Reinforcement Learning and Stochastic Optimization: A Unified Framework for Sequential Decisions

REINFORCEMENT LEARNING AND STOCHASTIC OPTIMIZATION Clearing the jungle of stochastic optimization Sequential decision problems, which consist of “decision, information, decision, information,” are ubiquitous, spanning virtually every human activity ranging from business applications, health (personal and public health, and medical decision making), energy, the sciences, all fields of engineering, finance, and e-commerce. The diversity of applications attracted the attention of at least 15 distinct fields of research, using eight distinct notational systems which produced a vast array of analytical tools. A byproduct is that powerful tools developed in one community may be unknown to other communities. Reinforcement Learning and Stochastic Optimization offers a single canonical framework that can model any sequential decision problem using five core components: state variables, decision variables, exogenous information variables, transition function, and objective function. This book highlights twelve types of uncertainty that might enter any model and pulls together the diverse set of methods for making decisions, known as policies, into four fundamental classes that span every method suggested in the academic literature or used in practice. Reinforcement Learning and Stochastic Optimization is the first book to provide a balanced treatment of the different methods for modeling and solving sequential decision problems, following the style used by most books on machine learning, optimization, and simulation. The presentation is designed for readers with a course in probability and statistics, and an interest in modeling and applications. Linear programming is occasionally used for specific problem classes. The book is designed for readers who are new to the field, as well as those with some background in optimization under uncertainty. Throughout this book, readers will find references to over 100 different applications, spanning pure learning problems, dynamic resource allocation problems, general state-dependent problems, and hybrid learning/resource allocation problems such as those that arose in the COVID pandemic. There are 370 exercises, organized into seven groups, ranging from review questions, modeling, computation, problem solving, theory, programming exercises and a "diary problem" that a reader chooses at the beginning of the book, and which is used as a basis for questions throughout the rest of the book.

Table of Contents:
Preface xxv Acknowledgments xxxi Part I – Introduction 1 1 Sequential Decision Problems 3 1.1 The Audience 7 1.2 The Communities of Sequential Decision Problems 8 1.3 Our Universal Modeling Framework 10 1.4 Designing Policies for Sequential Decision Problems 15 1.5 Learning 20 1.6 Themes 21 1.7 Our Modeling Approach 27 1.8 How to Read this Book 27 1.9 Bibliographic Notes 33 Exercises 34 Bibliography 38 2 Canonical Problems and Applications 39 2.1 Canonical Problems 39 2.2 A Universal Modeling Framework for Sequential Decision Problems 64 2.3 Applications 69 2.4 Bibliographic Notes 85 Exercises 90 Bibliography 93 3 Online Learning 101 3.1 Machine Learning for Sequential Decisions 102 3.2 Adaptive Learning Using Exponential Smoothing 110 3.3 Lookup Tables with Frequentist Updating 111 3.4 Lookup Tables with Bayesian Updating 112 3.5 Computing Bias and Variance* 118 3.6 Lookup Tables and Aggregation* 121 3.7 Linear Parametric Models 131 3.8 Recursive Least Squares for Linear Models 136 3.9 Nonlinear Parametric Models 140 3.10 Nonparametric Models* 149 3.11 Nonstationary Learning* 159 3.12 The Curse of Dimensionality 162 3.13 Designing Approximation Architectures in Adaptive Learning 165 3.14 Why Does It Work?