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Reinforcement Learning and Stochastic Optimization

A Unified Framework for Sequential Decisions

Powell, Warren B.

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1. Auflage März 2022
1136 Seiten, Hardcover
Wiley & Sons Ltd

ISBN: 978-1-119-81503-7
John Wiley & Sons

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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.

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
Warren B. Powell, PhD, is Professor Emeritus of Operations Research and Financial Engineering at Princeton University, where he taught for 39 years. He was the founder and Director of CASTLE Laboratory, a research unit that works with industrial partners to test new ideas found in operations research. He supervised 70 graduate students and post-docs, with whom he wrote over 250 papers. He is currently the Chief Analytics Officer of Optimal Dynamics, a lab spinoff that is taking his research to industry.

W. B. Powell, Princeton University (NJ)