r in Eq. (10.46). The backward recursion given by Eqs. (10.45) and (10.47) proceeds all the way backward starting at t = T − 1 to the present t = 0. At each time step, the problem of maximization over possible actions amounts to convex optimization which is done analytically using Eq. (10.45), which is then substituted into Eq. (10.47) for the current time step. Note that such simplicity of action optimization in the Bellman optimality equation is not encountered very often in other SOC problems. As Eq. (10.47) provides the backward recursion directly for the optimal Q-function, neither continuous nor discrete action space representation is required in our setting, as the action in this equation is always just one optimal action. If we deal with a finite-state QLBS model, then the values of the optimal time-t Q-function for each node are obtained directly from sums of values of the next-step expectation in various states at time t + 1, times one-step probabilities to reach these states. The end result of the backward recursion for the action-value function is its current value. According to our definition of the option price (10.16), it is exactly the negative of the optimal Q-function. We therefore obtain the following expression for the fair ask option price in our approach, which we can refer to as the QLBS option price: (QLBS)

Ct

& ' (St , ask) = −Qt St , at% .

(10.48)

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10 Applications of Reinforcement Learning

It is interesting to note that while in the original BSM model the price and the hedge for an option are given by two separate expressions, in the QLBS model, they are parts of the same expression (10.48) simply because its option price is the (negative of the) optimal Q-function, whose second argument is by construction the optimal action—which corresponds to the optimal hedge in the setting of the QLBS model. Equations (10.48) and (10.45) that give, respectively, the optimal price and the optimal hedge for the option, jointly provide a complete solution of the QLBS model (when the dynamics are known) that generalizes the classical BSM model towards a non-asymptotic case t > 0, while reducing to the latter in the strict BSM limit t → 0. In the next section, we will see how they can be implemented.

4.4 DP Solution: Monte Carlo Implementation In practice, the backward recursion expressed by Eqs. (10.45) and (10.47) is solved in a Monte Carlo setting, where we use N simulated (or real) paths for the state variable Xt . In addition, we assume that we have chosen a set of basis functions { n (x)}. % We & can% 'then expand the optimal action (hedge) at (Xt ) and optimal Q-function % Qt Xt , at in basis functions, with time-dependent coefficients: at% (Xt )

=

M

φnt

n (Xt )

n

,

Q%t

M & ' % Xt , at = ωnt

n (Xt ) .

(10.49)

n

Coefficients φnt and ωnt are computed recursively backward in time for t = T − 1, . . . , 0. First, we find coefficients φnt of the optimal action expansion. This is found by minimization of the following quadratic functional that is obtained by replacing the expectation in Eq. (10.42) by a MC estimate, dropping all at -independent terms, substituting the expansion (10.49) for at , and changing the overall sign to convert maximization into minimization: ⎛ -2 ⎞ , N MC

⎝− ˆ kt+1 − Gt (φ) = φnt n Xtk Stk + γ λ $ φnt n Xtk Sˆtk ⎠ . k=1

n

n

(10.50) This formulation automatically ensures averaging over market scenarios at time t. Minimization of Eq. (10.50) with respect to coefficients φnt produces a set of linear equations: M m

(t) A(t) nm φmt = Bn , n = 1, . . . , M

(10.51)

4 The QLBS Model

369

where A(t) nm :=

N MC n

Xtk

m

2 Xtk Sˆtk

k=1

Bn(t)

:=

1 k k k k ˆ ˆ S $t+1 St + n Xt 2γ λ t

N MC k=1

(10.52)

which produces the solution for the coefficients of expansions of the optimal action at% (Xt ) in a vector form: φ %t = A−1 t Bt ,

(10.53)

where At and Bt are a matrix and vector, respectively, with matrix elements given by Eq. (10.52). Note a similarity between this expression and the general relation (10.45) for the optimal action. Once the optimal action at% at time t is found in terms of its coefficients (10.53), we turn to the problem of finding coefficients ωnt of the basis function expansion (10.49) for the optimal Q-function. To this end, the one-step Bellman optimality equation (10.40) for at = at% is interpreted as regression of the form & ' Rt Xt , at% , Xt+1 + γ max Q%t+1 (Xt+1 , at+1 ) = Q%t (Xt , at% ) + εt , at+1 ∈A

(10.54)

where εt is a random noise at time t with mean zero. Clearly, taking expectations of both sides of (10.54), we recover Eq. (10.40) with at = at% ; therefore, Eqs. (10.54) and (10.40) are equivalent in expectations when at = at% . Coefficients ωnt are therefore found by solving the following least square optimization problem: Ft (ω) =

N MC

,

&

Rt Xt , at% , Xt+1

'

+ γ max

at+1 ∈A

k=1

Q%t+1 (Xt+1 , at+1 ) −

M

ωnt

2 k . n Xt

n

(10.55)

Introducing another pair of a matrix Ct and a vector Dt with elements (t) Cnm :=

N MC n

Xtk

m

Xtk

(10.56)

k=1

Dn(t)

:=

N MC k=1

* ) & ' k % % Rt Xt , at , Xt+1 + γ max Qt+1 (Xt+1 , at+1 ) . n Xt at+1 ∈A

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10 Applications of Reinforcement Learning

We obtain the vector-valued solution for optimal weights ωt defining the optimal Q-function at time t: ωt% = C−1 t Dt .

(10.57)

Equations (10.53) and (10.57) computed jointly and recursively for t = T −1, . . . , 0 provide a practical implementation of the backward recursion scheme of Sect. 4.3 in a continuous-space setting using expansions in basis functions. This approach can be used to find optimal price and optimal hedge when the dynamics are known.

•

? Multiple Choice Question 2

Select all the following correct statements: a. The coefficients of expansion of the Q-function in the QLBS model are obtained in the DP solution from the Bellman equation interpreted as a classification problem, which is solved using deep learning. b. The coefficients of expansion of the Q-function in the QLBS model are obtained in the DP solution from the Bellman equation interpreted as a regression problem, which is solved using least square minimization. c. The DP solution requires rewards to be observable. d. The DP solution computes rewards as a part of the hedge optimization.

4.5 RL Solution for QLBS: Fitted Q Iteration When the transition probabilities and reward functions are not known, the QLBS model can be solved using reinforcement learning. In this section, we demonstrate this approach using a version of Q-learning that is formulated for continuous stateaction spaces and is known as Fitted Q Iteration. Our setting assumes a batch-mode learning, when we only have access to some historically collected data. The data available is given by a set of NMC trajectories for the underlying stock St (expressed as a function of Xt using Eq. (10.26)), hedge position at , instantaneous reward Rt , and the next-time value Xt+1 : (n)

Ft

=

(n) (n) (n) (n) Xt , at , Rt , Xt+1

T −1 t=0

, n = 1, . . . , NMC .

(10.58)

We assume that such dataset is available either as a simulated data, or as a real historical stock price data, combined with real trading data or artificial data that would track the performance of a hypothetical stock-and-cash replicating portfolio for a given option.

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371

A starting point of the Fitted Q Iteration (FQI) (Ernst et al. 2005; Murphy 2005) method is a choice of a parametric family of models for quantities of interest, namely optimal action and optimal action-value function. We use linear architectures where functions sought are linear in adjustable parameters that are next optimized to find the optimal action and action-value function. We use the same set of basis functions { n (x)} as we used above in Sect. 4.4. As the optimal Q-function Q%t (Xt , at ) is a quadratic function of at , we can represent it as an expansion in basis functions, with time-dependent coefficients parameterized by a matrix Wt : ⎞⎛ ⎛ * W11 (t) W12 (t) · · · W1M (t) ) 1 ⎜ Q%t (Xt , at ) = 1, at , at2 ⎝ W21 (t) W22 (t) · · · W2M (t) ⎠ ⎝ 2 W31 (t) W32 (t) · · · W3M (t)

1 (Xt )

⎞

⎟ .. ⎠ . M (Xt )

:= ATt Wt (Xt ) := ATt UW (t, Xt ).

(10.59)

Equation (10.59) is further re-arranged to convert it into a product of a parameter vector and a vector that depends on both the state and the action: Q%t (Xt , at ) = ATt Wt (X) =

M 3

Wt * At ⊗ T (X)

i=1 j =1

= Wt · vec At ⊗ T (X) := Wt (Xt , at ) .

ij

(10.60)

Here * and ⊗ stand for an element-wise (Hadamard) and outer (Kronecker) product of two matrices, respectively. The vector of time-dependent parameters Wt is obtained by concatenating columns of matrix Wt , and similarly, (Xt , at ) = ' & vec At ⊗ T (X) denotes a vector obtained by concatenating columns of the outer product of vectors At and (X). Coefficients Wt can now be computed recursively backward in time for t = T − 1, . . . , 0. To this end, the one-step Bellman optimality equation (10.40) is interpreted as regression of the form Rt (Xt , at , Xt+1 ) + γ max Q%t+1 (Xt+1 , at+1 ) = Wt (Xt , at ) + εt , at+1 ∈A

(10.61)

where εt is a random noise at time t with mean zero. Equations (10.61) and (10.40) are equivalent in expectations, as taking the expectation of both sides of (10.61), we recover (10.40) with function approximation (10.59) used for the optimal Qfunction Q%t (x, a). Coefficients Wt are therefore found by solving the following least square optimization problem:

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10 Applications of Reinforcement Learning

Lt (Wt ) =

N MC

*2 ) % Rt (Xt , at , Xt+1 ) + γ max Qt+1 (Xt+1 , at+1 ) − Wt (Xt , at ) . at+1 ∈A

k=1

(10.62) Note that this relation holds for a general off-model, off-policy setting of the Fitted Q Iteration method of RL. Performing minimization, we obtain

Wt% = S−1 t Mt ,

(10.63)

where (t) := Snm

N MC

n Xtk , atk m Xtk , atk

(10.64)

k=1

Mn(t)

:=

N MC k=1

n

Xtk , atk

)

* k k k % k Rt Xt , at , Xt+1 + γ max Qt+1 Xt+1 , at+1 at+1 ∈A

To perform the maximization step in the second equation in (10.64) analytically, note that because coefficients Wt+1 and hence vectors UW (t + 1, Xt+1 ) := Wt+1 (Xt+1 ) (see Eq. (10.59)) are known from the previous step, we have & ' (0) (1) % % = UW (t + 1, Xt+1 ) + at+1 UW (t + 1, Xt+1 ) Q%t+1 Xt+1 , at+1 & % '2 a (2) + t+1 UW (t + 1, Xt+1 ) . (10.65) 2 % , it would be We emphasize here that while this is a quadratic expression in at+1 % wrong to use a point of its maximum as a function of at+1 as such an optimal value in Eq. (10.65). This would amount to using the same dataset to estimate both the &optimal action ' and the optimal Q-function, leading to an overestimation % in Eq. (10.64), due to Jensen’s inequality and convexity of of Q%t+1 Xt+1 , at+1 the max(·) function. The correct approach for using Eq. (10.65) is to input a value % of at+1 computed using the analytical solution Eq. (10.45) (implemented in the sample-based approach in Eq. (10.53)), applied at the previous time step. Due to the availability of the analytical optimal action (10.45), a potential overestimation problem—a classical problem of Q-learning that is sometimes addressed using such methods as Double Q-learning (van Hasselt 2010)—is avoided in the QLBS model, leading to numerically stable results. Equation (10.63) gives the solution for the QLBS model in a model-free and off-policy setting, via its reliance on Fitted Q Iteration which is a model-free and off-policy algorithm (Ernst et al. 2005; Murphy 2005).

4 The QLBS Model

373

•

? Multiple Choice Question 3

Select all the following correct statements: a. Unlike the classical Black–Scholes model, the discrete-time QLBS model explicitly prices mis-hedging risk of the option because it maximizes the Q-function which incorporates mis-hedging risk as a penalty. b. Counting by the number of parameters to learn, the RL setting for the QLBS model has more unknowns, but also a higher dimensionality of data (more features per observation) than the DP setting. c. The BS solution is recovered from the RL solution in the limit t → 0 and λ → 0. d. The RL solution is recovered from the BS solution in the limit t → ∞ and λ → ∞.

4.6 Examples Here we illustrate the performance of the QLBS model using simulated stock price histories St with the initial stock price S0 = 100, stock drift μ = 0.05, and volatility σ = 0.15. Option maturity is T = 1 year, and a risk-free rate is r = 0.03. We consider an ATM (“at-the-money”) European put option with strike K = 100. Rehedges are performed bi-weekly (i.e., t = 1/24). We use N = 50, 000 Monte Carlo scenarios of the stock price trajectory and report results obtained with two MC runs (each having N paths), where the error reported is equal to one standard deviation calculated from these runs. In our experiments, we use pure risk-based hedges, i.e. omit the second term in the numerator in Eq. (10.45), for ease of comparison with the BSM model. We use 12 basis functions chosen to be cubic B-splines on a range of values of Xt between the smallest and largest values observed in a dataset. In our experiments below, we pick the Markowitz risk-aversion parameter λ = 0.001. This provides a visible difference of QLBS prices from BS prices, while being not too far away from BS prices. The dependence of the ATM option price on λ is shown in Fig. 10.1. Simulated path and solutions for optimal hedges, portfolio values, and Qfunction values corresponding to the DP solution of Sect. 4.4 are illustrated in Fig. 10.2. The resulting QLBS ATM put option price is 4.90 ± 0.12 (based on two MC runs), while the BS price is 4.53. We first report results obtained with on-policy learning with λ = 0.001. In this case, optimal actions and rewards computed as a part of a DP solution are used as inputs to the Fitted Q Iteration algorithm of Sect. 4.5 and the IRL method of Sect. 10.2, in addition to the paths of the underlying stock. Results of two MC

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Fig. 10.1 The ATM put option price vs risk-aversion parameter. The time step is t = 1/24. The horizontal red line corresponds to the continuous-time BS model price. Error bars correspond to one standard deviation of two MC runs

batches with Fitted Q Iteration algorithm of Sect. 4.5 are shown (respectively, in the left and right columns, with a random selection of a few trajectories) in Fig. 10.3. Similar to the DP solution, we add a unit matrix with a regularization parameter of 10−3 to invert matrix Ct in Eq. (10.63). Note that because here we use onpolicy the resulting optimal Q-function Q%t (Xt , at ) and its optimal value & learning, ' % % Qt Xt , at are virtually identical in the graph. The resulting QLBS RL put price is 4.90 ± 0.12 which is identical to the DP value. As expected, the IRL method of Sect. 10.2 produces the same result. In the next set of experiments we consider off-policy learning. The risk-aversion parameter is λ = 0.001. To generate off-policy data, we multiply, at each time step, optimal hedges computed by the DP solution of the model by a random uniform number in the interval [1 − η, 1 + η], where 0 < η < 1 is a parameter controlling the noise level in the data. We will consider the values of η = [0.15, 0.25, 0.35, 0.5] to test the noise tolerance of our algorithms. Rewards corresponding to these suboptimal actions are obtained using Eq. (10.35). In Fig. 10.4 we show results obtained for off-policy learning with 10 different scenarios of sub-optimal actions obtained by random perturbations of a fixed simulated dataset. Note that the impact of suboptimality of actions in recorded data is rather mild, at least for a moderate level of noise. This is as expected as long as Fitted Q Iteration is an off-policy algorithm. This implies that when dataset is large enough, the QLBS model can learn even from data with purely random actions. In particular, if the stock prices are log-normal, it can learn the BSM model itself. Results of two MC batches for off-policy learning with the noise parameter η = 0.5 with Fitted Q Iteration algorithm are shown in Fig. 10.5.

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Fig. 10.2 The DP solution for the ATM put option on a subset of MC paths

4.7 Option Portfolios Thus far we have only considered the problem of hedging and pricing of a single European option by an option seller that does not have any pre-existing option portfolio. Here we outline a simple generalization to the case when the option seller does have such a pre-existing option portfolio, or alternatively if she seeks to sell a few options simultaneously. In this case, she is concerned with consistency of pricing and hedging of all options in her new portfolio. In other words, she has to solve the notorious volatility smile problem for her particular portfolio. Here we outline how she can solve it using the QLBS model, illustrating the flexibility and data-driven nature of the model. Such flexibility facilitates adaptation to arbitrary consistent volatility surfaces.

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Fig. 10.3 The RL solution (Fitted Q Iteration) for on-policy learning for the ATM put option on a subset of MC paths for two MC batches

Assume the option seller has a pre-existing portfolio of K options with market prices C1 , . . . , CK . All these options reference an underlying state vector (market) Xt which can be high-dimensional such that each particular option Ci with i = 1, . . . , K references only one or a few components of market state Xt . Alternatively, we can add vanilla option prices as components of the market state Xt . In this case, our dynamic replicating portfolio would include vanilla options, along with underlying stocks. Such hedging portfolio would provide a dynamic generalization of static option hedging for exotics introduced by Carr et al. (1988). We assume that we have a historical dataset F which includes N observations of trajectories of tuples of vector-valued market factors, actions (hedges), and rewards (compare with Eq. (10.58)):

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377 Optimal option price vs noise in action data

5.4

On-policy value = 5.02

Optimal option price

5.2 5.0 4.8 4.6 4.4 4.2 4.0 0.15

0.20

0.25

0.30 0.35 Noise level

0.40

0.45

0.50

Fig. 10.4 Means and standard deviations of option prices obtained with off-policy FQI learning with data obtained by randomization of DP optimal actions by multiplying each optimal action by a uniform random variable in the interval [1 − η, 1 + η] for η = [0.15, 0.25, 0.35, 0.5]. Error bars are obtained with 10 scenarios for each value of η. The horizontal red line shows the value obtained with on-policy learning corresponding to η = 0

(n)

Ft

=

(n) (n) (n) (n) Xt , at , Rt , Xt+1

T −1 t=0

, n = 1, . . . , N.

(10.66)

We now assume that the option seller seeks to add to this pre-existing portfolio another (exotic) option Ce (or alternatively, she seeks to sell a portfolio of options C1 , . . . , CK , Ce ). Depending on whether the exotic option Ce was traded before in the market or not, there are two possible scenarios. We shall analyze these scenarios one by one. In the first case, the exotic option Ce was previously traded in the market (by the seller herself, or by someone else). As long as its deltas and related P&L impacts marked by a trading desk are available, we can simply extend vectors of actions (n) (n) at and rewards Rt in Eq. (10.66), and then proceed with the FQI algorithm of Sect. 4.5 (or with the IRL algorithm of Sect. 10.2, if rewards are not available). The outputs of the algorithm will be the optimal price Pt of the whole option portfolio, plus optimal hedges for all options in the portfolio. Note that as long as FQI is an off-policy algorithm, it is quite forgiving to human or model errors: deltas in the data should not even be perfectly mutually consistent (see single-option examples in the previous section). But of course, the more consistency in the data, the less data is needed to learn an optimal portfolio price Pt . Once the optimal time-zero value P0 of the total portfolio C1 , . . . , CK , Ce is computed, a market-consistent price for the exotic option is simply given by a subtraction:

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Fig. 10.5 The RL solution (Fitted Q Iteration) for off-policy learning with noise parameter η = 0.5 for the ATM put option on a subset of MC paths for two MC batches

Ce = P0 −

K

Ci .

(10.67)

i=1

Note that by construction, the price Ce is consistent with all option prices C1 , . . . , CK and all their hedges, to the extent they are consistent between themselves (again, this is because Q-learning is an off-policy algorithm). Now consider a different case, when the exotic option Ce was not previously traded in the market, and therefore there are no available historical hedges for this option. This can be handled by the QLBS model in essentially the same way as in the previous case. Again, because Q-learning is an off-policy algorithm, it means that a delta and a reward of a proxy option Ce (that was traded before) to Ce could be used in the scheme just described in lieu of their actual values for option Ce .

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Consistent with common intuition, this will just slow down the learning, so that more data would be needed to compute the optimal price and hedge for the exotic Ce . On the other hand, the closer the traded proxy Ce to the actual exotic Ce the option seller wants to hedge and price, the more it helps the algorithm on the data demand side.