** 166 3.15 Bibliographic Notes 174 Exercises 176 Bibliography 180 4 Introduction to Stochastic Search 183 4.1 Illustrations of the Basic Stochastic Optimization Problem 185 4.2 Deterministic Methods 188 4.3 Sampled Models 193 4.4 Adaptive Learning Algorithms 202 4.5 Closing Remarks 210 4.6 Bibliographic Notes 210 Exercises 212 Bibliography 218 Part II – Stochastic Search 221 5 Derivative-Based Stochastic Search 223 5.1 Some Sample Applications 225 5.2 Modeling Uncertainty 228 5.3 Stochastic Gradient Methods 231 5.4 Styles of Gradients 237 5.5 Parameter Optimization for Neural Networks* 242 5.6 Stochastic Gradient Algorithm as a Sequential Decision Problem 247 5.7 Empirical Issues 248 5.8 Transient Problems* 249 5.9 Theoretical Performance* 250 5.10 Why Does it Work? 250 5.11 Bibliographic Notes 263 Exercises 264 Bibliography 270 6 Stepsize Policies 273 6.1 Deterministic Stepsize Policies 276 6.2 Adaptive Stepsize Policies 282 6.3 Optimal Stepsize Policies* 289 6.4 Optimal Step sizes for Approximate Value Iteration* 297 6.5 Convergence 300 6.6 Guidelines for Choosing Stepsize Policies 301 6.7 Why Does it Work* 303 6.8 Bibliographic Notes 306 Exercises 307 Bibliography 314 7 Derivative-Free Stochastic Search 317 7.1 Overview of Derivative-free Stochastic Search 319 7.2 Modeling Derivative-free Stochastic Search 325 7.3 Designing Policies 330 7.4 Policy Function Approximations 333 7.5 Cost Function Approximations 335 7.6 VFA-based Policies 338 7.7 Direct Lookahead Policies 348 7.8 The Knowledge Gradient (Continued)* 362 7.9 Learning in Batches 380 7.10 Simulation Optimization* 382 7.11 Evaluating Policies 385 7.12 Designing Policies 394 7.13 Extensions* 398 7.14 Bibliographic Notes 409 Exercises 412 Bibliography 424 Part III – State-dependent Problems 429 8 State-dependent Problems 431 8.1 Graph Problems 433 8.2 Inventory Problems 439 8.3 Complex Resource Allocation Problems 446 8.4 State-dependent Learning Problems 456 8.5 A Sequence of Problem Classes 460 8.6 Bibliographic Notes 461 Exercises 462 Bibliography 466 9 Modeling Sequential Decision Problems 467 9.1 A Simple Modeling Illustration 471 9.2 Notational Style 476 9.3 Modeling Time 478 9.4 The States of Our System 481 9.5 Modeling Decisions 500 9.6 The Exogenous Information Process 506 9.7 The Transition Function 515 9.8 The Objective Function 518 9.9 Illustration: An Energy Storage Model 523 9.10 Base Models and Lookahead Models 528 9.11 A Classification of Problems* 529 9.12 Policy Evaluation* 532 9.13 Advanced Probabilistic Modeling Concepts** 534 9.14 Looking Forward 540 9.15 Bibliographic Notes 542 Exercises 544 Bibliography 557 10 Uncertainty Modeling 559 10.1 Sources of Uncertainty 560 10.2 A Modeling Case Study: The COVID Pandemic 575 10.3 Stochastic Modeling 575 10.4 Monte Carlo Simulation 581 10.5 Case Study: Modeling Electricity Prices 589 10.6 Sampling vs. Sampled Models 595 10.7 Closing Notes 597 10.8 Bibliographic Notes 597 Exercises 598 Bibliography 601 11 Designing Policies 603 11.1 From Optimization to Machine Learning to Sequential Decision Problems 605 11.2 The Classes of Policies 606 11.3 Policy Function Approximations 610 11.4 Cost Function Approximations 613 11.5 Value Function Approximations 614 11.6 Direct Lookahead Approximations 616 11.7 Hybrid Strategies 620 11.8 Randomized Policies 626 11.9 Illustration: An Energy Storage Model Revisited 627 11.10 Choosing the Policy Class 631 11.11 Policy Evaluation 641 11.12 Parameter Tuning 642 11.13 Bibliographic Notes 646 Exercises 646 Bibliography 651 Part IV – Policy Search 653 12 Policy Function Approximations and Policy Search 655 12.1 Policy Search as a Sequential Decision Problem 657 12.2 Classes of Policy Function Approximations 658 12.3 Problem Characteristics 665 12.4 Flavors of Policy Search 666 12.5 Policy Search with Numerical Derivatives 669 12.6 Derivative-Free Methods for Policy Search 670 12.7 Exact Derivatives for Continuous Sequential Problems* 677 12.8 Exact Derivatives for Discrete Dynamic Programs** 680 12.9 Supervised Learning 686 12.10 Why Does it Work? 687 12.11 Bibliographic Notes 690 Exercises 