4.8 Possible Extensions So far we have presented the QLBS model in its most basic setting where it is applied to a single European vanilla option such as a put or call option. This framework can be extended or generalized along several directions. Here we overview them, in the order of increasing complexity of changes that would be needed on top of the basic computational framework presented above. The simplest extension of the basic QLBS setting is to apply it to European options with a non-vanilla terminal payoff, e.g. to a straddle option. Clearly, the only change needed to the basic QLBS setting in this case would be a different terminal condition for the action-value function. The second extension that is easy to incorporate in the basic QLBS setting are early exercise features for options. This can be added to the QLBS model in much the same way as they are implemented in the American Monte Carlo method of Longstaff and Schwartz. Namely, in the backward recursion, at each time step where an early option exercise is possible, the optimal action-value function is obtained by comparing its value from the next time step continued to the current time step with an intrinsic option value. The latter is defined as a payoff from an immediate exercise of the option, see also Exercise 10.2. One more possible extension involves capturing higher moments of the replicating portfolio. This assumes using a non-quadratic utility function. One approach is to use an exponential utility function as was outlined above (see also in Halperin (2018)). On the computational side, using a non-quadratic utility gives rise to the need to solve a convex optimization problem at each time step, instead of quadratic optimization. The basic QLBS framework can also be extended by incorporating transaction costs. This requires re-defining the state and action spaces in the problem. As in the presence of transaction costs holding cash is not equivalent to holding a stock, for this case we can use changes in the stock holding as action variables, while the current stock holding and the stock market price should now be made parts of a state vector. Depending on a functional model for transaction cost, the resulting optimization problem can be either quadratic (if both the reward and transaction cost functions are quadratic in action), or convex, if both these functions are convex. Finally, the basic framework can be generalized to a multi-asset setting, including option portfolios. The main challenge of such task would be to specify a good set of basis functions. In multiple dimensions, this might be a challenging problem. Indeed, a simple method to form a basis in a multi-dimensional space is to take a

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direct (cross) product of individual bases, but this produces an exponential number of basis functions. As a result, such a naive approach becomes intractable beyond a rather low (< 10) number of dimensions. Feature selection in high-dimensional spaces is a general problem in machine learning, which is not specific to reinforcement learning or the QLBS approach. The latter can benefit from methods developed in the literature. Rather than pursuing this direction, we now turn to a different and equally canonical finance application, namely the multi-period optimization of stock portfolios. We will show that such a multi-asset setting may entirely avoid the need to choose basis functions.

5 G-Learning for Stock Portfolios 5.1 Introduction In this section, we consider a multi-dimensional setting with a multi-asset investment portfolio. Specifically, we consider a stock portfolio, although similar methods can be used for portfolios of other assets, including options. As we mentioned above in this chapter, one challenge with scaling to multiple dimensions with reinforcement learning is the computational cost and the problem of under-sampling due to the curse of dimensionality. Another potential (and related) issue is the pronounced importance of noise in data. With finite samples, estimations of functions such as the action-value function or the policy function with noisy high-dimensional data can become quite noisy themselves. Rather than relying on deterministic policies as in Q-learning, we may prefer to work with probabilistic methods where such noise can be captured. A framework that is presented below is designed as a probabilistic approach that scales to a very high-dimensional setting. Again, for ease of exposition, we consider methods for quadratic (Markowitz) reward functions; however, the approach can be generalized to include other reward (utility) functions. Our approach is based on a probabilistic extension of Q-learning known in the literature as “G-learning.” While G-learning was initially formulated for finite MDPs, here we extend it to a continuous-state and continuous-action case. For an arbitrary reward function, this requires relying on a set of pre-specified basis functions, or using universal function approximators (e.g., neural networks) to represent the action-value function. However, as we will see below, when a reward function is quadratic, neither approach is needed, and the portfolio optimization procedure is semi-analytic.

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5.2 Investment Portfolio We adopt the notation and assumption of the portfolio model suggested by Boyd et al. (2017). In this model, dollar values of positions in n assets i = 1, . . . , n are denoted as a vector xt with components (xt )i for a dollar value of asset i at the beginning of period t. In addition to assets xt , an investment portfolio includes a risk-free bank cash account bt with a risk-free interest rate rf . A short position in any asset i then corresponds to a negative value (xt )i < 0. The vector of mean of bid and ask prices of assets at the beginning of period t is denoted as pt , with (pt )i > 0 being the price of asset i. Trades ut are made at the beginning of interval t, so that asset values x+ t immediately after trades are deterministic: x+ t = xt + ut .

(10.68)

vt = 1T xt + bt ,

(10.69)

The total portfolio value is

where 1 is a vector of ones. The post-trade portfolio is therefore + T + T + vt+ = 1T x+ t + bt = 1 (xt + ut ) + bt = vt + 1 ut + bt − bt .

(10.70)

We assume that all rebalancing of stock positions are financed from the bank cash account (additional cash costs related to the trade will be introduced below). This imposes the following “self-financing” constraint: 1T ut + bt+ − bt = 0,

(10.71)

which simply means that the portfolio value remains unchanged upon an instantaneous rebalancing of the wealth between the stock and cash: vt+ = vt .

(10.72)

The post-trade portfolio, vt+ and cash are invested at the beginning of period t until the beginning of the next period. The return of asset i over period t is defined as (rt )i =

(pt+1 )i − (pt )i , i = 1, . . . , n. (pt )i

(10.73)

Asset positions at the next time period are then given by + xt+1 = x+ t + rt ◦ xt ,

(10.74)

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where ◦ denotes an element-wise (Hadamard) product, and rt ∈ Rn is the vector of asset returns from period t to period t + 1. The next-period portfolio value is then obtained as follows: vt+1 = 1T xt+1 = =

(1 + rt )T x+ t T (1 + rt ) (xt

(10.75) (10.76) + ut ).

(10.77)

Given a vector of returns rt in period t, the change of the portfolio value in excess of a risk-free growth is vt := vt+1 − (1 + rf )vt = (1 + rt )T (xt + ut ) + (1 + rf )bt+ −(1 + rf )1T xt − (1 + rf )bt = (rr − rf 1)T (xt + ut ),

(10.78)

where in the second equation we used Eq. (10.71).

5.3 Terminal Condition As we generally assume a finite-horizon portfolio optimization with a finite investment horizon T , we must supplement the problem with a proper terminal condition at time T . For example, if the investment portfolio should track a given benchmark portfolio (e.g., the market portfolio), a terminal condition is obtained from the requirement that at time T , all stock positions should be equal to the actual observed weights of B stocks in the benchmark xB T . This implies that xT = xT . By Eq. (10.68), this fixes the action uT at the last time step: uT = xM T − xT −1 .

(10.79)

Therefore, action uT at the last step is deterministic and is not subject to optimization that should be applied to T remaining actions uT −1 , . . . , u0 . Alternatively, the goal of the investment portfolio can be maximization of a riskadjusted cumulative reward of the portfolio. In this case, an appropriate terminal condition could be xT = 0, meaning that any remaining long stock positions should be converted to cash at time T .

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383

5.4 Asset Returns Model We assume the following linear specification of one-period excess asset returns: rt − rf 1 = Wzt − MT ut + εt ,

(10.80)

where zt is a vector of predictors with factor loading matrix W, M is a matrix of permanent market impacts with a linear impact specification, and εt is a vector of residuals with E [εt ] = 0, V [εt ] = r ,

(10.81)

where E [·] denotes an expectation with respect to the physical measure P. Equation (10.80) specifies stochastic returns rt , or equivalently the next-step stock prices, as driven by external signals zt , control (action) variables ut , and uncontrollable noise εt . Though they enter “symmetrically” in Eq. (10.80), two drivers of returns zt and ut play entirely different roles. While signals zt are completely external for the agent, actions ut are controlled degrees of freedom. In our approach, we will be looking for optimal controls ut for the market-wise portfolio. When we set up a proper optimization problem, we solve for an optimal action ut . As will be shown in this section, this optimal control turns out to be a linear function of xt , plus noise.

5.5 Signal Dynamics and State Space Our approach is general and works for any set of predictors zt that might be relevant at the time scale of portfolio rebalancing periods t. For example, for daily portfolio trading with time steps t ( 1/250, predictors zt may include news and various market indices such as VIX and MSCI indices. For portfolio trading on monthly or quarterly steps, additional predictors can include macroeconomic variables. In the opposite limit of intra-day or high-frequency trading, instead of macroeconomic variables, variables derived from the current state of the limit order book (LOB) might be more useful. As a general rule, a predictor zt may be of interest if it satisfies three requirements: (i) it correlates with equity returns, (ii) is predictable itself, to a certain degree (e.g., it can be a mean-reverting process); and (iii) its characteristic times τ are larger than the time step t. In particular, for a mean-reverting signal zt , a mean reversion parameter κ gives rise to a characteristic time scale τ ( 1/κ. The last requirement simply means that if τ , t and the mean level of zt is zero, then fluctuations of zt will be well described by a stationary white noise process, and thus will be indistinguishable from the white noise term that is already present in Eq. (10.80). It is for this reason that it would be futile to, e.g., include

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any features derived from the LOB for portfolio construction designed for monthly rebalancing. For dynamics of signals zt , similar to Garleanu and Pedersen (2013), we will assume a simple multivariate mean-reverting Ornstein–Uhlenbeck (OU) process for a K-component vector zt : zt+1 = (I −

) ◦ zt + εtz ,

(10.82)

where εtz ∼ N (0, z ) is the noise term, and is a diagonal matrix of mean reversion rates. It is convenient to form an extended state vector yt of size N + K by concatenating vectors xt and zt : yt =

xt . zt

(10.83)

The extended vector, yt , describes a full state of the system for the agent that has some control of its x-component, but no control of its z-component.

5.6 One-Period Rewards We first consider an idealized case when there are no costs of taking action ut at time step t. An instantaneous random reward received upon taking such action is obtained by substituting Eq. (10.80) in Eq. (10.78):

T (0) Rt (yt , ut ) = Wzt − MT ut + ε t (xt + ut ) .

(10.84)

In addition to this reward that would be obtained in an ideal friction-free market, we must add (negative) rewards received due to instantaneous market impact and transaction fees.6 Furthermore, we must include a negative reward due to risk in a newly created portfolio position at time t + 1. Following Boyd et al. (2017), we choose a simple quadratic measure of such risk penalty, as the variance of the instantaneous reward (10.84) conditional on the new state xt + ut , multiplied by the risk-aversion parameter λ: " ! (0) (yt , ut ) = −λV Rt (yt , ut ) xt + ut = −λ(xt + ut )T r (xt + ut ). (10.85) To specify negative rewards (costs) of an instantaneous market impact and transaction costs, it is convenient to represent each action uti as a difference of two − non-negative action variables u+ ti , uti ≥ 0: (risk)

Rt

6 We

assume no short sale positions in our setting, and therefore do not include borrowing costs.

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385

− + − + − uti = u+ ti − uti , |uti | = uti + uti , uti , uti ≥ 0,

(10.86)

− so that uti = u+ ti if uti > 0 and uti = −uti if uti < 0. The instantaneous market impact and transaction costs are then given by the following expressions: (impact)

Rt

(f ee)

Rt

T − − T (yt , ut ) = −xTt + u+ t − xt ut − xt ϒzt −T − (yt , ut ) = −ν +T u+ ut . t −ν

(10.87)

Here + , − , ϒ and ν + , ν − are, respectively, matrix-valued and vector-valued parameters which in the simplest case can be parameterized in terms of single scalars multiplied by unit vectors or matrices. Combining Eqs. (10.84, (10.85), (10.87), we obtain our final specification of a risk- and cost-adjusted instantaneous reward function for the problem of optimal portfolio liquidation: (0)

(risk)

Rt (yt , ut ) = Rt (yt , ut ) + Rt

(impact)

(yt , ut ) + Rt

(f ee)

(yt , ut ) + Rt

− The expected one-step reward given action ut = u+ t − ut is given by

(yt , ut ). (10.88)

(impact) (f ee) (0) (risk) (yt , ut ) + Rt (yt , ut ) + Rt (yt , ut ), Rˆ t (yt , ut ) = Rˆ t (yt , ut ) + Rt (10.89) where

! "

T & ' − − xt + u+ Rˆ t(0) (yt , ut ) = Et,u Rt(0) (yt , ut ) = Wzt − MT (u+ t − ut ) t − ut , (10.90) and where Et,u [·] := E [·|yt , ut ] denotes averaging over next-periods realizations of market returns. Note that the one-step expected reward (10.89) is a quadratic form of its inputs. We can write it more explicitly using vector notation: ˆ t , at ) = yTt Ryy yt + aTt Raa a + aTt Ray yt + aTt Ra , R(y

(10.91)

where ) at = Ray =

u+ t u− t

*

−M − λr M + λr −λr W − ϒ , Ryy = , M + λr −M − λr 0 0 + W −M − 2λr − + ν , . (10.92) = − , R a M + 2λr − − ν+ W

, Raa =

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5.7 Multi-period Portfolio Optimization Multi-period portfolio optimization is equivalently formulated either as maximization of risk- and cost-adjusted returns, as in the Markowitz portfolio model, or as minimization of risk- and cost-adjusted trading costs. The latter specification is usually used in problems of optimal portfolio liquidation. The multi-period risk- and cost-adjusted reward maximization problem is defined as ! " T −1 t −t ˆ maximize Et γ (y , a ) (10.93) R t t t t =t where Rˆ t (yt , at ) = yTt Ryy yt + aTt Raa a + aTt Ray yt + aTt Ra ) +* ut ≥ 0, w.r.t. at = u− t − subject to xt + u+ t − ut ≥ 0.

Here 0 < γ ≤ 1 is a discount factor. Note that the sum over future periods t = [t, . . . , T − 1] does not include the last period t = T , because the last action is fixed by Eq. (10.79). The last constraint in Eq. (10.93) is appropriate for a long-only portfolio and can be replaced by other constraints, for example, a constraint on the portfolio leverage. With any (or both) of these constraints, the problem belongs to the class of convex optimization with constraints, and thus can be solved in a numerically efficient way (Boyd et al. 2017). An equivalent cost-focused formulation is obtained by flipping the sign of the above problem and re-phrasing it as minimization of trading costs Cˆ t (yt , at ) = −Rˆ t (yt , at ): minimize Et

!

T −1 t −t ˆ Ct (yt , at ) t =t γ

where Cˆ t (yt , at ) = −Rˆ t (yt , at ),

" (10.94) (10.95)

subject to the same constraints as in (10.93).

5.8 Stochastic Policy Note that the multi-period portfolio optimization problem (10.93) assumes that an optimal policy that determines actions at is a deterministic policy that can also be described as a delta-like probability distribution & ' π(at |yt ) = δ at − a%t (yt ) ,

(10.96)

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387

where the optimal deterministic action a%t (yt ) is obtained by maximization of the objective (10.93) with respect to controls at . But the actual trading data may be sub-optimal, or noisy at times, because of model mis-specifications, market timing lags, human errors, etc. Potential presence of such sub-optimal actions in data poses serious challenges, if we try to assume deterministic policy (10.96) that assumes the action chosen is always an optimal action. This is because such events should have zero probability under these model assumptions, and thus would produce vanishing path probabilities if observed in data. Instead of assuming a deterministic policy (10.96), stochastic policies described by smoothed distributions π(at |yt ) are more useful for inverse problems such as the problem of inverse portfolio optimization. In this approach, instead of maximization with respect to deterministic policy/action at , we re-formulate the problem as maximization over probability distributions π(at |yt ): maximize Eqπ

! T −1 t =t

γ t −t Rˆ t (yt , at )

" (10.97)

ˆ t , at ) = yTt Ryy yt + aTt Raa a + aTt Ray yt + aTt Ra where R(y $T −1 w.r.t. qπ (x, ¯ a|y ¯ 0 ) = π(a0 |y0 ) t=1 π(at |yt )P (yt+1 |yt , at ) % subject to dat π (at |yt ) = 1. Here Eqπ [·] denotes expectations with respect to path probabilities defined according to the third line in Eqs. (10.97). Note that due to inclusion of a quadratic risk penalty in the risk-adjusted return, ˆ t , at ), the original problem of risk-adjusted return optimization is re-stated in R(x Eq. (10.97) as maximizing the expected cumulative reward in the standard MDP setting, thus making the problem amenable to a standard risk-neutral approach of MDP models. Such simple risk adjustment based on one-step variance penalties was suggested in a non-financial context by Gosavi (2015) and used in a reinforcement learning based approach to option pricing in Halperin (2018, 2019). Another comment that is due here is that a probabilistic approach to actions in portfolio trading appears, on many counts, a more natural approach than a formalism based on deterministic policies. Indeed, even in a simplest one-period setting, because the Markowitz-optimal solution for portfolio weights is a function of estimated stock means and covariances, they are in fact random variables. Yet the probabilistic nature of portfolio optimization is not recognized as such in the Markowitz-type single-period or multi-period optimization settings such as (10.93). A probabilistic portfolio optimization formulation was suggested in a one-period setting by Marschinski et al. (2007).

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5.9 Reference Policy We assume that we are given a probabilistic reference (or prior) policy π0 (at |yt ) which should be decided upon prior to attempting the portfolio optimization (10.97). Such policy can be chosen based on a parametric model, past historic data, etc. We will use a simple Gaussian reference policy ) * 'T −1 & ' 1& 1 # ˆ ˆ π0 (at |yt ) = exp − 2 at − a(yt ) p at − a(yt ) , (2π )N p (10.98) where aˆ (yt ) can be a deterministic policy chosen to be a linear function of a state vector yt : ˆ0 +A ˆ 1 yt . aˆ (yt ) = A

(10.99)

A simple choice of parameters in (10.98) could be to specify them in terms of only ˆ 0 = aˆ 0 1|A| and A ˆ 1 = aˆ 1 1|A|×|A| , where |A| is the two scalars aˆ 0 , aˆ 1 as follows: A size of vector at , 1A and 1A×A are, respectively, a vector and matrix made of ones. The scalars aˆ 0 and aˆ 1 would then serve as hyperparameters in our setting. Similarly, covariance matrix p for the prior policy can be taken to be a simple matrix with constant correlations ρp and constant variances σp . As will be shown below, an optimal policy has the same Gaussian form as the ˆ 0, A ˆ 1 , and p . These updates will prior policy (10.98), with updated parameters A be computed iteratively starting with their initial values defining the prior (10.98). ˆ (k) , Respectively, updates at iteration k will be denoted by upper subscripts, e.g. A 0 (k) ˆ . A 1 Furthermore, it turns out that a linear dependence on yt at iteration k, driven by ˆ (k) arises even if we set A ˆ1 = A ˆ (0) = 0 in the prior (10.98). Such the value of A 1 1 choice of a state-independent prior π0 (at |yt ) = π0 (at ), although not very critical, reduces the number of free parameters in the model by two, as well as simplifies some of the analyses below, and hence will be assumed going forward. It also makes it unnecessary to specify the value of y¯ t in the prior (10.98) (equivalently, we can initialize it at zero). The final set of hyperparameters defining the prior (10.98) therefore includes only three values of aˆ 0 , ρa , p .

5.10 Bellman Optimality Equation Let Vt% (yt )

= max E π(·|y)

T −1 t =t

γ

t −t

ˆ Rt (yt , at ) yt .

(10.100)

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389

The optimal state-value function Vt% (xt ) satisfies the Bellman optimality equation % (yt+1 ) . Vt% (yt ) = max Rˆ t (yt , at ) + γ Et,at Vt+1 at

(10.101)

The optimal policy π % can be obtained from V % as follows: % πt% (at |yt ) = arg max Rˆ t (yt , at ) + γ Et,at Vt+1 (yt+1 ) . at

(10.102)

The goal of reinforcement learning (RL) is to solve the Bellman optimality equation based on samples of data. Assuming that an optimal value function is found by means of RL, solving for the optimal policy π % takes another optimization problem as formulated in Eq. (10.102).

5.11 Entropy-Regularized Bellman Optimality Equation We start by reformulating the Bellman optimality equation using a Fenchel-type representation: Vt% (yt ) = max

π(·|y)∈P

% π(at |yt ) Rˆ t (yt , at ) + γ Et,at Vt+1 (yt+1 ) .

(10.103)

at ∈At

7 6 Here P = π : π ≥ 0, 1T π = 1 denotes a set of all valid distributions. Equation (10.103) is equivalent to the original Bellman optimality equation (10.100), because for any x ∈ Rn , we have maxi∈{1,...,n} xi = maxπ ≥0,||π ||≤1 π T x. Note that while we use discrete notations for simplicity of presentation, all formulas below can be equivalently expressed in continuous notations by replacing sums by integrals. For brevity, we will denote the expectation Eyt+1 |yt ,at [·] as Et,a [·] in what follows. The one-step information cost of a learned policy π(at |yt ) relative to a reference policy π0 (at |yt ) is defined as follows (Fox et al. 2015): g π (y, a) = log

π(at |yt ) . π0 (at |yt )

(10.104)

Its expectation with respect to the policy π is the Kullback–Leibler (KL) divergence of π(·|yt ) and π0 (·|yt ): π(at |yt ) . Eπ g π (y, a) yt = KL[π ||π0 ](yt ) := π(at |yt ) log π 0 (at |yt ) a

(10.105)

t

The total discounted information cost for a trajectory is defined as follows:

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10 Applications of Reinforcement Learning

I π (y) =

T

γ t −t E g π (yt , at ) yt = y .

(10.106)

t =t

The free energy function Ftπ (yt ) is defined as the value function (10.103) augmented by the information cost penalty (10.106): 1 π I (yt ) β T 1 = γ t −t E Rˆ t (yt , at ) − g π (yt , at ) . β

Ftπ (yt ) = Vtπ (yt ) −

(10.107)

t =t

Note that β in Eq. (10.107) serves as the “inverse temperature” parameter that controls a tradeoff between reward optimization and proximity to the reference policy, see below. The free energy, Ftπ (yt ), is the entropy-regularized value function, where the amount of regularization can be tuned to the level of noise in the data.7 The reference policy, π0 , provides a “guiding hand” in the stochastic policy optimization process that we now describe. A Bellman equation for the free energy function Ftπ (yt ) is obtained from (10.107): Ftπ (yt )

= Ea|y

π 1 π ˆ Rt (yt , at ) − g (yt , at ) + γ Et,a Ft+1 (yt+1 ) . β

(10.108)

For a finite-horizon setting, Eq. (10.108) should be supplemented by a terminal condition FTπ (yT ) = Rˆ T (yT , aT ) (10.109) aT =−xT −1

(see Eq. (10.79)). Eq. (10.108) can be viewed as a soft probabilistic relaxation of the Bellman optimality equation for the value function, with the KL information cost penalty (10.104) as a regularization controlled by the inverse temperature β. In addition to such a regularized value function (free energy), we will next introduce an entropy regularized Q-function.

7 Note

that in physics, free energy is defined with a negative sign relative to Eq. (10.107). This difference is purely a matter of a sign convention, as maximization of Eq. (10.107) can be re-stated as minimization of its negative. Using our sign convention for the free energy function, we follow the reinforcement learning and information theory literature.

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391

5.12 G-Function: An Entropy-Regularized Q-Function Similar to the action-value function, we define the state-action free energy function Gπ (x, a) as (Fox et al. 2015) π (10.110) (yt+1 ) yt , at Gπt (yt , at ) = Rˆ t (yt , at ) + γ E Ft+1 ⎡ ⎤ * ) T 1 = Rˆ t (yt , at ) + γ Et,a ⎣ γ t −t−1 Rˆ t (yt , at ) − g π (yt , at ) ⎦ β = Et,a

T t =t

t =t+1

γ

t −t

) * 1 π Rˆ t (yt , at ) − g (yt , at ) , β

where in the last equation we used the fact that the first action at in the G-function is fixed, and hence g π (yt , at ) = 0 when we condition on at = a. If we now compare this expression with Eq. (10.107), we obtain the relation between the G-function and the free energy Ftπ (yt ): Ftπ (yt ) =

at

π(at |yt ) 1 . π(at |yt ) Gπt (yt , at ) − log β π0 (at |yt )

(10.111)

This functional is maximized by the following distribution π(at |yt ): π(at |yt ) = Zt =

π 1 π0 (at |yt )eβGt (yt ,at ) Zt π (y

π0 (at |yt )eβGt

t ,at )

(10.112)

.

at

The free energy (10.111) evaluated at the optimal solution (10.112) becomes Ftπ (yt ) =

π 1 1 log Zt = log π0 (at |yt )eβGt (yt ,at ) . β β a

(10.113)

t

Using Eq. (10.113), the optimal action policy can be written as follows : π (y

π(at |yt ) = π0 (at |yt )eβ (Gt

π t ,at )−Ft (yt )

).

(10.114)

Equations (10.113), (10.114), along with the first form of Eq. (10.110) repeated here for convenience: π (yt+1 ) yt , at , Gπt (yt , at ) = Rˆ t (yt , at ) + γ Et,a Ft+1

(10.115)

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10 Applications of Reinforcement Learning

constitute a system of equations that should be solved self-consistently by backward recursion for t = T − 1, . . . , 0, with terminal conditions GπT (yT , aT ) = Rˆ T (yT , aT )

(10.116)

FTπ (yT ) = GπT (yT , aT ) = Rˆ T (yT , aT ). Equations (10.113, 10.114, 10.115) (Fox et al. 2015) constitute a system of equations that should be solved self-consistently for π(at |yt ), Gπt (yt , at ), and Ftπ (yt ). Before proceeding with methods of solving it, we want to digress on an alternative interpretation of entropy regularization in Eq. (10.107), that can be useful later in the book.

•

> Adversarial Interpretation of Entropy Regularization

A useful alternative interpretation of the entropy regularization term in Eq. (10.107) can be suggested using its representation as a Legendre–Fenchel transform of another function (Ortega and Lee 2014): −

π(at |yt ) 1 = min π(at |yt ) log (−π(at |yt ) (1 + C(at , yt )) β a π0 (at |yt ) C(at ,yt ) a t t

(10.117) + π0 (at |yt )eβC(at ,yt ) ,

where C(at , yt ) is an arbitrary function. Equation (10.117) can be verified by direct minimization of the right-hand side with respect to C(at , yt ). Using this representation of the KL term, the free energy maximization problem (10.111) can be re-stated as a max–min problem Ft% (yt )= max min π

C

π(at |yt ) Gπt (yt , at )−C(at , yt )−1 +π0 (at |yt )eβC(at ,yt ) .

at

(10.118) The imaginary adversary’s optimal cost obtained from (10.118) is C % (at , yt ) =

π(at |yt ) 1 log . β π0 (at |yt )

(10.119)

Similar to Ortega and Lee (2014), one can check that this produces an indifference solution for the imaginary game between the agent and its adversarial environment where the total sum of the optimal G-function and the (continued)

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393

optimal adversarial cost (10.119) is constant: G%t (yt , at )+C % (at , yt ) = const, which means that the game of the original agent and its adversary is in a Nash equilibrium . Therefore, portfolio optimization in a stochastic environment by a single agent is mathematically equivalent to studying a Nash equilibrium in a twoparty game of our agent with an adversarial counterparty with an exponential budget given by the last term in Eq. (10.118).

5.13 G-Learning and F-Learning In the RL setting when rewards are observed, the system Eqs. (10.113, 10.114, 10.115) can be reduced to one non-linear equation. Substituting the augmented free energy (10.113) into Eq. (10.115), we obtain ⎡

⎤ π γ ˆ t , at ) + Et,a ⎣ log Gπt (y, a) = R(y π0 (at+1 |yt+1 )eβGt+1 (yt+1 ,at+1 ) ⎦ . β a t+1

(10.120) This equation provides a soft relaxation of the Bellman optimality equation for the action-value Q-function, with the G-function defined in Eq. (10.110) being an entropy-regularized Q-function (Fox et al. 2015). The “inverse-temperature” parameter β in Eq. (10.120) determines the strength of entropy regularization. In particular, if we take β → ∞, we recover the original Bellman optimality equation for the Q-function. Because the last term in (10.120) approximates the max(·) function when β is large but finite, Eq. (10.120) is known, for the special case of a uniform reference policy π0 , as “soft Q-learning”. For finite values β < ∞, in a setting of reinforcement learning with observed rewards, Eq. (10.120) can be used to specify G-learning (Fox et al. 2015): an off-policy time-difference (TD) algorithm that generalizes Q-learning to noisy environments where an entropy-based regularization might be needed. The Glearning algorithm of Fox et al. (2015) was specified in a tabulated setting where both the state and action space are finite. In our case, we deal with high-dimensional continuous state and action spaces. Respectively, we cannot rely on a tabulated Glearning and need to specify a functional form of the action-value function, or use a non-parametric function approximation such as a neural network to represent its values. An additional challenge is to compute a multi-dimensional integral (or a sum) over all next-step actions in Eq. (10.120). Unless a tractable parameterization is used for π0 and Gt , repeated numerical integration of this integral can substantially slow down the learning.

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10 Applications of Reinforcement Learning

•

> G-Learning vs Q-Learning

– Q-learning is an off-policy method with a deterministic policy. – G-learning is an off-policy method with a stochastic policy. Because the G-learning operates with stochastic policies, it gives rise to generative models. G-learning can be considered as an entropy-regularized Q-learning.

Another possible approach is to bypass the G-function (i.e., the entropy-regulated Q-function) altogether, and proceed with the Bellman optimality equation for the free energy F-function (10.107). In this case, we have a pair of equations for Ftπ (yt ) and π(at |yt ): ˆ t , at ) − 1 g π (yt , at ) + γ Et,a F π (yt+1 ) Ftπ (yt ) = Ea|x R(y t+1 β π(at |yt ) =

π 1 ˆ π0 (at |yt )eR(yt ,at )+γ Et,a Ft+1 (yt+1 ) . Zt

(10.121)

Here the first equation is the Bellman equation (10.108) for the F-function, and the second equation is obtained by substitution of Eq. (10.115) into Eq. (10.112). Also note that the normalization constant Zt in Eq. (10.121) is in general different from the normalization constant in Eq. (10.112). We will return to solutions of G-learning with continuous states and actions in the next chapter where we will address inverse reinforcement learning.

•

> G-Learning for Stationary Problems

For time-stationary (infinite-horizon) problems, the “soft Q-learning” (Glearning) equation (10.120) becomes (continued)

5 G-Learning for Stock Portfolios

395

γ π ˆ a) + Gπ (y, a) = R(y, ρ(y |y, a) log π0 (a |y )eβG (y ,a ) β y

a

(10.122) This is a non-linear integral equation. For example, if both the state and action space are one-dimensional, the resulting integral equation is two-dimensional. Therefore, computationally, G-learning for time-stationary problems amounts to solution of a non-linear integral equation (10.122). Existing numerical methods could be used to address this problem, see also Exercise 10.4.

5.14 Portfolio Dynamics with Market Impact A state equation for the portfolio vector xt is obtained using Eqs. (10.74) and (10.80): xt+1 = xt + ut + rt ◦ (xt + ut )

= xt + ut + rf 1 + Wzt − MT ut + εt ◦ (xt + ut )

(10.123)

= (1 + rf )(xt + ut ) + diag (Wzt − Mut ) (xt + ut ) + ε(xt , ut ) Here we assumed that the matrix M of market impacts is diagonal with elements μi , and set M = diag (μi ) , ε(xt , ut ) := εt ◦ (xt + ut )

(10.124)

Eq. (10.123) shows that the dynamics are non-linear in controls ut due to the market impact ∼ M. More specifically, when friction parameters μi > 0, the state equation is linear in xt , but quadratic in controls ut . In the limit μ → 0, the dynamics become linear. On the other hand, the reward (10.91) is quadratic for either case μi = 0 or μi > 0. The fact that the dynamics are non-linear (quadratic) when μi > 0 has farreaching implications both on the practical (computational) and theoretical side. Before discussing the non-linear case, we want to first analyze a simpler case when μi = 0, i.e. market impact effects are neglected, and the dynamics are linear. When dynamics are linear while rewards are quadratic, the problem of optimal portfolio management with a deterministic policy amounts to a well-known linear quadratic regulator (LQR) whose solution is known from control theory. In the next section we present a probabilistic version of the LQR problem that is particularly well suited for dynamic portfolio optimization.

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10 Applications of Reinforcement Learning

5.15 Zero Friction Limit: LQR with Entropy Regularization When the market impact is neglected, so that μi = 0 for all i, the portfolio optimization problem simplifies because dynamics become linear with the following state equation: ' & xt+1 = 1 + rf + Wzt + εt ◦ (xt + ut ).

(10.125)

We can equivalently write it as follows: xt+1 = At (xt + ut ) + (xt + ut ) ◦ εt

(10.126)

' & At = A(zt ) = diag 1 + rf + Wzt

(10.127)

where

Unlike the previous section where we assumed proportional transaction costs, here we assume convex transaction costs ηuTt Cut , where η is a transaction cost parameter and C is a matrix which can be taken to be a diagonal unit matrix. We neglect other costs such as holding costs. The expected one-step reward at time t is then given by the following expression: Rˆ t (xt , ut ) = (xt + ut )T Wzt − λ (xt + ut )T r (xt + ut ) − ηuTt Cut .

(10.128)

If we assume that our problem is to maximize the risk-adjusted return of the portfolio for a pre-specified time horizon T , then the natural terminal condition for xT would be to set xT = 0, meaning that all stock positions should be converted to cash at maturity of the portfolio. This implies that the last action is deterministic rather than stochastic and is determined by the stock holding at time T − 1: uT −1 = xT − xT −1 = −xT −1 .

(10.129)

The last reward is therefore a quadratic functional of xT −1 : Rˆ T −1 = −ηuTT −1 CuT −1 .

(10.130)

As the last action is deterministic, optimization amounts to choosing the remaining T − 1 portfolio adjustments u0 , . . . , uT −2 . We now show that reinforcement learning with G-learning can be solved semianalytically in this setting using Gaussian time-varying policies (GTVP) . We start by specifying a functional form of the value function as a quadratic form of xt : (0) Ftπ (xt ) = xTt F(xx) xt + xTt F(x) t t + Ft ,

(10.131)

5 G-Learning for Stock Portfolios (xx)

(x)

397

(0)

where Ft , Ft , Ft are parameters that depend on time both explicitly (for finite horizon problems) and implicitly, via their dependence on the signals zt . As for the last time step we have FTπ−1 (xT −1 ) = Rˆ T −1 , using Eqs. (10.130) and (10.131), we obtain the terminal conditions for coefficients in Eq. (10.131): (xx)

(x)

(0)

FT −1 = −ηC, FT −1 = 0, FT −1 = 0.

(10.132)

For an arbitrary time step t = T − 2, . . . , 0, we use Eq. (10.126) and independence of noise terms for xt and zt to compute the conditional expectation of the next-period F-function in Eq. (10.115) as follows:

π (xx) (xx) (xt+1 ) = (xt + ut )T ATt F¯ t+1 At + r ◦ F¯ t+1 (xt + ut ) Et,a Ft+1 + (xt + ut )T ATt F¯ t+1 + F¯ t+1 , (x)

(0)

(10.133)

! " (xx) (xx) (x) (0) where F¯ t+1 := Et Ft+1 , and similarly for F¯ t+1 and F¯ t+1 . Importantly, this is a ˆ t , at ) quadratic function of xt and ut . Combining it with the quadratic reward R(x in (10.131) in the Bellman equation (10.115), we see that the action-value function Gπt (xt , ut ) should also be a quadratic function of xt and ut : (0) T (x) Gπt (xt , ut ) = xTt Q(xx) xt + uTt Q(uu) ut + uTt Q(ux) xt + uTt Q(u) t t t t + xt Qt + Qt , (10.134) where

(xx) (xx) (xx) Qt = −λr + γ ATt F¯ At + r ◦ F¯ t+1

(uu)

= −ηC + Qt

(ux)

= 2Qt

Qt Qt

t+1

(xx)

(xx)

(10.135)

T ¯ (x) Q(x) t = Wzt + γ At Ft+1 (u)

Qt

(x)

= Qt

(0) Q(0) t = γ Ft+1 .

Having computed the G-function Gπt (xt , ut ) in terms of its coefficients (10.135), the F-function for the current step can be found using Eq. (10.113) which we repeat here in terms of the original variables xt , ut , and replacing summation by integration: Ftπ (xt ) =

1 log β

(

π (x

π0 (ut |xt )eβGt

t ,ut )

dut .

We assume that a reference policy π0 (ut |xt ) is Gaussian:

(10.136)

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10 Applications of Reinforcement Learning

π0 (ut |xt ) = #

1

− 12 (ut −uˆ t ) p−1 (ut −uˆ t ) T

e (2π )n p

,

(10.137)

where the mean value uˆ t is a linear function of the state xt : uˆ t = u¯ t + v¯ t xt .

(10.138)

Here u¯ t and v¯ t are parameters that can be considered time-independent (so that the time index can be omitted) in the prior distribution (10.137). The reason we keep the time label is that, as we will see shortly, the optimal policy obtained from Glearning with linear dynamics (10.126) is also a Gaussian that can be written in the same form as (10.137) but with updated parameters u¯ t and v¯ t that will become time-dependent due to their dependence on signals zt . If no constraints are imposed on ut , integration over ut in Eq. (10.136) with a Gaussian reference policy π0 can be easily performed analytically as long as Gπt (xt , ut ) is quadratic in ut .8 Using the n-dimensional Gaussian integration formula gives: 9

( e

− 12 xT Ax+xT B n

d x=

(2π )n 1 BT A−1 B , e2 |A|

(10.139)

where |A| denotes the determinant of matrix A. Using this relation to calculate the integral in Eq. (10.136) and introducing auxiliary parameters Ut = βQ(ux) + p−1 v¯ t t (u)

Wt = βQt

+ p−1 u¯ t

(10.140)

(uu) ¯ p = −1

, p − 2βQt

we find that the resulting F -function has the same structure as in Eq. (10.131), where the coefficients are now computed in terms of coefficients of the Q-function (see Exercise 10.3): (xx)

Ftπ (xt ) = xTt Ft (xx)

Ft

8 As

(x)

(0)

xt + xTt Ft + Ft

1 T ¯ −1 (xx) ¯ Ut p Ut − v¯ Tt −1 v = Qt + t p 2β

in the present formulation actions are constrained by the self-financing condition, an independent Gaussian integration may produce inaccurate results. For a constrained version of the integral with a constraint on the sum of variables, see Exercise 10.6. In the next section we will present a case where an unconstrained Gaussian integration works better.

5 G-Learning for Stock Portfolios (x)

Ft

(0)

Ft

399

1 T ¯ −1 ¯t Ut p Wt − v¯ Tt −1 (10.141) p u β

' 1 & 1 T ¯ −1 (0) ¯ p . ¯ Wt p Wt − u¯ Tt −1 log p + log u − = Qt + t p 2β 2β (x)

= Qt

+

Finally, the optimal policy for the given step can be found using Eq. (10.114) which we again rewrite here in terms of the original variables xt , ut : π (x

π(ut |xt ) = π0 (ut |xt )eβ (Gt

π t ,ut )−Ft (xt )

).

(10.142)

As Gπt (xt , ut ) is a quadratic function of ut , this produces a Gaussian policy π(ut |xt ): T ˜ −1 1 ˜ t −˜vt xt ) − 1 (ut −u˜ t −˜vt xt ) p (ut −u π(ut |xt ) = : , e 2 ˜ (2π )n p

(10.143)

with updated parameters (uu) −1 ˜ −1

p = p − 2βQt

˜ p −1 ¯ t + βQ(u) u˜ t = t p u

(ux) ˜ p −1 ¯ v˜ t = . v + βQ t t p

(10.144)

Therefore, policy optimization with the entropy-regularized LQR with signals expressed by Eq. (10.142) amounts to the Bayesian update of the prior distribu˜p tion (10.137) with parameters updates u¯ t , v¯ t , p to the new values u˜ t , v˜ t , defined in Eqs. (10.144). These quantities depend on time via their dependence on zt . Note that the third of equations (10.144) indicates that even if we started with v¯ t = 0 (meaning a state-independent mean level in Eq. (10.137)), the optimal policy has a mean that is linear in state xt as long as Q(ux) = 0. Therefore, t the entropy-regularized LQR produces a Gaussian optimal policy whose mean is a linear function of the state xt . This provides a probabilistic generalization of a regular (deterministic) LQR where an optimal policy itself is a linear function of the state. Another interesting point to note about the Bayesian update (10.144) is that ¯ v¯ , the updated values even if we start with time-independent values u¯ t , v¯ t = u, (u) u˜ t , v˜ t will be time-dependent as parameters Q(ux) and Q depend on time via t t their dependence on signals zt , see Eqs. (10.135). For a given time step t, the G-learning algorithm keeps iterating between the policy optimization step that updates policy parameters according to Eq. (10.144)

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10 Applications of Reinforcement Learning

for fixed coefficients of the F - and G-functions, and the policy evaluation step that involves Eqs. (10.134, 10.135, 10.141) and solves for parameters of the F - and Gfunctions given policy parameters. At convergence of this iteration for time step t, Eqs. (10.134, 10.135, 10.141) and (10.143) together solve one step of G-learning for the entropy-regularized linear dynamics. The calculation then proceeds by moving to the previous step t → t − 1, and repeating the calculation. To accelerate convergence, optimal parameters of the policy at time t can be used as parameters of a prior distribution for time step t − 1. The whole procedure is then continued all the way back to the present time. Note that as parameters in Eq. (10.141) depend on the signals zt , their expected next-step values will have to be computed as indicated in Eq. (10.133).

•

? Multiple Choice Question 4

Select all the following correct statements: a. In G-learning, the conventional action-value function and value recovered from F- and G-functions in the limit β → 0. b. In G-learning, the conventional action-value function and value recovered from F- and G-functions in the limit β → ∞. c. Linearization of portfolio dynamics with G-learning is needed to linear G-function and F-function. d. Linearization of portfolio dynamics with G-learning is needed to quadratic G-function and F-function.

function are function are get a locally get a locally

5.16 Non-zero Market Impact: Non-linear Dynamics As we just demonstrated, in the limit of vanishing market impact parameters, dynamic portfolio optimization using G-learning with quadratic rewards and Gaussian reference policies amounts to a probabilistic version of a linear quadratic regulator (LQR) and provides a convenient and fast semi-analytical calculation method to optimize an investment policy for a multi-asset and multi-period portfolio. This provides a probabilistic and multi-period RL-based generalization of the classical Markowitz portfolio optimization problem. If we turn the market impact parameters μi on, this leads to drastic changes in the problem: it is no longer analytically tractable due to non-linearity of dynamics for μi > 0. The state equation in this case is xt+1 = (1 + rf )(xt + ut ) + diag (Wzt − Mut ) (xt + ut ) + ε(xt , ut ), where M = diag (μi ) and ε(xt , ut ) := εt ◦ (xt + ut ) (see Eqs. (10.123), (10.124)).