691 Bibliography 698 13 Cost Function Approximations 701 13.1 General Formulation for Parametric CFA 703 13.2 Objective-Modified CFAs 704 13.3 Constraint-Modified CFAs 714 13.4 Bibliographic Notes 725 Exercises 726 Bibliography 729 Part V – Lookahead Policies 731 14 Exact Dynamic Programming 737 14.1 Discrete Dynamic Programming 738 14.2 The Optimality Equations 740 14.3 Finite Horizon Problems 747 14.4 Continuous Problems with Exact Solutions 750 14.5 Infinite Horizon Problems* 755 14.6 Value Iteration for Infinite Horizon Problems* 757 14.7 Policy Iteration for Infinite Horizon Problems* 762 14.8 Hybrid Value-Policy Iteration* 764 14.9 Average Reward Dynamic Programming* 765 14.10 The Linear Programming Method for Dynamic Programs** 766 14.11 Linear Quadratic Regulation 767 14.12 Why Does it Work?** 770 14.13 Bibliographic Notes 783 Exercises 783 Bibliography 793 15 Backward Approximate Dynamic Programming 795 15.1 Backward Approximate Dynamic Programming for Finite Horizon Problems 797 15.2 Fitted Value Iteration for Infinite Horizon Problems 804 15.3 Value Function Approximation Strategies 805 15.4 Computational Observations 810 15.5 Bibliographic Notes 816 Exercises 816 Bibliography 821 16 Forward ADP I: The Value of a Policy 823 16.1 Sampling the Value of a Policy 824 16.2 Stochastic Approximation Methods 835 16.3 Bellman’s Equation Using a Linear Model* 837 16.4 Analysis of TD(0), LSTD, and LSPE Using a Single State* 842 16.5 Gradient-based Methods for Approximate Value Iteration* 845 16.6 Value Function Approximations Based on Bayesian Learning* 852 16.7 Learning Algorithms and Atepsizes 855 16.8 Bibliographic Notes 860 Exercises 862 Bibliography 864 17 Forward ADP II: Policy Optimization 867 17.1 Overview of Algorithmic Strategies 869 17.2 Approximate Value Iteration and Q-Learning Using Lookup Tables 871 17.3 Styles of Learning 881 17.4 Approximate Value Iteration Using Linear Models 886 17.5 On-policy vs. off-policy learning and the exploration–exploitation problem 888 17.6 Applications 894 17.7 Approximate Policy Iteration 900 17.8 The Actor–Critic Paradigm 907 17.9 Statistical Bias in the Max Operator* 909 17.10 The Linear Programming Method Using Linear Models* 912 17.11 Finite Horizon Approximations for Steady-State Applications 915 17.12 Bibliographic Notes 917 Exercises 918 Bibliography 924 18 Forward ADP III: Convex Resource Allocation Problems 927 18.1 Resource Allocation Problems 930 18.2 Values Versus Marginal Values 937 18.3 Piecewise Linear Approximations for Scalar Functions 938 18.4 Regression Methods 941 18.5 Separable Piecewise Linear Approximations 944 18.6 Benders Decomposition for Nonseparable Approximations** 946 18.7 Linear Approximations for High-Dimensional Applications 956 18.8 Resource Allocation with Exogenous Information State 958 18.9 Closing Notes 959 18.10 Bibliographic Notes 960 Exercises 962 Bibliography 967 19 Direct Lookahead Policies 971 19.1 Optimal Policies Using Lookahead Models 974 19.2 Creating an Approximate Lookahead Model 978 19.3 Modified Objectives in Lookahead Models 985 19.4 Evaluating DLA Policies 992 19.5 Why Use a DLA? 997 19.6 Deterministic Lookaheads 999 19.7 A Tour of Stochastic Lookahead Policies 1005 19.8 Monte Carlo Tree Search for Discrete Decisions 1009 19.9 Two-Stage Stochastic Programming for Vector Decisions* 1018 19.10 Observations on DLA Policies 1024 19.11 Bibliographic Notes 1025 Exercises 1027 Bibliography 1031 Part VI – Multiagent Systems 1033 20 Multiagent Modeling and Learning 1035 20.1 Overview of Multiagent Systems 1036 20.2 A Learning Problem – Flu Mitigation 1044 20.3 The POMDP Perspective* 1059 20.4 The Two-Agent Newsvendor Problem 1062 20.5 Multiple Independent Agents – An HVAC Controller Model 1067 20.6 Cooperative Agents – A Spatially Distributed Blood Management Problem 1070 20.7 Closing Notes 1074 20.8 Why Does it Work? 1074 20.9 Bibliographic Notes 1076 Exercises 1077 Bibliography 1083 Index 1085