6 RL for Wealth Management

401

When dynamics are non-linear, one possibility is to iteratively linearize the dynamics, similar to the working of an extended Kalman filter. For problems with deterministic policies, an iterative quadratic regulator (IQR) can be used to linearize the dynamics around a reference path at each step of a policy iteration (Todorov and Li 2005). Other methods can be applied when working with stochastic policies. In particular, Halperin and Feldshteyn (2018) explore a variational EM algorithm where Gaussian random hidden variables are used to linearize dynamics, providing a probabilistic version of the IQR. Such approaches, needed when the dynamics are non-linear, are more technically involved that the linear LQR case, and the reader is referred to the literature for details. Beyond and above of purely computational issues, non-linearity of dynamics leads to important theoretical implications. The problem of optimal control is to find the optimal action u%t as a function of the state xt . When this is done, the resulting expression can be substituted back to obtain non-linear open-loop dynamics (i.e., the dynamics where action variables are substituted by their optimal values). As will be discussed later in this book, the ensuing non-linearity of dynamics might have important consequences for capturing the behavior of real markets.

6 RL for Wealth Management 6.1 The Merton Consumption Problem Our two previous use cases for reinforcement learning in quantitative finance, namely option pricing and dynamic portfolio optimization, operate with selffinancing portfolios. Recall that self-financing portfolios are asset portfolios that do not admit cash injections or withdrawals at any point in a lifetime of a portfolio, except at its inception and closing. For these classical financial problems, the assumption of a self-financing portfolio is well suited and reasonably matches actual financial practices. There is yet another wide class of classical problems in finance where a selffinancing portfolio is not a good starting point for modeling. Financial planning and wealth management are two good examples of such problems. Indeed, a typical investor in a retirement plan makes periodic investments in the portfolio while being employed and periodically draws from the account when in retirement. In addition to adding or withdrawing capital to the portfolio, the investor can also re-balance the portfolio by selling and buying different assets (e.g., stocks). The problem of optimal consumption with an investment portfolio is frequently referred to as the Merton consumption problem, after the celebrated work of Robert Merton who considered this problem as a problem of continuous-time optimal control with log-normal dynamics for asset prices (Merton 1971). As optimization in problems involving cash injections instead of cash withdrawals formally corresponds to a sign change of one-step consumption in the Merton formulation,

402

10 Applications of Reinforcement Learning

we can collectively refer to all types of wealth management problems involving injections or withdrawals of funds at intermediate time steps as a generalized Merton consumption problem. We shall begin with a simple example which involves a combination of asset allocation and consumption under a specific choice of utility of wealth and lifetime consumption. Example 10.9

Discrete time Merton consumption with CRRA utility

To illustrate the basic setup, let us consider the problem of allocating wealth between a risky asset and a risk-free asset, in addition to wealth consumption. The optimal consumption problem is formulated in a discrete-time, finitehorizon setting rather than the classical continuous-time approach of Merton (1971). Following Cheung and Yang (2007), we will assume that the investment horizon T ∈ N is fixed. We further assume that at the beginning of each time period, an investor can decide the allocation of wealth between the risky asset and the level of consumption, which should be non-negative and less than her total wealth at that time. Denote the wealth of the investor at time t as Wt , and the random return from the risk assets in time period [t, t +1] is denoted as Rt . The time t consumption level is denoted as ct ∈ [0, Wt ]. After consumption, a proportion αt ∈ [0, 1] of the remaining amount will be invested in the risky asset and the rest in the risk-free asset. We refer to these constraints as the “budget constraints.” The discrete-time wealth evolution equation is given by Wt+t = (Wt − ct )[(1 − αt )Rt + αt Rf t].

(10.145)

with an initial positive wealth W0 at time t = 0. The sequence of maps (C, α) = {(c0 , α0 ), . . . , (cT −1 , αT −1 )} that satisfies the budget constraints is called the “investment-consumption strategy.” The expected sum of the rewards for consumption (i.e., a utility of consumption) with a terminal reward is used as a criterion to measure the performance of an investment-consumption strategy. In RL, we are free to choose any reward function which is concave in the actions. Writing down the optimization problem: max

(c0 ,α0 ),...,(CT −1 ,αT −1

E[

T −1

γ t R(Wt , (ct , αt ), Wt+1 ) + γ T R(WT )|W0 = w],

t=0

(10.146) (continued)

6 RL for Wealth Management

Example 10.9

403

(continued)

the optimal investment strategy is given by Vt (w) =

T −1

E[

max

(ct ,αt ),...,(CT −1 ,αT −1

γ s−t R(Ws , (cs , αs ), Ws+1 )+γ T −t R(WT )|W0 = w],

s=t

(10.147) which can be solved from the Bellman Equation for the value function updates:

Vt (w) = max E[γ Vt+1 (Wt+1 )|Wt = w], ∀t ∈ {0, . . . , T − 1}, (ct ,αt )

(10.148)

and some terminal condition for VT (w). A common choice of utility function, which leads to closed-form solutions, is to choose a constant relative risk aversion (CRRA) utility function of the form U (x) = γ1 x γ , with γ ∈ (0, 1). Then the state function reduces to Vt (w) =

"1−γ wγ ! 1/(1−γ ) 1 + Y H , (γ ) t γ

(10.149)

with optimal consumption, linear in the wealth: cˆt (w) =

w

(1 + (γ Yt Ht )1/(1−γ ) )

≤ w,

(10.150)

where the expected returns of the fund under optimal allocation at time t are

Yt = E[(αt∗ Rt + (1 − αt∗ )Rf )γ ]

(10.151)

and the recurrent variable is given by the recursion relation

Ht = 1 + (γ Yt+1 Ht+1 )1/(1−γ ) ,

(10.152)

and we assume HT = 0. Note that there is a unique αt∗ ∈ [0, 1] such that Yt is maximized. To show that Yt is concave in αt∗ , we see that γ (γ − 1) (αt∗ E[Rt ] + (1 − αt∗ )Rf )γ

−2

(E[Rt ] − Rf )2 ≤ 0,

(10.153) (continued)

404

10 Applications of Reinforcement Learning

Example 10.9

(continued)

since γ (γ −1) < 0, (E[Rt ]−Rf )2 ≥ 0, and (αt∗ E[Rt ]+(1−αt∗ )Rf )γ −2 > 0 for non-negative risk-free rates and average stock returns. In the absence of transaction costs, clearly when the expected return of the risky asset is above Rf , we allocate the wealth to the risky asset and when the expected return is below, we choose to entirely invest in the risk-free account. We can therefore simplify the optimization problem under the CRRA utility function to solve for the optimal consumption from Eq. (10.150). Figure 10.6 illustrates the optimal consumption under simulated stock prices. The optimal allocation is not shown here but alternates between 0 and 1 depending on whether the mean return of the risk asset is, respectively, above or below the risk-free rate. The analytic approach of Cheung and Yang (2007) in the above example is limited to the choice of utility function and single asset portfolio. One can solve the same problem with flexibility in the choice of utility functions using the LSPI algorithm, but such an approach does not extend to higher dimensional portfolios. We therefore turn to a G-learning approach which scales to high-dimensional portfolios while providing some flexibility in the choice of utility functions. In the following section, we will consider a class of wealth management problems: optimization of a defined contribution retirement plan, where cash is injected (rather than withdrawn) at each time step. Instead of relying on a utility of consumption, along the lines of the approach just previously described, we will adopt a more “RL-native” approach by directly specifying one-step rewards. Another difference is that, as in the previous section, we define actions as absolute (dollar-valued) changes of asset positions, instead of defining them in fractional terms, as in the Merton approach. As we will see shortly, this enables a simple transformation into an unconstrained optimization problem and provides a semianalytical solution for a particular choice of the reward function.

30

c *t (optimal consumption)

160

St

140 120 100 80

25 20 15 10 5 0

2

4

6 t (a)

8

10

10

20

30 Wealth (W) (b)

40

50

Fig. 10.6 Stock prices are simulated using an Euler scheme over a one-year horizon. At each of ten periods shown by ten separate lines, the optimal consumption is estimated using the closedform formula in Eq. (10.150). The optimal investment is monotonically increasing in time. (a) Simulated stock prices. (b) Optimal consumption against wealth

6 RL for Wealth Management

405

6.2 Portfolio Optimization for a Defined Contribution Retirement Plan Here we consider a simplified model for retirement planning. We assume a discretetime process with T steps, so that T is the (integer-valued) time horizon. The investor/planner keeps the wealth in N assets, with xt being the vector of dollar values of positions in different assets at time t, and ut being the vector of changes in these positions. We assume that the first asset with n = 1 is a risk-free bond, and other assets are risky, with uncertain returns rt whose expected values are r¯ t . The covariance matrix of return is r of size (N − 1) × (N − 1). Note that our notation in this section is different from the previous section where xt was used to denote a vector of risky asset holding values. Optimization of a retirement plan involves optimization of both regular contributions to the plan and asset allocations. Let ct be a cash installment in the plan at time t. The pair (ct , ut ) can thus be considered the action variables in a dynamic optimization problem corresponding to the retirement plan. We assume that at each time step t, there is a pre-specified target value Pˆt+1 of a portfolio at time t + 1. We assume that the target value Pˆt+1 at step t exceeds the next-step value Vt+1 = (1 + rt )(xt + ut ) of the portfolio, and we want to impose a penalty for under-performance relative to this target. To this end, we can consider the following expected reward for time step t: Rt (xt , ut , ct ) = −ct − λEt

Pˆt+1 − (1 + rt )(xt + ut ) − uTt ut . +

(10.154)

Here the first term is due to an installment of amount ct at the beginning of time period t, the second term is the expected negative reward from the end of the period for under-performance relative to the target, and the third term approximates transaction costs by a convex functional with the parameter matrix and serves as a L2 regularization. The one-step reward (10.154) is inconvenient to work with due to the rectified non-linearity (·)+ := max(·, 0) under the expectation. Another problem is that decision variables ct and ut are not independent but rather satisfy the following constraint: N

utn = ct ,

(10.155)

n=1

which simply means that at every time step, the total change in all positions should equal the cash installment ct at this time. We therefore modify the one-step reward (10.154) in two ways: we replace the first term using Eq. (10.155) and approximate the rectified non-linearity by a quadratic function. The new one-step reward is

406

10 Applications of Reinforcement Learning

Rt (xt , ut ) = −

N

utn − λEt

Pˆt+1 − (1 + rt )(xt + ut )

2

− uTt ut .

n=1

(10.156) The new reward function (10.156) is attractive on two counts. First, it explicitly resolves the constraint (10.155) between the cash injection ct and portfolio allocation decisions, and thus converts the initial constrained optimization problem into an unconstrained one. This differs from the Merton model where allocation variables are defined as fractions of the total wealth, and thus are constrained by construction. The approach based on dollar-measured actions both reduces the dimensionality of the optimization problem and makes it unconstrained. When the unconstrained optimization problem is solved, the optimal contribution ct at time t can be obtained from Eq. (10.155). The second attractive feature of the reward (10.156) is that it is quadratic in actions ut and is therefore highly tractable. On the other hand, the well-known disadvantage of quadratic rewards (penalties) is that they are symmetric, and penalize both scenarios Vt+1 ' Pˆt+1 and Vt+1 , Pˆt+1 , while in fact we only want to penalize the second class of scenarios. To mitigate this drawback, we can consider target values Pˆt+1 that are considerably higher than the time-t expectation of the next-period portfolio value. In what follows we assume this is the case, otherwise the value of Pˆt+1 can be arbitrary. For example, one simple choice could be to set the target portfolio as the current portfolio growing with a fixed and sufficiently high return. We note that a quadratic loss specification relative to a target time-dependent wealth level is a popular choice in the recent literature on wealth management. One example is provided by Lin et al. (2019) who develop a dynamic optimization approach with a similar squared loss function for a defined contribution retirement plan. A similar approach that relies on a direct specification of a reward based on a target portfolio level is known as “goal-based wealth management” (Browne 1996; Das et al. 2018).

•

> Goal-Based Wealth Management

Mean–variance Markowitz optimization remains one of the most commonly used tools in wealth management. Portfolio objectives in this approach are defined in terms of expected returns and covariances of asset in the portfolio, which may not be the most natural formulation for retail investors. Indeed, the latter typically seek specific financial goals for their portfolios. For example, a contributor to a retirement plan may demand that the value of their portfolio (continued)

6 RL for Wealth Management

407

at the age of his or her retirement be at least equal to, or preferably larger than, some target value PT . Goal-based wealth management offers some interesting perspectives into optimal structuring of wealth management plans such as retirement plans or target date funds. The motivation for operating in terms of wealth goals can be more intuitive (while still tractable) than the classical formulation in terms of expected excess returns and variances. To see this, let VT be the final wealth in the portfolio, and PT be a certain target wealth level at the horizon T . The goal-based wealth management approach of Browne (1996) and Das et al. (2018) uses the probability P [VT − PT ≥ 0] of final wealth VT to be above the target level PT as an objective for maximization by an active portfolio management. This probability is the same as the price of a binary option on the terminal wealth VT with strike PT : P [VT − PT ≥ 0] = Et 1VT >PT . Instead of a utility of wealth such as a power or logarithmic utility, this approach uses the price of this binary option as the objective function. This ideacan also be modified by using a call option-like expectation Et (VT − PT )+ , instead of a binary option. Such an expectation quantifies how much the terminal wealth is expected to exceed the target, rather than simply providing the probability of such an event.

The square loss reward specification is very convenient, as we have already seen on many occasions in this chapter, as it allows one to construct optimal policies semi-analytically. Here we will show how to build a semi-analytical scheme for computing optimal stochastic consumption-investment policies for a retirement plan—the method is sufficiently general for either a cumulation or de-cumulation phase. For other specifications of rewards, numerical optimization and function approximations (e.g., neural networks) would be required. The expected reward (10.156) can be written in a more explicit form if we denote asset returns as rt = r¯ t + ε˜ t where the first component r¯0 (t) = rf is the risk-free rate (as the first asset is risk-free), and ε˜ t = (0, ε t ), where ε t is an idiosyncratic noise with covariance r of size (N − 1) × (N − 1). Substituting this expression in Eq. (10.156), we obtain 2 − uTt 1 + 2λPˆt+1 (xt + ut )T (1 + r¯ t ) Rt (xt , ut ) = −λPˆt+1

ˆ t (xt + ut ) − uTt ut , − λ (xt + ut )T

(10.157)

ˆ t = 0 0 + (1 + r¯ t )(1 + r¯ t )T .

0 r

(10.158)

where we defined

408

10 Applications of Reinforcement Learning

The quadratic one-step reward (10.157) has a similar structure to the rewards we considered in the previous section, see, e.g., Eq. (10.128). In contrast to the setting in Sect. 5.15, instead of a self-financing portfolio, here we deal with a portfolio with periodic cash installments ct . However, because the latter are related to allocation decision variables by the constraint (10.155), the resulting quadratic reward (10.157) has the same quadratic structure as the linear LQR reward (10.128).

6.3 G-Learning for Retirement Plan Optimization As we have just mentioned, the quadratic one-step reward (10.157) is very similar to the reward Eq. (10.128) which we considered in Sect. 5.15 for a self-financing 2 in (10.157) portfolio. The main difference is the presence of the first term −λPˆt+1 which is independent of a state or action. This term does not impact the policy optimization task, and can be trivially accounted for, if needed, e.g., to compute the total reward, by a direct summation from all time steps in the plan lifeline. As in Sect. 5.15, we use a similar semi-analytical formulation of G-learning with Gaussian time-varying policies (GTVP). We start by specifying a functional form of the value function as a quadratic form of xt : (xx)

Ftπ (xt ) = xTt Ft (xx)

(x)

(x)

xt + xTt Ft

(0)

+ Ft ,

(10.159)

(0)

where Ft , Ft , Ft are parameters that can depend on time via their dependence on the target values Pˆt+1 and the expected returns r¯ t (in the formulation in Sect. 5.15, the latter were encoded in signals zt ). The dynamic equation now reads (compare with Eq. (10.126)) At := diag (1 + r¯ t ) , ε˜ t := (0, ε t ) (10.160) Coefficients of the value function (10.159) are computed backward in time starting from the last maturity t = T − 1. For t = T − 1, the quadratic reward (10.157) can be optimized analytically by the following action: xt+1 = At (xt + ut ) + (xt + ut ) ◦ ε˜ t ,

˜ ˜ −1 ˆ uT −1 = T −1 PT − T −1 xT −1 ,

(10.161)

˜ T −1 and P˜ T as follows: where we defined parameters ˜ T −1 := ˆ T −1 + 1 , P˜ T := PˆT (1 + r¯ T −1 ) − 1 .

λ 2λ

(10.162)

Note that the optimal action is a linear function of the state, as in our previous section. Another interesting point to note is that the last term ∼ that describes

6 RL for Wealth Management

409

convex transaction costs in Eq. (10.157) produces regularization of matrix inversion in Eq. (10.161). As for the last time step we have FTπ−1 (xT −1 ) = Rˆ T −1 , coefficients

(x) (0) F(xx) T −1 , FT −1 , FT −1 can be computed by plugging Eq. (10.161) back in Eq. (10.157), and comparing the result with Eq. (10.159) with t = T − 1. This provides terminal conditions for parameters in Eq. (10.159): −1

−1

ˆ T −1 ˜ T −1 ˆ T −1 T −1 − ˜ T −1 ˆ T −1 FT −1 = −λTT −1 (xx)

ˆ T ˆ ˜ −1 ¯ T −1 ) F(x) T −1 = T −1 T −1 1 + 2λPT T −1 (1 + r −1

−1

−1

ˆ T −1 ˜ T −1 P˜ T ˆ T −1 ˜ T −1 P˜ T + 2 ˜ T −1 − 2λTT −1

(10.163)

(0) ˜ ˆ ˜ −1 ˜ −1 ¯ T −1 )T FT −1 = −λPˆT2 − P˜ TT T −1 1 + 2λPT (1 + r T −1 PT −1

−1

−1

−1

˜ T −1 P˜ T , . ˜ T −1 ˆ T −1 ˜ T −1 P˜ T − P˜ TT ˜ T −1 − λP˜ TT ˜ −1 ˆ where we defined T −1 := I − T −1 T −1 . For an arbitrary time step t = T − 2, . . . , 0, we use Eq. (10.160) to compute the conditional expectation of the nextperiod F-function in the Bellman equation (10.115) as follows:

π (xx) ˜ r ◦ F¯ (xx) (xt + ut ) (xt+1 ) = (xt + ut )T ATt F¯ t+1 At + Et,a Ft+1 t+1 (x) (0) ˜ r := 0 0 ,(10.164) + (xt + ut )T ATt F¯ t+1 + F¯t+1 , 0 r ! " (xx) ¯ (x) ¯ (0) where F¯ (xx) F := E t t+1 t+1 , and similarly for Ft+1 and Ft+1 . This is a quadratic ˆ t , at ) in function of xt and ut and has the same structure as the quadratic reward R(x Eq. (10.157). Plugging both expressions in the Bellman equation π (yt+1 ) yt , at Gπt (yt , at ) = Rˆ t (yt , at ) + γ Et,a Ft+1 we see that the action-value function Gπt (xt , ut ) should also be a quadratic function of xt and ut : (xx)

Gπt (xt , ut ) = xTt Qt

(xu)

xt + xTt Qt

(uu)

ut + uTt Qt

(x)

ut + xTt Qt

(u)

+ uTt Qt

(0)

+ Qt , (10.165)

where (xx)

Qt

ˆ t + γ ATt F¯ (xx) At + ˜ r ◦ F¯ (xx) = −λ t+1 t+1

Q(xu) = 2Q(xx) t t (uu)

Qt

(xx)

= Qt

−

(10.166)

410

10 Applications of Reinforcement Learning (x)

(x) = 2λPˆt+1 (1 + r¯ t ) + γ ATt F¯ t+1

(u)

= Qt

(0)

(0) 2 = −λPˆt+1 + γ Ft+1 .

Qt Qt

Qt

(x)

−1

Note that the quadratic action-value function in Eq. (10.165) is similar to Eq. (10.134)—the only difference is the specification of the parameters. Beyond different expressions for coefficients of the value function Ft (xt ) and action-value function Gt (xt , ut ) and a different terminal condition, the rest of calculations to perform one step of G-learning is the same as in Sect. 5.15. The F-function for the current step can be found using Eq. (10.136) repeated again here: Ftπ (xt ) =

1 log β

(

π (x

π0 (ut |xt )eβGt

t ,ut )

(10.167)

dut .

A reference policy π0 (ut |xt ) is Gaussian: π0 (ut |xt ) = #

1

− 12 (ut −uˆ t ) p−1 (ut −uˆ t ) T

e (2π )n p

,

(10.168)

where the mean value uˆ t is a linear function of the state xt : uˆ t = u¯ t + v¯ t xt .

(10.169)

Again as in Sect. 5.15, integration over ut in Eq. (10.167) is performed analytically using Eq. (10.139). The difference is that in Sect. 5.15 we considered a selffinancing asset portfolio, which constrains actions ut . Ignoring such a constraint can produce numerical inaccuracies. In contrast, in the present case we do not impose constraints on actions ut ; therefore, an unconstrained multivariate Gaussian integration should be superior in this case. Remarkably, this implies that once the decision variables are chosen appropriately, portfolio optimization for wealth management tasks may in a sense be an easier problem than portfolio optimization with self-financing. Performing the Gaussian integration and comparing the resulting expression with Eq. (10.159), we obtain for its coefficients: (xx)

Ftπ (xt ) = xTt Ft (xx)

Ft

F(x) t (0)

Ft

(x)

(0)

xt + xTt Ft + Ft

1 T ¯ −1 (xx) ¯ Ut p Ut − v¯ Tt −1 v = Qt + t p 2β

1 T ¯ −1 ¯t Ut p Wt − v¯ Tt −1 (10.170) = Q(x) t + p u β

' 1 & 1 T ¯ −1 (0) ¯ p , ¯ Wt p Wt − u¯ Tt −1 log p + log − u = Qt + t p 2β 2β

6 RL for Wealth Management

411

where we use the auxiliary parameters (ux)

Ut = βQt

(u)

Wt = βQt

+ p−1 v¯ t

+ p−1 u¯ t

(10.171)

(uu) ¯ p = −1

. p − 2βQt

The optimal policy for the given step can be found using Eq. (10.142) repeated again here: π (x

π(ut |xt ) = π0 (ut |xt )eβ (Gt

π t ,ut )−Ft (xt )

).

(10.172)

Using here the quadratic action-value function (10.165) produces a new Gaussian policy π(ut |xt ): π(ut |xt ) = :

1

e n ˜ (2π ) p

T ˜ −1 ˆ t −˜vt xt ) − 12 (ut −u˜ t −˜vt xt ) p (ut −u

,

(10.173)

where (uu) ˜ p−1 = −1 p − 2βQt

˜ p −1 ¯ t + βQ(u) u˜ t = t p u

˜ p −1 ¯ t + βQ(ux) v˜ t = t p v

(10.174)

Therefore, policy optimization for G-learning with quadratic rewards and Gaussian reference policy amounts to the Bayesian update of the prior distribution (10.168) ˜ p defined in with parameters updates u¯ t , v¯ t , p to the new values u˜ t , v˜ t , Eqs. (10.174). These quantities depend on time via their dependence on the targets Pˆt and expected asset returns r¯ t . As in Sect. 5.15, for a given time step t, the G-learning algorithm keeps iterating between the policy optimization step that updates policy parameters according to Eq. (10.174) for fixed coefficients of the F - and G-functions, and the policy evaluation step that involves Eqs. (10.165, 10.166, 10.170) and solves for parameters of the F - and G-functions given policy parameters. Note that convergence of ˜ −1 iterations for u˜ t , v˜ t is guaranteed as p p < 1. At convergence of iteration for time step t, Eqs. (10.165, 10.166, 10.170) and (10.143) together solve one step of G-learning. The calculation then proceeds by moving to the previous step t → t −1, and repeating the calculation, all the way back to the present time.

412

10 Applications of Reinforcement Learning

The additional step needed from G-learning for the present problem is to find the optimal cash contribution for each time step by using the budget constraint (10.155). As G-learning produces Gaussian random actions ut , Eq. (10.155) implies that the time-t optimal contribution ct is Gaussian distributed with mean c¯t = 1T (u¯ t + v¯ t xt ). The expected optimal contribution c¯t thus has a part ∼ u¯ t that is independent of the portfolio value, and a part ∼ v¯ t that depends on the current portfolio. This is similar, e.g., to a linear specification of the defined contribution with a deterministic policy in Lin et al. (2019). It should be noted that in practice, we may want to impose some constraints on cash installments ct . For example, we could impose band constraints 0 ≤ ct ≤ cmax with some upper bound cmax . Such constraints can be easily added to the framework. To this end, we need to replace the exactly solvable unconstrained least squares problem with a constrained least squares problem. This can be done without a substantial increase of computational time using efficient off-the-shell convex optimization software. An illustration of an optimal solution trajectory obtained without enforcing any constraints is shown in Fig. 10.7 which presents simulation results for a portfolio of 100 assets with 30 time steps. For the specific choice of model parameters used in this example, the model optimally chooses to invest approximately equal contributions around $500 that slightly increase towards the end of the plan without enforcing constraints, which is achieved by setting a high target portfolio. However, in a more general setting adding constraints might be desirable. Optimal cash installment and portfolio value 30000

optimal cash installments expected portfolio value realized target portfolio

25000 20000 15000 10000 5000 0 0

5

10

15 Time Steps

20

25

30

Fig. 10.7 An illustration of G-learning with Gaussian time-varying policies (GTVP) for a retirement plan optimization using a portfolio with 100 assets

7 Summary

413

6.4 Discussion To summarize, we have shown how the same G-learning method with quadratic rewards that we used in the previous section to construct an optimal policy for dynamic portfolio optimization can also be used for wealth management problems, provided we use absolute (dollar-nominated) asset position changes as action variables and choose a reward function which is quadratic in these actions. As shown in Sect. 5.15, we found that G-learning applied in the current setting with a quadratic reward and Gaussian reference policy gives rise to an entropy-regulated LQR as a tool for wealth management tasks. This approach results in a Gaussian optimal policy whose mean is a linear function of the state xt . The method we presented enables extensions to other formulations including constrained versions or other specifications of the reward function. One possibility is to use the definition (10.154) with the constraint (10.155), which provides an example of a non-quadratic concave reward. Such cases should be dealt with using flexible function approximations for the action-value function such as neural networks.

7 Summary The main underlying idea of this chapter was to show that reinforcement learning provides a very natural framework for some of most classical problems of quantitative finance: option pricing and hedging, portfolio optimization, and wealth management problems. As trading in individual assets (or pairs, or buckets of assets) is a particular case of the general portfolio optimization problem, we can say that this list covers most cases for quantitative trading or planning problems in finance. We saw that for option pricing with reinforcement learning, batch-mode Qlearning can be used as a way to produce a distribution-free discrete-time approach to pricing and hedging of options. In the simplest case when transaction costs and market impact are neglected, as in the classical Black–Scholes model, the reinforcement learning approach is semi-analytical and only requires solving linear matrix equations. In other cases, for example, if the exponential utility is used to enforce a strict no-arbitrage, analytic quadratic optimization should be replaced by numerical convex optimization. We then presented a multivariate and probabilistic extension of Q-learning known as G-learning, as a tool for using reinforcement learning for portfolio optimization with multiple assets. Unlike the previous case of the QLBS model that neglects transaction costs and market impact, the G-learning approach captures these effects. When the reward function is quadratic as in the Markowitz mean– variance approach and market impact is neglected, the G-learning approach again yields a semi-analytical solution to the dynamic portfolio optimization, that is given by a probabilistic version of the classical linear quadratic regulator (LQR).

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10 Applications of Reinforcement Learning

For other cases (e.g., when market impact is incorporated, or when rewards are not quadratic), the G-learning approach should rely on more involved numerical optimization methods and/or use function approximations such as neural networks. In addition to demonstrating how G-learning can be applied for portfolio management, we also showed how it can be used for tasks of wealth management, which differ from the former case by intermediate cash-flows as additional controls. We showed that both these classical financial problems can be treated using the same computational method (G-learning) with different parameter specifications. Therefore, the reinforcement learning approach is capable of providing a unified approach to these classes of problems, which are traditionally treated as different problems because of a different definition of control variables. What we showed is that when using absolute (dollar-nominated) decision variables, both problems can be treated in the same way. Moreover, with this approach wealth management problems turn out to be simpler, not harder, than traditional portfolio optimization problems, as they amount to unconstrained optimization.9 While our presentation in this chapter used general RL methods (Q-learning and G-learning), we mostly focused on cases with quadratic rewards which enable semi-analytical and hence easily understandable computational methods. Different reward specifications are possible, of course, but they require relying on function approximations (e.g., neural networks—giving rise to deep reinforcement learning) and numerical optimization for optimizing parameters of these networks. Furthermore, other methods of reinforcement learning (e.g., LSPI, policy-gradient methods, actor-critic, etc.) can also be used, see, e.g., Sato (2019) for a review.

8 Exercises Exercise 10.1 Derive Eq. (10.46) that gives the limit of the optimal action in the QLBS model in the continuous-time limit. Exercise 10.2 Consider the expression (10.121) for optimal policy obtained with G-learning π(at |yt ) =

π 1 ˆ π0 (at |yt )eR(yt ,at )+γ Et,at Ft+1 (yt+1 ) Zt

where the one-step reward is quadratic as in Eq. (10.91): ˆ t , at ) = yTt Ryy yt + aTt Raa a + aTt Ray yt + aTt Ra . R(y

9 Or,

if we want to put additional constraints on the resulting cash-flows, to optimization with one constraint, instead of two constraints as in the Merton approach.

8 Exercises

415

Howdoes this relation simplify in two cases: (a) when the conditional expectation π (y Et,a Ft+1 t+1 ) does not depend on the action at , and (b) when the dynamics are linear in at as in Eq. (10.125)? Exercise 10.3 Derive relations (10.141). Exercise 10.4 Consider G-learning for a time-stationary case, given by Eq. (10.122): ˆ a) + Gπ (y, a) = R(y,

γ π ρ(y |y, a) log π0 (a |y )eβG (y ,a ) β y

a

Show that the high-temperature limit β → 0 of this equation reproduces the fixedpolicy Bellman equation for Gπ (y, a) where the policy coincides with the prior policy, i.e. π = π0 . Exercise 10.5 Consider policy update equations for G-learning given by Eqs. (10.174): (uu) ˜ p−1 = −1 p − 2βQt

(u) ˜ p −1 ¯ u˜ t = u + βQ t t p

(ux) ˜ p −1 ¯ v v˜ t = + βQ t t p

(a) Find the limiting forms of these expressions in the high-temperature limit β → 0 and low-temperature limit β → ∞. (b) Assuming that we know the stable point (u¯ t , v¯ t ) of these iterative equations, ˜ p , invert them to find parameters of Q-function in as well as the covariance (uu) (ux) terms of stable point values u¯ t , v¯ t . Note that only parameters Qt , Qt , and (u) (xx) (x) Qt can be recovered. Can you explain why parameters Qt and Qt are lost in this procedure? (Note: this problem can be viewed as a prelude to the topic of inverse reinforcement learning covered in the next chapter.) Exercise 10.6*** The formula for an unconstrained Gaussian integral in n dimensions reads 9

( e

− 12 xT Ax+xT B n

d x=

(2π )n 1 BT A−1 B . e2 |A|

¯ with a parameter X ¯ is imposed on the Show that when a constraint ni=1 xi ≤ X integration variables, a constrained version of this integral reads

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10 Applications of Reinforcement Learning

,

( e

− 12 xT Ax+xT B

¯ − θ X

n

9

xi d x = n

i=1

-, , ¯ BT A−1 1 − X (2π)n 1 BT A−1 B 2 e 1−N √ |A| 1T A−1 1

where N (·) is the cumulative normal distribution. Hint: use the integral representation of the Heaviside step function 1 ε→0 2π i

θ (x) = lim

(

∞ −∞

eizx dz. z − iε

Appendix Answers to Multiple Choice Questions Question 1 Answer: 2, 3. Question 2 Answer: 2, 4. Question 3 Answer: 1, 2, 3. Question 4 Answer: 2, 4.

Python Notebooks This chapter is accompanied by two notebooks which implement the QLBS model for option pricing and optimal hedging, and G-learning for wealth management. Further details of the notebooks are included in the README.md file.

References Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654. Boyd, S., Busetti, E., Diamond, S., Kahn, R., Koh, K., Nystrup, P., et al. (2017). Multi-period trading via convex optimization. Foundations and Trends in Optimization, 1–74. Browne, S. (1996). Reaching goals by a deadline: digital options and continuous-time active portfolio management. https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/841/ sidbrowne_deadlines.pdf.

References

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Carr, P., Ellis, K., & Gupta, V. (1988). Static hedging of exotic options. Journal of Finance, 53(3), 1165–1190. Cerný, A., & Kallsen, J. (2007). Hedging by sequential regression revisited. Working paper, City University London and TU München. Cheung, K. C., & Yang, H. (2007). Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete and Continuous Dynamical Systems, 8(2), 315–332. Das, S. R., Ostrov, D., Radhakrishnan, A., & Srivastav, D. (2018). Dynamic portfolio allocation in goals-based wealth management. https://papers.ssrn.com/sol3/papers.cfm?abstract_id= 3211951. Duan, J. C., & Simonato, J. G. (2001). American option pricing under GARCH by a Markov chain approximation. Journal of Economic Dynamics and Control, 25, 1689–1718. Ernst, D., Geurts, P., & Wehenkel, L. (2005). Tree-based batch model reinforcement learning. Journal of Machine Learning Research, 6, 405–556. Föllmer, H., & Schweizer, M. (1989). Hedging by sequential regression: An introduction to the mathematics of option trading. ASTIN Bulletin, 18, 147–160. Fox, R., Pakman, A., & Tishby, N. (2015). Taming the noise in reinforcement learning via soft updates. In 32nd Conference on Uncertainty in Artificial Intelligence (UAI). https://arxiv.org/ pdf/1512.08562.pdf. Garleanu, N., & Pedersen, L. H. (2013). Dynamic trading with predictable returns and transaction costs. Journal of Finance, 68(6), 2309–2340. Gosavi, A. (2015). Finite horizon Markov control with one-step variance penalties. In Conference Proceedings of the Allerton Conferences, Allerton, IL. Grau, A. J. (2007). Applications of least-square regressions to pricing and hedging of financial derivatives. PhD. thesis, Technische Universit"at München. Halperin, I. (2018). QLBS: Q-learner in the Black-Scholes(-Merton) worlds. Journal of Derivatives 2020, (to be published). Available at https://papers.ssrn.com/sol3/papers.cfm?abstract_ id=3087076. Halperin, I. (2019). The QLBS Q-learner goes NuQLear: Fitted Q iteration, inverse RL, and option portfolios. Quantitative Finance, 19(9). https://doi.org/10.1080/14697688.2019. 1622302, available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3102707. Halperin, I., & Feldshteyn, I. (2018). Market self-learning of signals, impact and optimal trading: invisible hand inference with free energy, (or, how we learned to stop worrying and love bounded rationality). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3174498. Lin, C., Zeng, L., & Wu, H. (2019). Multi-period portfolio optimization in a defined contribution pension plan during the decumulation phase. Journal of Industrial and Management Optimization, 15(1), 401–427. https://doi.org/10.3934/jimo.2018059. Longstaff, F. A., & Schwartz, E. S. (2001). Valuing American options by simulation - a simple least-square approach. The Review of Financial Studies, 14(1), 113–147. Markowitz, H. (1959). Portfolio selection: efficient diversification of investment. John Wiley. Marschinski, R., Rossi, P., Tavoni, M., & Cocco, F. (2007). Portfolio selection with probabilistic utility. Annals of Operations Research, 151(1), 223–239. Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4), 373–413. Merton, R. C. (1974). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183. Murphy, S. A. (2005). A generalization error for Q-learning. Journal of Machine Learning Research, 6, 1073–1097. Ortega, P. A., & Lee, D. D. (2014). An adversarial interpretation of information-theoretic bounded rationality. In Proceedings of the Twenty-Eighth AAAI Conference on AI. https://arxiv.org/abs/ 1404.5668. Petrelli, A., Balachandran, R., Siu, O., Chatterjee, R., Jun, Z., & Kapoor, V. (2010). Optimal dynamic hedging of equity options: residual-risks transaction-costs. working paper. Potters, M., Bouchaud, J., & Sestovic, D. (2001). Hedged Monte Carlo: low variance derivative pricing with objective probabilities. Physica A, 289, 517–525.

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Sato, Y. (2019). Model-free reinforcement learning for financial portfolios: a brief survey. https:// arxiv.org/pdf/1904.04973.pdf. Schweizer, M. (1995). Variance-optimal hedging in discrete time. Mathematics of Operations Research, 20, 1–32. Todorov, E., & Li, W. (2005). A generalized iterative LQG method for locally-optimal feedback control of constrained nonlinear stochastic systems. In Proceeding of the American Control Conference, Portland OR, USA, pp. 300–306. van Hasselt, H. (2010). Double Q-learning. Advances in Neural Information Processing Systems. http://papers.nips.cc/paper/3964-double-q-learning.pdf. Watkins, C. J. (1989). Learning from delayed rewards. Ph.D. Thesis, Kings College, Cambridge, England. Watkins, C. J., & Dayan, P. (1992). Q-learning. Machine Learning, 8(179–192), 3–4. Wilmott, P. (1998). Derivatives: the theory and practice of financial engineering. Wiley.

Chapter 11

Inverse Reinforcement Learning and Imitation Learning

This chapter provides an overview of the most popular methods of inverse reinforcement learning (IRL) and imitation learning (IL). These methods solve the problem of optimal control in a data-driven way, similarly to reinforcement learning, however with the critical difference that now rewards are not observed. The problem is rather to learn the reward function from the observed behavior of an agent. As behavioral data without rewards is widely available, the problem of learning from such data is certainly very interesting. This chapter provides a moderatelevel technical description of the most promising IRL methods, equips the reader with sufficient knowledge to understand and follow the current literature on IRL, and presents examples that use simple simulated environments to evaluate how these methods perform when the “ground-truth” rewards are known. We then present use cases for IRL in quantitative finance which include applications in trading strategy identification, sentiment-based trading, option pricing, inference of portfolio investors, and market modeling.

1 Introduction One of the challenges faced by researchers when applying reinforcement learning to solve real-world problems is how to choose the reward function. In general, a reward function should encourage a desired behavior of an agent, but often there are multiple approaches to specify it. For example, assume that we seek to solve an index tracking problem, where the task is to replicate a certain market index target (e.g., the S&P 500) Pt by a smaller portfolio of stocks whose value at time t

target target 2 is Pttrack . We can view expressions Pttrack − Pt , or or Pttrack − Pt

target track − Pt as possible reward functions. Clearly the optimal choice of the Pt +

reward here is equivalent to the optimal choice of an expected risk-adjusted return

© Springer Nature Switzerland AG 2020 M. F. Dixon et al., Machine Learning in Finance, https://doi.org/10.1007/978-3-030-41068-1_11

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in the corresponding multi-period portfolio optimization problem. Therefore, the choice of a reward function is as non-unique as the choice of a risk function for portfolio optimization. Challenges of defining a good reward function to teach an agent to perform a certain task are well known in other fields that use machine learning. For example, teaching physical robots to perform simple (for humans) tasks such as carrying a cup of coffee between two tables by hand-engineering a reward function in a multidimensional space of robot joints’ positions and velocities at each time step may be as difficult as defining the execution policy directly, without relying on any reward function. Therefore the need to pre-specify a reward function considerably limits the applicability of reinforcement learning for many cases of practical interest. In finance, traders do not often think in terms of any specific utility (reward) function, but rather in terms of strategies (or policies, using the language of RL). In response to such practical challenges, researchers in machine learning developed a number of alternatives to the classical setting of dynamic programming and reinforcement learning that do not require a reward (utility) function to be specified. Learning to act without a reward function is known as learning from demonstrations or learning from behavior. As behavioral data is often produced in abundance (think of GPS monitoring data, cell phone data, web browsing data, etc.), the notion of learning from observed behavior of other agents (humans or machines) is certainly appealing, and has a wide range of potential industrial and business applications. But what exactly does it mean to learn from demonstrations? One possible answer to this question is that it means learning the optimal policy from the observed behavior given only observations of actions produced by this policy (or a similar one). This is called imitation learning. This is similar to the batch-mode RL that we considered in the previous chapter, but without knowledge of the rewards. That is, we observe a series of states and actions taken by an agent, and our task is to find the optimal policy solely from this data. As with batch-mode RL, this is an inference problem of a distribution (policy) from data. In contrast to batch-mode RL, however, the problem of learning from demonstrations is ill-defined. Indeed for any particular trajectory of states and actions, there exist an infinite number of policies consistent with this trajectory. The same holds for any finite set of observed trajectories. The problem of learning a policy from a finite set of observed trajectories is therefore an ill-posed inverse problem.

•

> Ill-Posed Inverse Problems

Ill-posed inverse problems usually exhibit an infinite number of solutions or no solution at all. Classical examples of such inverse problems include the problems of restoring a signal after passing a filter and contaminated by (continued)

1 Introduction

421

an additive noise. A simple financial example is implying the risk-neutral distribution p(sT |s0 ) of future stock prices from observed prices of European options. If F (sT , K) is a discounted payoff of a liquid European option with maturity T , the observed market mid-price of the option can be written as ( C(st , K) =

dst p(sT |s0 )F (sT , K) + εt ,

where εt stands for an observation noise. Having market prices for a finite number of quoted options is not sufficient to reconstruct a conditional density p(sT |s0 ) in a model-independent way, as a model-independent specification is equivalent to an infinite number of observations. The inverse problem is ill-posed in the sense that it does not have a unique solution. Various forms of regularization are needed in order to select the “best” solution. For example, one may use Lagrange multipliers to enforce constraints on the entropy of p(sT |s0 ) or its KL divergence with some reference (“prior”1 ). What constitutes the “best” solution of the inverse problem is therefore only specified after a regularization function is chosen. A good choice of a regularization function may be problem- or domain-specific and go beyond a KL regularization or, e.g., a simple L2 or L1 regularization. Essentially, as the choice of regularization amounts to the choice of a “regularization model,” we may assert that a purely model-independent method for inverse problems does not exist.

We may be tempted to reason that such a problem is scarcely different from the conventional setting of supervised learning. Indeed, we could treat the observed states and actions in a training dataset as, respectively, features and outputs. We could then try to directly construct a policy as either a classifier or a regressor, considering each action as a separate observation. This approach is indeed possible and is known as behavioral cloning. While it is simple (any supervised method can be used), it also has a number of serious drawbacks that render it impractical. The main problem is that it often does not generalize well to new unseen states. This is simply because as an action policy is estimated from each individual state using supervised learning, it passes no information on how these states can be related to each other. This can be contrasted with TD methods (such as Q-learning) that use transitions as observations. As 1 It

is convenient to regard the reference distribution as a prior, however it is not strictly a prior in the context of Bayesian estimation. MaxEnt or Minimum Cross-Entropy finds a distribution that is compatible with all available constraints and minimizes a KL distance to this reference distribution. In contrast, Bayesian learning involves using Bayes’ rule to incrementally update the posterior.

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11 Inverse Reinforcement Learning and Imitation Learning

a result, generalization of the policy to unseen states obtained using supervised learning with behavioral cloning loses any connection to the actual dynamics of the environment. If such a learned policy is executed over multiple steps, errors can compound, and an induced state distribution can shift away from the actual one used in demonstrations. Therefore, a combination of different methods is usually required when rewards are not available. Such a multi-faceted approach sounds reasonable in principle but the devil is in the detail. For example, we could use a recurrent neural network to capture the state dynamics, and use a feedforward network to directly parameterize the policy. Parameters of both networks would then be learned from the data using, e.g., stochastic gradient descent methods. The main potential problem with such an approach is that it may not be easily portable to other environments, when the model is used with dynamics that are different from those used for model training. But in an approach similar to the one described above, dynamics and the learned policy are intertwined in complex ways. Therefore, we can expect that a learned policy would become sub-optimal once the dynamics (environment) changes. On the other hand, a reward function is portable, as it depends only on states and actions, but is not concerned with how these states are reached. If we find a way to learn the reward function from demonstrated behavior, this function would be portable to other environments, as it expresses properties of the agent but not of the environment. The idea of learning the reward function as a condensed and portable representation of an agent’s preferences was suggested by Russell (1998). Methods centered around this idea have collectively became known as inverse reinforcement learning (IRL). Clearly, if the reward function is found from IRL, then the optimal policy with any new environment can be found using the conventional (direct) RL. In this chapter, we provide an overview of the most popular methods of IRL, as well as methods of imitation learning that do not rely on a learned reward function. We hasten to add that so far, IRL was only adopted for financial applications in a handful of publications, despite several successful applications in robotics and video games. Nevertheless, we believe that methods of IRL are potentially very useful for quantitative finance. Both because the whole field of IRL is still nascent and keeps evolving, and because there are only a few published papers on using IRL for financial applications, this chapter is mostly focused on theoretical concepts of IRL. Our task in this chapter is three-fold: (i) provide a reasonably high-level description of the most promising IRL methods; (ii) equip the reader with enough knowledge to understand and follow the current literature on IRL; and (iii) present use cases for IRL in quantitative finance including applications to trading strategy identification, sentiment-based trading, option pricing, inference of portfolio investors, and market modeling. Chapter Objectives This chapter will review some of the most pertinent aspects of inverse reinforcement learning and their application to finance:

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– Introduce methods of inverse reinforcement learning (IRL) and imitation learning (IL); – Provide a review of recent adversarial approaches to IRL and IL; – Introduce IRL methods which can surpass a demonstrator; and – Review existing and potential applications of IRL and IL in quantitative finance. The chapter is accompanied by a notebook comparing various IRL methods for the financial cliff walking problem. See Appendix “Python Notebooks” for further details.

2 Inverse Reinforcement Learning The key idea of Russell (1998) is that the reward function should provide the most succinct representation of agents’ preferences while being transferable between both environments and agents. Before we turn to finance applications, imagine for a moment examples in everyday life—an adaptive smart home that learns from the habits of its occupants in scheduling different tasks such as pre-heating food, placing orders to buy other food, etc. In autonomous cars, the control system could learn from drivers’ preferences to set up an autonomous driving style that would be comfortable to a driver when taken for a ride as a passenger. In marketing applications, knowing preferences of customers or potential buyers, quantified as their utility functions, can inform marketing strategies tuned to their perceived preferences. In financial applications, knowing the utility of a counterparty may be useful in bilateral trading, e.g. over-the-counter (OTC) trades in derivatives or credit default swaps. Other financial applications of IRL, such as option pricing, will be discussed later in this chapter once we present the most popular IRL methods. Just as reinforcement learning is rooted in dynamic programming, IRL has also its analog (or predecessor) in inverse optimal control (ICO). As with IRL, the objective of ICO is to learn the cost function. However, in the ICO setting, the dynamics and optimal policy are assumed to be known. Faithful to a datadriven approach of (direct) reinforcement learning, IRL does not assume that state dynamics or policy functions are known, and instead constructs an empirical distribution.2 Inverse reinforcement learning (IRL) therefore provides a useful extension (or inversion, hence justifying its name) of the (direct) RL paradigm. In the context of batch-mode learning used in the previous chapter, the setting of IRL is nearly identical to that of RL (see Eq. (10.58)), except that there is no information about the rewards:

T −1 (n) (n) (n) (n) Ft = Xt , at , Xt+1 , n = 1, . . . , N. (11.1) t=0

2 Methods

of ICO that assume that dynamics are known are sometimes referred to as model-based IRL in the machine learning literature.

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11 Inverse Reinforcement Learning and Imitation Learning (n)

The objective of IRL is typically two-fold: (i) find the rewards Rt most consistent with observed states and action and (ii) find the optimal policy and action-value function (as in RL). One can distinguish between on-policy IRL and off-policy IRL. In the former case, we know that observed actions were optimal actions. In the latter case, observed actions may not necessarily follow an optimal policy and can be suboptimal or noisy. In general, IRL is a harder problem than RL. Indeed, not only must the optimal policy be found from data, which is the same task as in RL, but under the additional complexity that the rewards are unobserved. It appears that information about rewards is frequently missing in many potential real-world applications of RL/IRL. In particular, this is typically the case when RL methods are applied to study human behavior, see, e.g., Liu et al. (2013). IRL is also widely used in robotics as a useful alternative to direct RL methods via training robots by demonstrations (Kober et al. 2013). It appears that IRL offers a very attractive, at least conceptually, approach for many financial applications that use rational agents involved in a sequential decision process, where no information about rewards received by an agent is available to a researcher. Some examples of such (semi-) rational agents would be retail or institutional investors, loan or mortgage borrowers, deposit or saving account holders, credit card holders, consumers of utilities such as cloud computing, mobile data, electricity, etc. In the context of trading applications, such an IRL setting may arise when a trader seeks to learn a strategy of a counterparty. She observes the counterparty’s actions in their bilateral trades, but not the counterparty’s rewards. Clearly, if she reverseengineered the most likely counterparty’s rewards from the observed actions to find the counterparty’s objective (strategy), she could use it to design her own strategy. This typifies an IRL problem. Example 11.1

IRL for Financial Cliff Walking

Consider our financial cliff walking (FCW) example from Chap. 9 where we presented it as a toy problem for RL control—an over-simplified model of household finance. Now let us consider an IRL formulation of this problem where we would be given a set of trajectories sampled from some policy, and attempt to find both the rewards and the policy. Clearly, as you will recall, the optimal policy for the FCW example is to deposit the minimum amount in the account at time t = 0, then take no further action until the very last step, at which point the account should be closed, with the reward of 10. However, sampling from this policy, possibly randomized by adding a random component, may miss the important penalty for breaching the bankruptcy level, and rather treat any examples of such events in the training data as occasional “sub-optimal” trajectories. As we will show in Sect. 9, the conventional IRL indeed misses the higher importance of not breaching the bankruptcy level than of achieving the final-step rewards.

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2.1 RL Versus IRL A very convenient concept for differentiating between the IRL and the direct RL problem is the occupancy measure ρπ (s, a) : S × A → R (see (Putterman 1994) and Exercise 9.6 in Chap. 9): ρπ (s, a|s0 ) = π(a|s)

∞

γ t Pr (st = s|π, s0 ) ,

(11.2)

t=0

where Pr (st = s|π ) is the probability density of the state s = st at time t following policy π . The occupancy measure is also a function of the current state s0 at time t = 0. The value function V = V (s0 ) of the current state can now be defined as an expectation of the reward function: ( V (s0 ) =

ρπ (s, a|s0 )r(s, a) dsda.

(11.3)

Recall from Exercise 9.1 that due to an invariance of the optimal policy under a common rescaling of all rewards r(s, a) → αr(s, a) for some fixed α > 0, the occupancy measure ρπ (s, a|s0 ) can be interpreted as a normalized probability density of state-action pairs, even though the correct normalization is not explicitly enforced in the definition (11.2). Here we arrive at the central difference between RL and IRL. In RL, we are given numerical values of an unknown reward function r(st , at ), where sampled trajectories τ = {st , at }Tt=0 of length T are obtained by using an unknown “expert” policy πE (for batch-mode RL), or alternatively for on-line RL by sampling from the environment when executing a model policy πθ (a|s). The problem of (direct) RL is to find an optimal policy π% , and hence an optimal measure ρ% that maximizes the expectation (11.3) given the sampled data in the form of tuplets (st , at , rt , st+1 ). Because we observe numerical rewards, a value function can be directly estimated from data using Monte Carlo methods. Note that the optimal measure ρ% cannot be an arbitrary probability density function, but rather should satisfy time consistency constraints imposed by model dynamics, known as Bellman flow constraints. Now compare this setting with IRL. In IRL, data consists of tuplets (st , at , st+1 ) without rewards rt . In other words, all we observe are trajectories—sequences of states and actions τ = {st , at }Tt=0 . In terms of maximization of the value function as the expected value (11.3), the IRL setting amounts to providing a set of pairs {st , at }Tt=0 that is assumed to be informative for a Monte Carlo based estimate of the expectation (11.3) of a (yet unknown) reward function r(st , at ). Clearly, to build any model of the reward function given the observed trajectories, we should assume that trajectories demonstrated are sufficiently representative of true dynamics, and that the expert policy used in the recorded data is optimal or at least “sufficiently close” to an optimal policy. If either of these assumptions do not

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hold, it is highly implausible that a “true” reward function can be recovered from such data. On the other hand, if demonstrated trajectories are obtained from an expert, actions taken should be optimal from the viewpoint of Monte Carlo sampling of the expectation (11.3). A simple model that relates observed actions with rewards and encourages taking optimal actions could be a stochastic policy π(a|s) ∼ exp (β(r(s, a) + F (s, a))), where β > 0 is a parameter (inverse temperature), r(s, a) is the expected reward for a single step, and F (s, a) is a function that incorporates information of future rewards into decision-making at time t. As we saw in the previous chapter, Maximum Entropy RL produces exactly this type a) + F (s, a) = Gπt (st , at ) = Eπ [r(st , at , st+1 )] + of policy, where r(s, π γ st+1 p(st+1 |st , at )Ft+1 (st+1 ) is the G-function (the “soft” Q-value), see also Exercise 9.13 in Chap. 9. Due to exponential dependence of this policy on instantaneous rewards r(st , at , st+1 ) or equivalently on expected rewards r(st , at ) = Eπ [r(st , at , st+1 )], the optimal policy optimizes the expected total reward. The idea of Maximum Entropy IRL (MaxEnt IRL) is to preserve the functional form of Boltzmann-like policies πθ (a|s) produced by MaxEnt RL, where θ is a vector of model parameters. As they are explicit functions of states and actions, we can now use them differently, as probabilities of data in terms of observed values of states st and actions at . Parameters of the MaxEnt Boltzmann policy πθ (a|s) can therefore be inferred using the standard maximum likelihood method. We shall return to MaxEnt IRL later in this chapter.

2.2 What Are the Criteria for Success in IRL? In the absence of a “ground truth” expected reward function r(s, a), what are the performance criteria for any IRL method that learns a parameterized reward policy rθ (s, a) ∈ R, where R is the space of all admissible reward functions? Recall that the task of IRL is to learn both the reward function and optimal policy π and hence the state-action occupation measure ρπ from the data. Therefore, once both functions are learned, we can use them both to compute the value function obtained with these inferred reward and policy functions. The quality of the IRL method would thus be determined by the value function obtained using these reward and policy functions. We conclude that performance criteria for IRL involve solving a direct RL problem with a learned reward function. Moreover, we can make maximization of the expected reward an objective of IRL, such that each iteration over parameters θ specifying a parameterized expected reward function rθ (s, a) ∈ R will involve solving a direct RL problem with a current reward function. But such an IRL method could easily become very time-consuming and infeasible in practice for problems with large state-action space, where a direct RL problem would become computationally intensive.

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427

Some methods of imitation learning or IRL avoid the need to solve a direct RL problem in an inner loop. For example, MaxEnt IRL methods that we mentioned above reduce IRL to a problem of inference in a graphical (more precisely, exponential) model. This changes the computational framework, but produces another computational burden, as it requires estimation of a normalization constant Z of a MaxEnt policy (also known as a partition function). Early versions of MaxEnt IRL for discrete state-action spaces computed such normalization constants using dynamic programming (Ziebart et al. 2008), see later in this chapter for more details. While more recent MaxEnt IRL approaches rely on different methods of estimation of the partition function Z (e.g., using importance sampling), this suggests that improving the computational efficiency of IRL methods by excluding RL from the internal loop is a hard problem. Such an approach is addressed by GAIL and related methods that will be presented later in this chapter.

2.3 Can a Truly Portable Reward Function Be Learned with IRL? Recall the basic premise of reinforcement learning is that a reward function is the most compact form of expressing preferences of an agent. As an expected reward function r(st , at ) depends only on the current state and actions and does not depend on the dynamics, once it specified, it can be used with direct RL to find an optimal policy for any environment. In the setting of IRL, we do not observe rewards but rather learn them from an observed behavior. As expected rewards are only functions of states and actions, and are independent of an environment, it appears appealing to try to estimate them by fitting a parameterized action policy πθ to observations, and conjecture that the inferred reward would produce optimal policies in environments different from the one used for learning. The main question, of course, is how realistic is such a conjecture? Note that this question is different (and in a sense harder) than the aforementioned problem of ill-posedness of IRL. As an inverse problem, IRL is sensitive to the choice of a regularization method. A particular regularization may address the problem of incomplete data for learning a function, but it does not ensure that a reward learned with one environment would produce an optimal policy when an agent with a learned reward is deployed in a new environment that differs from an environment used for learning. The concept of reward shaping suggested by Ng and Russell in 1999 (see Exercise 9.5 in Chap. 9) suggests that the problem of learning a portable (robust) reward function from demonstrations can be challenging. The main result of this analysis is that the optimal policy remains unchanged under the following transformation of the instantaneous reward function r(s, a, s ):

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r˜ (s, a, s ) = r(s, a, s ) + γ (s ) −

(s)

(11.4)

for an arbitrary function : S × A → R. As we will see next, the reward shape invariance of reinforcement learning may indicate that the problem of learning robust (portable) rewards may be quite challenging and infeasible to solve in a model-independent way. To see this, assume that we have two MDPs M and M that have identical rewards and differ only in transition probabilities that we will denote as T and T , respectively. A simple example would be deterministic dynamics T (s, a) → s . If a “true” reward function is of the form (11.4), it can be expressed as r(s, a, s ) + γ T (T (s, a)) − (s). If the new dynamics is such that T (s, a) = T (s, a), such reward will not be in the equivalence class of shape invariance for dynamics T (Fu et al. 2019). On the other hand, the same argument suggests an approach for constructing a reward function that could be transferred to other environments. To this end, we could simply constrain the inferred expected reward not to contain any additive contributions that would depend only on the state st . In other words, we can learn a reward function only up to an arbitrary additive function of the state. Due to the reward shape invariance of an optimal policy, this function is of no interest for finding the optimal policy, and thus can be set to zero for all practical purposes.

•

? Multiple Choice Question 1

Select all the following correct statements: a. The task of inverse reinforcement learning is to learn the dynamics from the observed behavior. b. The task of inverse reinforcement learning is to find the worst, rather than the best policy for the agent. c. The task of inverse reinforcement learning is to find both the reward function and the policy from observations of the states and actions of the agent.

3 Maximum Entropy Inverse Reinforcement Learning In learning from demonstrations, it is important to understand which assumptions are made regarding actions performed by an agent. As both the policy and rewards are unknown in the setting of IRL, we have to make additional assumptions when solving this problem. A natural assumption would be to expect that the demonstrated actions are optimal or close to optimal. In other words, it means that the agent acts optimally or close to optimally. If we assume that the agent follows a deterministic policy, then the above assumption implies that every action should be optimal. But this leaves little margin

3 Maximum Entropy Inverse Reinforcement Learning

429

for any errors resulting from model bias, noisy parameter estimations, etc. Under a deterministic policy model, a trajectory that contains a single sub-optimal action has an infinite negative log-likelihood—which exactly expresses the impossibility of such scenarios with deterministic policies. A more forgiving approach is to assume that the agent followed a stochastic policy πθ (a|s). Under such a policy, sub-optimal actions can be observed, though they are expected to be suppressed according to a model for πθ (a|s). Once a parametric model for the stochastic policy is specified, its parameters can be estimated using standard methods such as maximum likelihood estimation. In this section, we present a family of probabilistic IRL models with a particular exponential specification of a stochastic policy: πθ (a|s) =

1 erˆθ (s,a) , Zθ (s) = Zθ (s)

(

erˆθ (s,a) .

(11.5)

Here rˆθ (s, a) is some function that will be related to the reward and action-value functions of reinforcement learning. We have already seen stochastic policies with such exponential specification in Chap. 9 when we discussed “softmax in action” policies (see Eq. (9.38)), and in Chap. 10 when we introduced G-learning (see Eq. (10.114)). While in the previous sections a stochastic policy with an exponential parameterization such as Eq. (11.5) was occasionally referred to as a softmax policy, in the setting of inverse reinforcement learning, it is often referred to as a Boltzmann policy, in recognition of its links to statistical mechanics and work of Ludwig Boltzmann in the nineteenth century (see the box below). Methods leading to stochastic policies of the form (11.5) are generally based on maximization of entropy or minimization of the KL divergence in a space of parameterized action policies. The objective of this section is to provide an overview of such approaches.

•

> The Boltzmann Distribution in Statistical Mechanics

In statistical mechanics, exponential distributions similar to (11.5) appear when considering closed systems with a fixed composition, such as molecular gases, that are in thermal equilibrium with their environment. The first formulation was suggested by Ludwig Boltzmann in 1868 in his work that developed a probabilistic approach to molecular gases in thermal equilibrium. The Boltzmann distribution characterizes states of a macroscopic system, such as a molecular gas, in terms of their energies Ei , where index i enumerates possible states. When such a system is at equilibrium with its (continued)

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11 Inverse Reinforcement Learning and Imitation Learning

environment that has temperature T , the Boltzmann distribution gives the probabilities of different energy states in the following form: pi =

1 − kEiT e B , Z

where kB is a constant parameter called the Boltzmann constant, and Z is a normalization factor of the distribution, which is referred to as the partition function in statistical mechanics. The Boltzmann distribution can be obtained as a distribution which maximizes the entropy of the system H =−

pi log pi ,

i

where the sum is taken over all energy states accessible to the system, subject ¯ where E¯ is some average mean energy. The to the constraint i pi Ei = E, Boltzmann distribution was extensively investigated and generalized beyond molecular gases to a general setting of systems at equilibrium by Josiah Willard Gibbs in 1902. It is therefore often referred to as the Boltzmann– Gibbs distribution in physics. In particular, Gibbs introduced the notion of a statistical ensemble as an idealization consisting of virtual copies of a system, such that each copy represents a possible state that the real system might be in. The physics-based notion of an ensemble corresponds to the notion of a probability space in mathematics. The Boltzmann distribution arises for the so-called canonical ensemble that is obtained for a system with a fixed number of particles at thermal equilibrium with its environment (called a heat bath in physics) at a fixed temperature T . The Boltzmann–Gibbs distribution serves as the foundation for the modern approach to equilibrium statistical mechanics (Landau and Lifshitz 1980).

3.1 Maximum Entropy Principle The Maximum Entropy (MaxEnt) principle (Jaynes 1957) is a general and highly popular method for ill-posed inversion problems where a probability distribution should be learned from a finite set of integral constraints on this distribution. The main idea of the MaxEnt method for inference of distributions from data given constraints is that beyond matching such constraints, the inferred distribution

3 Maximum Entropy Inverse Reinforcement Learning

431

should be maximally non-informative, i.e. it should produce the highest possible uncertainty of a random variable described by this distribution. As an amount of uncertainty in a distribution can be quantified by its entropy, the MaxEnt method amounts to finding the distribution that maximizes the entropy while matching all available integral constraints. The MaxEnt principle provides a practical implementation of Laplace’s principle of insufficient reason, and has roots in statistical physics (Jaynes 1957). Here we show the working of the MaxEnt principle using an example of learning the action policy in a simple single-step reinforcement setting. Such a setting is equivalent to removing time from the problem. This simplifies the setup considerably, because all data can now be assumed i.i.d., and there is no need to consider future implications of a current action. The resulting time-independent version of the MaxEnt principle is a version originally proposed by Jaynes in 1957. To understand this approach, we shall elucidate it from a statistical mechanics perspective. Let π(a|s) be an action policy. Consider a one-step setting, with a single reward r(s, a) to be received for different combinations of (s, a). An optimal policy should % maximize the value function V π (s) = r(s, a)π(a|s)da. However, the value function is a linear functional of π(a|s) and does not exhibit an optimal value of π(a|s) per se. Assuming that the rewards r(s, a) are known, a concave optimization problem is obtained if we add an entropy regularization and consider the following functional: ( 1 1 π π F (s) := V (s) + H [π(a|s)] = π(a|s) r(s, a) − log π(a|s) da, β β (11.6) where 1/β is a regularization parameter. If we take the variational derivative of this expression with respect to π(a|s) and set it to zero, we obtain the optimal action policy (see Exercise 11.1) π(a|s) =

1 eβr(s,a) , Zβ (s) := Zβ (s)

( eβr(s,a) da.

(11.7)

Clearly, this expression assumes that the reward function r(s, a) is such that the integral defining the normalization factor Zβ (s) converges for all attainable values of s. Equation (11.7) has the same exponential form as Eq. (11.5) if we choose a parametric specification βr(s, a) = rθ (s, a). The expression (11.7) is obtained above as a solution of an entropy-regularized maximization problem. The same form can also be obtained in a different way, by maximizing the entropy of π(a|s) conditional on matching a given average reward r¯ (s). This is achieved by maximizing the following functional: F˜ π (s) = −

)(

( π(a|s) log π(a|s) + λ

* π(a|s)r(s, a)da − r¯ (s) ,

(11.8)

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11 Inverse Reinforcement Learning and Imitation Learning

where λ is a Lagrange multiplier. The optimal distribution is π(a|s) =

1 eλr(s,a) , Zλ (s) = Zλ (s)

( eλr(s,a) da.

(11.9)

This has the same form as (11.7) with β = 1/λ. On the other hand, for problems involving integral constraints, the value of λ can be fixed in terms of the expected reward r¯ . To this end, we substitute the solution (11.9) into Eq. (11.8) and minimize the resulting expression with respect to λ to give: min log Zλ − λ¯r (s). λ

(11.10)

This produces 1 Zλ (s)

( r(s, a)eλr(s,a) da = r¯ (s),

(11.11)

which is exactly the integral constraint on π(a|s). The optimal value of λ is found by solving Eq. (11.11) or equivalently by numerical minimization of (11.10). Note that this produces a unique solution as the optimization problem (11.10) is convex (see Exercise 11.1).

•

> Links with Statistical Mechanics

As especially suggested by the second derivation, Eq. (11.7) or (11.9) can also be seen as a Boltzmann distribution of a statistical ensemble with energies E(s, a) = −r(s, a) on a space of state-action pairs. In statistical mechanics, a distribution of states x ∈ X in a space of state X with energies E(x) for a canonical ensemble with a fixed average energy is given by the same Boltzmann form, albeit with a different functional form of the energy E. In statistical mechanics, parameter β has a specific form β = 1/(kB T ), where kB is the Boltzmann constant, and T is a temperature of the system. For this reason, parameter β in Eq. (11.7) is often referred to as the inverse temperature.

A direct generalization of the MaxEnt principle is given by the minimum crossentropy (MCE) principle that replaces the absolute entropy with a KL divergence with some reference distribution π0 (a|s). In this case, instead of (11.8), we consider the following KL-regularized value function:

3 Maximum Entropy Inverse Reinforcement Learning

( F (s) = π

433

π(a|s) 1 π(a|s) r(s, a) − log da. β π0 (a|s)

(11.12)

The optimal action policy for this case is π(a|s) =

1 π0 (a|s)eβr(s,a) , Zβ (s)

( Zβ (s) :=

π0 (a|s)eβr(s,a) da.

(11.13)

The common feature of all methods based on MaxEnt or MCE principles is therefore the appearance of an exponential energy-based probability distribution. For a simple single-step setting with reward r(s, a), the MaxEnt optimal policy is exponential in r(s, a). As we will see next, an extension of entropy-based analysis also produces an exponential specification for the action policy in a multi-step case.

3.2 Maximum Causal Entropy In general, reinforcement learning or inverse reinforcement learning involves trajectories that extend over multiple steps. In the most general form, we have a series of states ST = S0:T and a series of actions AT = A0:T with some trajectory length T > 1. Sequences of states and actions in an MDP can be thought of as two interacting random processes S0:T and A0:T . The problem of learning a policy can be viewed as a problem of inference of a distribution of actions A0:T given a distribution of states S0:T . Unlike MaxEnt inference problems for i.i.d. data, the time dependence of such problems requires some care. Indeed, a naive definition of a distribution of actions conditioned on states could involve conditional probabilities defined on the whole paths, such as P [ A0:T | S0:T ]. A problem with such a definition would be that it could violate causality, as conditioning on a whole path of states involves conditioning on the future. For Markov Decision Processes, actions at time t can depend only on the current state. If memory effects are important, they can be handled by using higher-order MDPs, or by switching to autoregressive models such as recurrent neural networks. Clearly, in both cases, to preserve causality, actions now (at time t) cannot depend on the future. When each variable at is conditioned only on a portion S0:t of all variables S0:T , the probability of A causally conditioned on S reads P

T AT ST = P ( At | S0:t , A0:t−1 ) .

(11.14)

t=0

Note that the standard definition of conditional probability would involve conditioning on the whole path S0:T —which would violate causality. The causal conditional

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11 Inverse Reinforcement Learning and Imitation Learning

' & probability (11.14) implies, in particular, that any joint distribution P AT , ST can ' & ' & ' & be factorized as P AT , ST = P AT ST P ST AT −1 . The causal entropy (Kramer 1998) is defined as follows: H

T

" ! H ( At || S0:t , A0:t ) . AT ST = EA,S − log P AT ST = t=0

' & T (11.15) In this section, we assume that the dynamics are Markovian, so that P S AT −1 $ = t P ( st+1 | st , at ). In addition, we assume a setting of an infinite-horizon MDP. For this case, we should use a discounted version of the causal entropy: H ( A0:∞ || S0:∞ ) =

T

γ t H ( At || S0:t , A0:t ) ,

(11.16)

t=0

with a discount factor γ ≤ 1. The causal entropy (11.16) (or (11.15), for a finite-horizon case) presents a natural extension of an entropy of a conditional distribution to a dynamic setting where conditional information changes over time. This is precisely the case for learning policies for Markov Decision Processes. In this setting, states st of a system can be considered conditioning information, while actions at are the subject of learning. For a first-order MDP, probabilities of actions at time t depend only on the state at time t, and the causal entropy of the action policy takes a simpler form that depends only on a policy distribution π(at |st ): H ( A0:∞ || S0:∞ ) =

∞ t=0

γ H ( at || st ) = −ES t

∞

( γ

t

π(at |st ) log π(at |st )dat ,

t=0

(11.17) where the expectation is taken over all future values of st . Importantly, because the process is Markov, this expectation depends only on marginal distributions of st at t = 0, 1, . . ., but not on their joint distribution. The causal entropy can be maximized under available constraints, providing an extension of the MaxEnt principle to dynamic processes. Let us assume that some feature functions F (S, A) are observed in a demonstration with T step and that feature functions are additive in time, i.e.& F (S, A) = t F (st , at ). In particular, ' a total reward obtained from a trajectory ST , AT is additive in time; therefore, it would fit such a choice. For additive feature functions, we have

∞ π π t EA,S [F (S, A)] = EA,S γ F (st , at ) . (11.18) t=0

3 Maximum Entropy Inverse Reinforcement Learning

435

Here EπA,S [·] denotes expectation w.r.t. a distribution over future states and actions induced by the policy π(at |st ) and transition dynamics of the system expressed via conditional transition probabilities P (st+1 |st , at ). Suppose we seek a policy π(a|s) which matches empirical feature expectations E˜ emp [F (S, A)] = Eemp

∞

γ F (st , at ) = t

t=0

T −1 1 t γ F (st , at ). T

(11.19)

t=0

Maximum Causal Entropy optimization can now be formulated as follows: argmax H π

AT ST

Subject to: EπA,S [F (S, A)] = E˜ A,S [F (S, A)] π(at |st ) = 1, π(at |st ) ≥ 0, ∀st . and

(11.20)

at

Here E˜ A,S [·] stands for the empirical feature expectation as in Eq. (11.19). In contrast to a single-step MaxEnt problem, constraints now refer to feature expectations collected over whole paths rather than single steps. Causality is not explicitly enforced in (11.21); however, a causally conditioned policy in an MDP factorizes $ π as ∞ t=0 t (at |st ). Therefore, using the factors π(at |st ) as decision variables, the policy π is forced to be causally conditioned. Equivalently, we can swap the objective and the constraints, and consider the following dual problem: argmax EπA,S [F (S, A)] − E˜ A,S [F (S, A)] π

Subject to: H AT ST = H¯ and π(at |st ) = 1, π(at |st ) ≥ 0, ∀st ,

(11.21)

at

where H¯ is some value of the entropy that is fixed throughout the optimization. Unlike the previous formulation that is non-concave and involves an infinite number of constraints, the dual formulation is concave, and involves only one constraint on the entropy (in addition to normalization constraints). The dual form (11.21) of the Max-Causal Entropy method can be used for both direct RL and IRL. We first consider applications of this approach to direct reinforcement learning problems where rewards are observed.

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11 Inverse Reinforcement Learning and Imitation Learning

•

? Multiple Choice Question 2

Select all the following correct statements: a. The Maximum Causal Entropy method provides an extension of the maximum entropy method for sequential decision-making inference which preserves causality relations between actions and future states. b. The dual form of Maximum Causality Entropy produces multiple solutions as it amounts to a non-concave optimization. c. A causality-conditioned policy with the Maximum Causality Entropy method is ensured by adding causality constraints. d. A causality-conditioned policy with the Maximum Causality Entropy method is ensured by the MDP factorization of the process.

3.3 G-Learning and Soft Q-Learning To apply the Max-Causal Entropy model (11.21) to reinforcement learning with observed rewards, we take expected instantaneous rewards r(st , at ) as features, i.e. set F (st , at ) = r(st , at ). Furthermore, for direct reinforcement learning, the second term in the objective function (11.21) depends only on the empirical measure but not the policy π(a|s), and therefore it can be dropped from the optimization objective function. Finally, we extend the Max-Causal Entropy method by switching to a KL divergence with some reference policy π0 (a|s) instead of using the entropy of π(a|s) as a regularization. The latter case can always be recovered from the former one by choosing a uniform reference density π0 (a|s). The Kullback–Leibler (KL) divergence of π(·|st ) and π0 (·|st ) is KL[π ||π0 ](st ) :=

at

π(at |st ) log

π(at |st ) = Eπ g π (s, a) st , π0 (at |st )

(11.22)

where g π (s, a) = log

π(at |st ) π0 (at |st )

(11.23)

is the one-step information cost of a learned policy π(at |st ) relative to a reference policy π0 (at |st ). The problem of policy optimization expressed by Eqs. (11.21), generalized here by using the KL divergence (11.22) instead of the causal entropy, can now be formulated as a problem of maximization of the following functional:

3 Maximum Entropy Inverse Reinforcement Learning

Ftπ (st )

=

T t =t

γ

t −t

437

1 π E r(st , at ) − g (st , at ) . β

(11.24)

where 1/β is a Lagrange multiplier. Note that we dropped the second term in the objective function (11.21) in this expression. The reason is that this term depends only on the reward function but not on the policy. Therefore, it can be omitted for the problem of direct reinforcement learning. Note however that it should be kept for IRL as we will discuss later. The expression (11.24) is a value function of a problem with a modified KLregularized reward r(st , at ) − β1 g π (st , at ) that is sometimes referred to as the free energy function. Note that β in Eq. (11.24) serves as the “inverse-temperature” parameter that controls a tradeoff between reward optimization and proximity to the reference policy. The free energy Ftπ (st ) is the entropy-regularized value function, where the amount of regularization can be calibrated to the level of noise in the data. The optimization problem (11.24) is exactly the one we studied in Chap. 10 where we presented G-learning. We recall a self-consistent set of equations that need to be solved jointly in G-learning: Ftπ (st ) =

π 1 1 log Zt = log π0 (at |st )eβGt (st ,at ) . β β a

(11.25)

t

π (s ,a )−F π (s ) t t t t

π(at |st ) = π0 (at |st )eβ (Gt

π Gπt (st , at ) = r(st , at ) + γ Et,a Ft+1

).

(11.26)

(st+1 ) st , at .

(11.27)

Here the G-function Gπt (st , at ) is a KL-regularized action-value function. Equations (11.25, 11.26, 11.27) constitute a system of equations that should be solved self-consistently for π(at |yt ), Gπt (yt , at ), and Ftπ (yt ) (Fox et al. 2015). For a finite-horizon problem of length T , the system can be solved by backward recursion for t = T − 1, . . . , 0, using appropriate terminal conditions at t = T . If we substitute the augmented free energy (11.25) into Eq. (11.27), we obtain ⎡

γ Gπt (s, a) = r(st , at )+ Et,a ⎣log β a

⎤ π0 (at+1 |st+1 )eβGt+1 (st+1 ,at+1 ) ⎦ . π

(11.28)

t+1

This equation is a soft relaxation of the Bellman optimality equation for the action-value function (Fox et al. 2015). The “inverse-temperature” parameter β in Eq. (11.28) determines the strength of entropy regularization. In particular, if we take β → ∞, we recover the original Bellman optimality equation for the Qfunction. Because the last term in (11.28) approximates the max(·) function when β is large but finite, Eq. (11.28) is known, for a special case of a uniform reference density π0 , as “soft Q-learning.”

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11 Inverse Reinforcement Learning and Imitation Learning

Note that we could also bypass the G-function altogether, and proceed with the Bellman optimality equation for the free energy F-function (11.24). In this case, we have a pair of equations for Ftπ (st ) and π(at |st ): π 1 Ftπ (st ) = Ea r(st , at ) − g π (st , at ) + γ Et,a Ft+1 (st+1 ) β π(at |st ) =

π 1 π0 (at |st )er(st ,at )+γ Et,a Ft+1 (st+1 ) . Zt

(11.29)

Equation (11.29) shows that one-step rewards r(st , at ) do not form an alternative specification of single-step action probabilities π(at |st ). Rather, a specification of π (s the sum r(st , at ) + γ Et,a Ft+1 t+1 ) is required (Ortega et al. 2015). However, in a special when dynamics are linear and rewards r(st , at ) are quadratic, the case π (s term Et,a Ft+1 t+1 ) has the same parametric form as the time-t reward r(st , at ); therefore, addition of this term amounts to a “renormalization” of parameters of the one-step reward function (see Exercise 11.3). When the objective is to learn a policy, such renormalized parameters can be directly learned from data, bypassing the need to separate them into the current reward and an expected future-reward. We see that the G-learning (or equivalently the Max-Causal Entropy) optimal policy (11.26) for an MDP formulation is still given by the familiar Boltzmann form as in (11.7), but with a different energy function Fˆ (st , at ), which is now given by the G-function. Unlike the previous single-step case, in a multi-period setting this function is not available in a closed form, but rather defined recursively by the self-consistent G-learning equations (11.25, 11.26, 11.27).

3.4 Maximum Entropy IRL The G-learning (Max-Causal Entropy) framework can be used for both direct and inverse reinforcement learning. Here we apply it for the task of IRL. In this section, we assume a time-homogeneous MDP with an infinite time horizon. We also assume that a time-stationary action-value function G(st , at ) is specified as a parametric model (e.g., a neural network) with parameters θ , so we write it as Gθ (st , at ). The objective of IRL inference is to learn parameters θ from data. We start with the G-learning equations (11.25) expressed in terms of the policy function π(at |st ) and the parameterized G-function Gθ (st , at ): πθ (at |st ) =

1 π0 (at |st )eβGθ (st ,at ) , Zθ (st ) := Zθ (st )

( π0 (at |st )eβGθ (st ,at ) dat ,

(11.30) where the G-function (a.k.a. a soft Q-function) satisfies a soft relation of the Bellman optimality equation

3 Maximum Entropy Inverse Reinforcement Learning

⎡ ⎤ γ Gθ (s, a) = r(st , at ) + Et,a ⎣log π0 (at+1 |st+1 )eβGθ (st+1 ,at+1 ) ⎦ . β a

439

(11.31)

t+1

Under MaxEnt IRL, the stochastic action policy (11.30) is used as a probabilistic model of observed data made of pairs (st , at ). A loss function in terms of parameters θ can therefore be obtained by applying the conventional maximum likelihood method to this model. To gain more insight, let us start with the likelihood of a particular path τ : P (τ ) = p(s0 )

T −1

πθ (at |st )P (st+1 |st , at )

t=0

= p(s0 )

T −1 t=0

1 π0 (at |st )eβGθ (st ,at ) P (st+1 |st , at ). Zθ (st )

(11.32)

Now we take a negative logarithm of this expression to get the negative loglikelihood, where we can drop contributions from the initial state distribution p(s0 ) and state transition probabilities P (st+1 |st , at ), as neither depends on the model parameters θ : L(θ ) =

T −1

(log Zθ (st ) − βGθ (st , at )) .

(11.33)

t=0

Minimization of this loss function with respect to parameters θ gives an optimal soft Q-function Gθ (st , at ). Once this function is found, if needed or desired, we could use Eq. (11.31) in reverse to estimate one-step expected rewards r(st , at ). Gradients of the loss function (11.33) can be computed as follows: * T −1 )( ∂Gθ (st , at ) ∂L(θ ) ∂Gθ (st , a) = da − πθ (a|st ) ∂θ ∂θ ∂θ t=0

=

∂Gθ (s, a) ∂Gθ (s, a) model − data . ∂θ ∂θ

(11.34)

The second term in this expression can be computed directly from the data for any given value of θ , and thus does not pose any issues. The problem is with the first term in (11.34) that gives the gradient of the log-partition function in Eq. (11.33). This term involves an integral over all possible actions at each step of the trajectory, computed with the probability density πθ (a|st ). For a discreteaction MDP, the integral becomes a finite sum that can be computed directly, but for continuous and possibly high-dimensional action spaces, an accurate calculation of this integral for a fixed value of θ might be time-consuming. Given that this

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11 Inverse Reinforcement Learning and Imitation Learning

integral should be evaluated multiple times during optimization of parameters θ , unless evaluated analytically, the computational burden of this step can be very high or even prohibitive. Leaving these computational issues aside for a moment, the intermediate conclusion is that the task of IRL can be solved using the maximum likelihood method for the action policy produced by G-learning, and without explicitly solving the Bellman optimality equation. The solution approach used here is to model the whole soft action-value function Gθ (st , at ) using a flexible function approximation method such as a neural network. As Gθ (st , at ) defines the action policy πθ (a|st ), by making inference of the policy, we can directly learn the soft action-value function. We note, however, that such apparent relief from the need to solve the (soft) Bellman optimality equation at each internal step of the IRL task has a flip side. In the form presented above, the MaxEnt IRL approach is identical to behavioral cloning with the G-function Gθ (st , at ) that is fitted to data in the form of pairs (st , at ). This is different from TD methods such as Q-learning or its “soft” versions that consider triplets of transitions (st , at , st+1 ), and thus capture dynamics of the system without estimating them explicitly. Therefore, all problems with behavioral cloning mentioned above in this chapter will also arise here. In particular, a soft value function using only pairs (st , at ) and maximum likelihood estimation could produce a G-function that would be compatible with the data, and yet produce implausible single-step reward functions when Eq. (11.31) is used at the last step with the estimated G-function. To preclude such potential problems arising, instead of using a parametric model for the G-function, we could directly specify a parametric single-step reward function rθ (st , at ). For example, with linear architectures, a reward function is linear in a set of K pre-specified features k (st , at ): rθ (st , at ) =

K

θk k (st , at ).

(11.35)

k=1

Alternatively, a reward function can be non-linear in parameters θ , and could be defined using, e.g., a neural network or a Gaussian process.

•

> IRL with Bounded Rewards

As we discussed in Chap. 9, reinforcement learning requires that rewards should be bounded from above by some value rmax . On the other hand, it also has certain invariances with respect to transformations of the reward function, namely the policy remains unchanged under affine transformations (continued)

3 Maximum Entropy Inverse Reinforcement Learning

441

of the reward function r(s, a) → ar(s, a) + b, where a > 0 and b are fixed parameters. We can use this invariance in order to fix the highest possible reward to be zero: rmax = 0 without any loss of generality. Let us assume the following functional form of the reward function: r(s, a) = log D(s, a),

(11.36)

where D(s, a) is another function of the state and action. Assume that the domain of function D(s, a) is a unit interval, i.e. 0 ≤ D(s, a) ≤ 1. In this case, the reward is bounded from above by zero, as required: −∞ < r(s, a) ≤ 0. Now, because 0 ≤ D(s, a) ≤ 1, we can interpret D(s, a) as a probability of a binary classifier. If D(s, a) is chosen to be the probability that the given action a in state s was generated by the expert, then according to Eq. (11.36), maximization of the reward corresponds to maximization of log-probability of the expert trajectory. Let us use a simple logistic regression model for D(s, a) = σ (θ (s, a)), where σ (x) is the logistic function, θ is a vector of model parameters of size K, and (s, a) is a vector of K basis functions. For this specification, we obtain the following parameterization of the reward function:

r(s, a) = − log 1 + e−θ (s,a) .

(11.37)

As one can check, this reward function is concave in a if basis functions k (s, a) are linear in a while having an arbitrary dependence on s (see Exercise 11.2). Therefore, such a reward can be used as an alternative to a linear specification (11.35) for risk-averse RL and IRL. We will return to the reward specification (11.36) below when we discuss imitation learning.

Once the reward function is defined, the parametric dependence of the G-function is fixed by the soft Bellman equation (11.31). The latter will also define gradients of the G-function that enter Eq. (11.34). The gradients can be estimated using samples from the true data-generating distribution πE and the model distribution πθ . Clearly, solving the IRL problem in this way would make estimated rewards rθ (st , at ) more consistent with dynamics than in the previous version that directly works with a parameterized G-function. However, this forces the IRL algorithm to solve the direct RL problem of finding the optimal soft action-value function Gθ (st , at ) at each step of optimization over parameters θ . Given that solving the direct RL problem even once might be quite time-consuming, especially in high-dimensional continuous action spaces, this can render the computational cost

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of directly inferring one-step rewards very high and impractical for real-world applications. Another and computationally more feasible option is to define the BC-like loss function (11.33) to be more consistent with the dynamics. We introduce a regularization that depends on observed triplet transitions (st , at , st+1 ). One simple idea in this direction is to add a regularization term equal to the squared Bellman error, i.e. the squared difference between the left- and right-hand sides of Eq. (11.31), where the one-step reward is set to be a fixed number, rather than a function of state and action. Such an approach was applied in robotics where it was called SQIL (Soft Q Imitation Learning) (Reddy et al. 2019). We will return to the topic of regularization in IRL in the next section, where we will consider IRL in the context of imitation learning. In passing, we shall consider other computational aspects of the IRL problem that persist with or without regularization.

3.5 Estimating the Partition Function After parameters θ are optimized, Eq. (11.31) can be used in order to estimate one-step expected rewards r(st , at ). In practice, computing the gradients of the resulting loss function involves integration over a (multi-dimensional) action space. This produces the main computational bottleneck of the MaxEnt IRL method. Note that the same bottleneck arises in direct reinforcement learning with G-learning Eqs. (11.25) that also involves computing integrals over the action space. One commonly used approach to numerically computing integrals involving probability distributions is importance sampling. If μ(a ˆ t |st ) is a sampling distribution, then the integral appearing in the gradient (11.34) can be evaluated as follows: ( πθ (at |st )

∂Gθ (st , at ) dat = ∂θ

( μ(a ˆ t |st )

πθ (at |st ) ∂Gθ (st , at ) dat , μ(a ˆ t |st ) ∂θ

(11.38)

which replaces integration with respect to the original distribution with the sampling distribution μ(a ˆ t |st ), with the idea that this distribution might be easier to sample from than the original probability density. When this distribution is used for sampling, the gradients ∂Gθ /∂θ are multiplied by the likelihood ratios πθ /μ. ˆ Importance sampling becomes more accurate when the sampling distribution μ(a ˆ t |st ) is close to the optimal action policy πθ (at |st ). This observation could be used to produce an adaptive sampling distribution μ(a ˆ t |st ). For example, we could envision a computational scheme with updates of the G-function according to the gradients * T −1 )( ∂Gθ (st , at ) ∂L(θ ) πθ (at |st ) ∂Gθ (st , a) = da − , μ(a ˆ t |st ) ∂θ μ(a ˆ t |st ) ∂θ ∂θ t=0

(11.39)

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443

which would be intermittent with updates of the sampling distribution μ(a ˆ t |st ) that would depend on values of θ from a previous iteration. Such methods are known in robotics as “guided cost learning” (Finn et al. 2016). We will discuss a related method in Sect. 5 where we consider alternative approaches to learning from demonstrations. Before turning to such advanced methods, we would like to present a tractable formulation of MaxEnt IRL where the partition function can be computed exactly so that approximations are not needed. Without too much loss of generality, we will present such a formulation in the context of a problem of inference of customer preferences and price sensitivity. Such a problem can also be viewed as a special case of a consumer credit problem. Similar examples in consumer credit might include prepaid household utility payment plans where the consumer prepays their utilities and is penalized for overage and lines of credit in payment processing and ATM services. Other examples include consumer loans and mortgages where different loan products are offered to the consumer, with varying interest rates and late payment penalties, and the user chooses when to make principal payments.

•

? Multiple Choice Question 3

Select all the following correct statements: a. Maximum Entropy IRL provides a solution of the self-consistent system of Glearning without access to the reward function that fits observable sequences of states and actions. b. “Soft Q-learning” is a method of relaxation of the Bellman optimality equation for the action-value function that is obtained from G-learning by adopting a uniform reference action policy. c. Maximum Entropy IRL assumes that all demonstrations are strictly optimal. d. Taking the limit β → ∞ in Maximum Entropy IRL is equivalent to assuming that all demonstrations are strictly optimal.

4 Example: MaxEnt IRL for Inference of Customer Preferences The previous section presented a general formulation of the MaxEnt IRL approach. While this approach can be formulated for both discrete and continuous stateaction spaces, for the latter case, computing the partition function is often the main computational burden in practical applications. These computational challenges should not overshadow the conceptual simplicity of the MaxEnt IRL approach. In this section we present a particularly simple version of this method which can be derived using quadratic rewards and Gaussian policies. We will present this formulation in the context of a problem of utmost interest in marketing, which is the problem of learning preferences and price sensitivities of customers of a recurrent utility service.

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We will also use this simple example to provide the reader with some intuition on the amount of data needed to apply IRL. As we will show below, caution should be exercised when applying IRL to real-world noisy data. In particular, using simulated examples, we will show how the observational noise, inevitable in any finite-sample data, can masquerade itself as an apparent heterogeneity of agents.

4.1 IRL and the Problem of Customer Choice Understanding customer choices, demand, and preferences, with customers being consumers or firms, is a central tenet in the marketing literature. One important class of such problems is dynamic consumer demand for recurrent utility-like plans and services such as cloud computing plans, internet data plans, utility plans (e.g., electricity, gas, phone), etc. Consumer actions in this settings extend over a period of time, such as the term of a contract or a period between regular payments for a plan, and can therefore be considered a multi-step decision-making problem. If customers are modeled as utility-maximizing rational agents, the problem is well suited for methods of inverse optimal control or inverse reinforcement learning. In the marketing literature, the inverse optimal control approach to learning the customer utility is often referred to as structural models, see, e.g., Marschinski et al. (2007). This approach has the advantage over purely statistical regression based models in its ability to discern true consumer choices and demand preferences from effects induced by particular marketing campaigns. This enables the promotion of new products and offers, whose attractiveness to consumers could then be assessed based on the learned consumer utility. Structural models view forward-looking consumers as rational agents maximizing their streams of expected utilities of consumption over a planning horizon rather than their one-step utility. Structural models typically specify a model for a consumer utility, and then estimate such a model using methods of dynamic programming and stochastic optimal control. Using the language of reinforcement learning, structural models require methods of dynamic programming or approximate dynamic programming using deterministic policies. As we mentioned earlier in this chapter, using deterministic policies to infer agents’ utilities may be problematic if the demonstrated behavior is suboptimal. A deterministic policy which assumes that each step should be strictly optimal, will assign a zero probability to any path that is not strictly optimal. This would rule out any data where the demonstrated behavior is expected to deviate in any way from a strictly optimal behavior. Needless to say, available data is almost always sub-optimal to a various extent. To relax the assumption of strict optimality for all demonstrations, structural models usually add a random component to the one-step customer utility, which is sometimes referred to as “user shocks.” An example of such an approach can be found in Xu et al. (2015) who applied it to infer reward (utility) functions of consumers of mobile data plans. While this enables sub-optimal trajectories,

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this approach requires optimization of the reward parameters using Monte Carlo simulation, where unobserved and simulated “user shocks” are added in the parameter estimation procedure. Instead of pursuing such an approach, MaxEnt IRL offers an alternative and more computationally efficient way to manage possible sub-optimality in data by using stochastic policies instead of deterministic policies. This approach provides some degree of tolerance to certain occasional, non-excessive, deviations from a strictly optimal behavior, which are described as rare fluctuations according to the model with optimal parameters. We will now present a simple parametric specification of the MaxEnt IRL method that we introduced in this chapter. As we will show, it leads to a very lightweight computational method, in comparison to Monte Carlo based methods for structural models.

4.2 Customer Utility Function More formally, consider a customer that purchased a single-service plan with the monthly price F , initial quota q0 , and price p to be paid for the unit of consumption upon breaching the monthly quota on the plan.3 We specify a single-step utility (reward) function of a customer at time t = 0, 1, . . . , T − 1 (where T is a length of a payment period, e.g., a month) as follows: 1 r(at , qt , dt ) = μat − βat2 + γ at dt − ηp(at − qt )+ + κqt 1at =0 . 2

(11.40)

Here at ≥ 0 is the daily consumption on day t, qt ≥ 0 is the remaining allowance at the start of day t, and dt is the number of remaining days until the end of the billing cycle, and we use a short notation x+ = max(x, 0) for any x. The fourth term in Eq. (11.40) is proportional to the payment p(at − qt )+ made by the customer once the monthly quota q0 is exhausted. Parameter η gives the price sensitivity of the customer, while parameters μ, β, γ specify the dependence of the user reward on the state-action variables qt , dt , at . Finally, the last term ∼ κqt 1at =0 gives the reward received upon zero consumption at = 0 at time t (here 1at =0 is an indicator function that is equal to one if at = 0, and is zero otherwise). Model calibration amounts to estimation of parameters η, μ, β, γ , κ given the history of the user’s consumption. Note that the reward (11.40) can be equivalently written as an expansion over K = 5 basis functions:

3 For

plans that do not allow breaching the quota q0 , the present formalism still applies by setting the price p to infinity.

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11 Inverse Reinforcement Learning and Imitation Learning

r(at , qt , dt ) = T (at , qt , dt ) =

K−1

θk

k (at , qt , dt ),

(11.41)

k=0

where 1 θ0 = μat , θ1 = − βat2 , θ2 = γ at dt , 2 θ3 = −ηp(at − qt )+ , θ4 = κqt 1at =0 (here X stands for the empirical mean of X), and the following set of basis K−1 is used: functions { k }k=0 0 (at , qt , dt )

= at /at ,

1 (at , qt , dt )

= at2 /at2 ,

2 (at , qt , dt )

= at dt /at dt ,

3 (at , qt , dt )

= (at − qt )+ /(at − qt )+

4 (at , qt , dt )

= qt 1at =0 /qt 1at =0 .

(11.42)

As we explained above, structural models attempt to reconcile deterministic policies and a possible sub-optimal behavior by adding random “user shocks” to the user utility. For example, such “user shocks” can be added to parameter μ. A drawback of such an approach is that for model estimation, it requires Monte Carlo simulation of user shock paths. This can be compared with MaxEnt IRL. Because MaxEnt IRL is a probabilistic approach that assigns probabilities to observed paths, it does not require introducing a random shock to the utility function in order to reconcile the model with a possible sub-optimal behavior. Therefore, MaxEnt IRL does not need Monte Carlo simulation to estimate parameters of the user utility, and instead can use standard maximum likelihood estimation (MLE). For the reward defined in Eq. (11.40), MLE amounts to a convex optimization with 5 variables, which can be performed efficiently using the standard off-the-shelf convex optimization software. Moreover, our specification (11.40) can be easily generalized by adding more basis functions while keeping the rest of the methodology intact.

4.3 Maximum Entropy IRL for Customer Utility We use an extension of the MaxEnt IRL called Relative Entropy IRL (Boularias et al. 2011) which replaces the uniform distribution in the MaxEnt method by a non-uniform benchmark (or “prior”) distribution π0 (at |qt , dt ). This produces the exponential single-step transition probability:

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447

P (qt+1 = qt − at , at |qt , dt ) := π(at |qt , dt )

(11.43)

π0 (at |qt , dt ) π0 (at |qt , dt ) exp (r(at , qt , dt )) = exp T (at , qt , dt ) , = Zθ (qt , dt ) Zθ (qt , dt )

where Zθ (qt , dt ) is a state-dependent normalization factor ( Zθ (qt , dt ) =

π0 (at |qt , dt ) exp T (at , qt , dt ) dat .

(11.44)

We note that most applications of MaxEnt IRL use multi-step trajectories as prime objects, and define the partition function Zθ on the space of trajectories. While the first applications of MaxEnt IRL calculated Zθ exactly for small discrete stateaction spaces as in (Ziebart et al. 2008, 2013), for large or continuous state-action spaces we resort to approximate dynamic programming, or other approximation methods. For example, the Relative Entropy IRL approach of (Boularias et al. 2011) uses importance sampling from a reference (“background”) policy distribution to calculate Zθ . It is this calculation that poses the main computational bottleneck for applications of MaxEnt/RelEnt IRL methods for large or continuous state-action spaces. With a simple piecewise-quadratic reward such as Eq. (11.40), we can proceed differently: we define state-dependent normalization factors Zθ (qt , dt ) for each time step. Because we trade a path-dependent “global” partition function Zθ for a local state-dependent factor Zθ (qt , dt ), we do not need to rely on exact or approximate dynamic programming to calculate this factor. This is similar to the approach of Boularias et al. (2011) (as it also relies on the Relative Entropy minimization), but in our case both the reference distribution π0 (at |qt , dt ) and normalization factor Zθ (qt , dt ) are defined on a single time step, and calculation of Zθ (qt , dt ) amounts to computing the integral (11.44). As we show below, this integral can be calculated analytically with a properly chosen distribution π0 (at |qt , dt ). We shall use a mixture of discrete and continuous distribution for the reference (“prior”) action distribution π0 (at |qt , dt ): π0 (at |qt , dt ) = ν¯ 0 δ(at ) + (1 − ν¯ 0 )π˜ 0 (at |qt , dt )Iat >0 ,

(11.45)

where δ(x) stands for the Dirac delta-function, and Ix>0 = 1 if x > 0 and zero otherwise. The continuous component π˜ 0 (at |qt , dt ) is given by a spliced Gaussian distribution ⎧

⎨ (1 − ω0 (qt , dt ))φ1 at , μ0 +γ0 dt , 1 if 0 < at ≤ qt β0 β0

, (11.46) π˜ 0 (at |qt , dt ) = ⎩ ω0 (qt , dt )φ2 at , μ0 +γ0 dt −η0 p , 1 if at ≥ qt β0 β0 where φ1 (at , μ1 , σ12 ) and φ2 (at , μ2 , σ22 ) are probability density functions of two truncated normal distributions defined separately for small and large daily consumption levels, 0 ≤ at ≤ qt and at ≥ qt , respectively (in particular, they both

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11 Inverse Reinforcement Learning and Imitation Learning

are separately normalized to one). The mixing parameter 0 ≤ ω0 (qt , dt ) ≤ 1 is determined by the continuity condition at at = qt : ) * ) * μ0 + γ0 dt 1 μ0 + γ0 dt − η0 p 1 = ω0 (qt , dt )φ2 qt , . (1−ω0 (qt , dt ))φ1 qt , , , β0 β0 β0 β0 (11.47) As this matching condition may involve large values of qt where the normal distribution would be exponentially small, in practice it is better to use it by taking logarithms of both sides:

1

. μ0 +γ0 dt −η0 p 1 μ0 +γ0 dt 1 − log φ q 1 + exp log φ2 qt , , , , 1 t β0 β0 β0 β0 (11.48) The prior mixing-spliced distribution (11.45), albeit represented in terms of simple distributions, leads to potentially quite complex dynamics that make intuitive sense and appear largely consistent with observed patterns of consumption. In particular, note that Eq. (11.46) indicates that large fluctuations at > qt are centered around dt a smaller mean value μ−γ βdt −ηp than the mean value μ−γ of smaller fluctuations β 0 < at ≤ qt . Both a reduction of the mean upon breaching the remaining allowance barrier and a decrease of the mean of each component with time appear quite intuitive in the current context. As will be shown below, a posterior distribution π(at |qt , dt ) inherits these properties while also further enriching the potential complexity of dynamics.4 The advantage of using the mixed-spliced distribution (11.45) as a reference distribution π0 (at |qt , dt ) is that the state-dependent normalization constant Zθ (qt , dt ) can be evaluated exactly with this choice: ω0 (qt , dt ) =

Zθ (qt , dt ) = ν¯ 0 eκqt + (1 − ν¯ 0 ) (I1 (θ, qt , dt ) + I2 (θ, qt , dt )) ,

(11.49)

where 9

2 ; (μ0 + μ + (γ0 + γ )dt )2 (μ0 + γ0 dt )2 β0 exp − β0 + β 2(β0 + β) 2β0

)dt −(β0 +β)qt √ 0 +γ )dt N − μ0 +μ+(γ0√+γ − N − μ0 +μ+(γ β0 +β β +β

0 × μ0 +γ√ μ0√ +γ0 dt 0 dt −β0 qt N − −N − β β

I1 (θ, qt , dt ) = (1 − ω0 (qt , dt ))

9

I2 (θ, qt , dt ) = ω0 (qt , dt )

4 In

2

β0 (μ0 + μ − (η0 + η)p + (γ0 + γ )dt )2 exp β0 + β 2(β0 + β)

(11.50)

particular, it promotes a static mixing coefficient ν0 to a state- and time-dependent variable νt = ν(qt , dt ).

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; 1 − N − μ0 +μ−(η0 +η)p+(γ √ 0 +γ )dt −(β0 +β)qt (μ0 − η0 p + γ0 dt )2 β0 +β

, − + ηpqt × 2β0 √ 0 dt −β0 qt 1 − N − μ0 −η0 p+γ β 0

where N (x) is the cumulative normal probability distribution. 7T 6 Probabilities of T -steps paths τi = ati , qti , dti t=0 (where i enumerates different user-paths) are obtained as products of single-step probabilities: P (τi ) =

π0 (at |qt , dt ) exp T (at , qt , dt ) ∼ exp T (τi ) (at , qt , dt ) . Zθ (qt , dt )

(at ,qt ,dt )∈τi

Here = the observed path τi : (τi ) (at , qt , dt )

(11.51) K−1 (τi ) k (at , qt , dt ) k=0

(τi ) k (at , qt , dt )

=

are cumulative feature counts along

(11.52)

k (at , qt , dt ).

(at ,qt ,dt )∈τi

Therefore, the total path probability in our model is exponential in the total reward along a trajectory, as in the “classical” MaxEnt IRL approach (Ziebart et al. 2008), while the pre-exponential factor is computed differently as we operate with onestep, rather than path probabilities. Parameters defining the exponential path probability distribution (11.51) can be estimated by the standard maximum likelihood estimation (MLE) method. Assume we have N historically observed single-cycle consumption paths, and assume these path probabilities are independent.5 The total likelihood of observing these data is L(θ ) =

N

i=1 (at ,qt ,dt )∈τi

π0 (at |qt , dt ) exp T (at , qt , dt ) . Zθ (qt , dt )

(11.53)

The negative log-likelihood is therefore, after omitting the term log π0 (at |qt , dt ) that does not depend on ,6 and rescaling by 1/N, ⎛ N 1 ⎝ 1 − log L(θ ) = N N i=1

5A

(qt ,dt )∈τi

log Zθ (qt , dt ) −

⎞ T (at , qt , dt )⎠

(at ,qt ,dt )∈τi

more complex case of co-dependencies between rewards for individual customers can be considered, but we will not pursue this approach here. 6 Note that Z (q , d ) still depends on π (a |q , d ), see Eq. (11.44). θ t