Optimal Learning
Wiley Series in Probability and Statistics Series

Everyday decisions are made without the benefit of accurate information. Optimal Learning develops the needed principles for gathering information to make decisions, especially when collecting information is time-consuming and expensive. Designed for readers with an elementary background in probability and statistics, the book presents effective and practical policies illustrated in a wide range of applications, from energy, homeland security, and transportation to engineering, health, and business.

This book covers the fundamental dimensions of a learning problem and presents a simple method for testing and comparing policies for learning. Special attention is given to the knowledge gradient policy and its use with a wide range of belief models, including lookup table and parametric and for online and offline problems. Three sections develop ideas with increasing levels of sophistication:

• Fundamentals explores fundamental topics, including adaptive learning, ranking and selection, the knowledge gradient, and bandit problems
• Extensions and Applications features coverage of linear belief models, subset selection models, scalar function optimization, optimal bidding, and stopping problems
• Advanced Topics explores complex methods including simulation optimization, active learning in mathematical programming, and optimal continuous measurements

Each chapter identifies a specific learning problem, presents the related, practical algorithms for implementation, and concludes with numerous exercises. A related website features additional applications and downloadable software, including MATLAB and the Optimal Learning Calculator, a spreadsheet-based package that provides an introduc­tion to learning and a variety of policies for learning.

Acknowledgments xix

1 The challenges of learning 1

1.1 Learning the best path 2

1.2 Areas of application 4

1.3 Major problem classes 12

1.4 The different types of learning 13

1.5 Learning from different communities 16

1.6 Information collection using decision trees 18

1.6.1 A basic decision tree 18

1.6.2 Decision tree for offline learning 20

1.6.3 Decision tree for online learning 21

1.6.4 Discussion 25

1.7 Website and downloadable software 26

1.8 Goals of this book 26

Problems 28

2 Adaptive learning 31

2.1 The frequentist view 32

2.2 The Bayesian view 33

2.2.1 The updating equations for independent beliefs 34

2.2.2 The expected value of information 36

2.2.3 Updating for correlated normal priors 38

2.2.4 Bayesian updating with an uninformative prior 41

2.3 Updating for non-Gaussian priors 42

2.3.1 The gamma-exponential model 43

2.3.2 The gamma-Poisson model 44

2.3.3 The Pareto-uniform model 45

2.3.4 Models for learning probabilities* 46

2.3.5 Learning an unknown variance* 49

2.4 Monte Carlo simulation 51

2.5 Why does it work?* 54

2.5.1 Derivation of ~_ 54

2.5.2 Derivation of Bayesian updating equations for independent beliefs 55

2.6 Bibliographic notes 57

Problems 57

3 The economics of information 61

3.1 An elementary information problem 61

3.2 The marginal value of information 65

3.3 An information acquisition problem 68

3.4 Bibliographic notes 70

Problems 70

4 Ranking and selection 71

4.1 The model 72

4.2 Measurement policies 75

4.2.1 Deterministic vs. sequential policies 75

4.2.2 Optimal sequential policies 76

4.2.3 Heuristic policies 77

4.3 Evaluating policies 81

4.4 More advanced topics* 83

4.4.1 An alternative representation of the probability space 83

4.4.2 Equivalence of using true means and sample estimates 84

4.5 Bibliographic notes 85

Problems 85

5 The knowledge gradient 89

5.1 The knowledge gradient for independent beliefs 90

5.1.1 Computation 91

5.1.2 Some properties of the knowledge gradient 93

5.1.3 The four distributions of learning 94

5.2 The value of information and the S-curve effect 95

5.3 Knowledge gradient for correlated beliefs 98

5.4 The knowledge gradient for some non-Gaussian distributions 103

5.4.1 The gamma-exponential model 104

5.4.2 The gamma-Poisson model 107

5.4.3 The Pareto-uniform model 108

5.4.4 The beta-Bernoulli model 109

5.4.5 Discussion 111

5.5 Relatives of the knowledge gradient 112

5.5.1 Expected improvement 113

5.5.2 Linear loss* 114

5.6 Other issues 116

5.6.1 Anticipatory vs. experiential learning 117

5.6.2 The problem of priors 118

5.6.3 Discussion 120

5.7 Why does it work?* 121

5.7.1 Derivation of the knowledge gradient formula 121

5.8 Bibliographic notes 125

Problems 126

6 Bandit problems 139

6.1 The theory and practice of Gittins indices 141

6.1.1 Gittins indices in the beta-Bernoulli model 142

6.1.2 Gittins indices in the normal-normal model 145

6.1.3 Approximating Gittins indices 147

6.2 Variations of bandit problems 148

6.3 Upper confidence bounding 149

6.4 The knowledge gradient for bandit problems 151

6.4.1 The basic idea 151

6.4.2 Some experimental comparisons 153

6.4.3 Non-normal models 156

6.5 Bibliographic notes 157

Problems 157

7 Elements of a learning problem 163

7.1 The states of our system 164

7.2 Types of decisions 166

7.3 Exogenous information 167

7.4 Transition functions 168

7.5 Objective functions 168

7.5.1 Designing versus controlling 168

7.5.2 Measurement costs 170

7.5.3 Objectives 170

7.6 Evaluating policies 175

7.7 Discussion 177

7.8 Bibliographic notes 178

Problems 178

8 Linear belief models 181

8.1 Applications 182

8.1.1 Maximizing ad clicks 182

8.1.2 Dynamic pricing 184

8.1.3 Housing loans 184

8.1.4 Optimizing dose response 185

8.2 A brief review of linear regression 186

8.2.1 The normal equations 186

8.2.2 Recursive least squares 187

8.2.3 A Bayesian interpretation 188

8.2.4 Generating a prior 189

8.3 The knowledge gradient for a linear model 191

8.4 Application to drug discovery 192

8.5 Application to dynamic pricing 196

8.6 Bibliographic notes 200

Problems 200

9 Subset selection problems 203

9.1 Applications 205

9.2 Choosing a subset using ranking and selection 206

9.2.1 Setting prior means and variances 207

9.2.2 Two strategies for setting prior covariances 208

9.3 Larger sets 209

9.3.1 Using simulation to reduce the problem size 210

9.3.2 Computational issues 212

9.3.3 Experiments 213

9.4 Very large sets 214

9.5 Bibliographic notes 216

Problems 216

10 Optimizing a scalar function 219

10.1 Deterministic measurements 219

10.2 Stochastic measurements 223

10.2.1 The model 223

10.2.2 Finding the posterior distribution 224

10.2.3 Choosing the measurement 226

10.2.4 Discussion 229

10.3 Bibliographic notes 229

Problems 229

11 Optimal bidding 231

11.1 Modeling customer demand 233

11.1.1 Some valuation models 233

11.1.2 The logit model 234

11.2 Bayesian modeling for dynamic pricing 237

11.2.1 A conjugate prior for choosing between two demand curves 237

11.2.2 Moment matching for non-conjugate problems 239

11.2.3 An approximation for the logit model 242

11.3 Bidding strategies 244

11.3.1 An idea from multi-armed bandits 245

11.3.2 Bayes-greedy bidding 245

11.3.3 Numerical illustrations 247

11.4 Why does it work?* 251

11.4.1 Moment matching for Pareto prior 251

11.4.2 Approximating the logistic expectation 252

11.5 Bibliographic notes 253

Problems 254

12 Stopping problems 255

12.1 Sequential probability ratio test 255

12.2 The secretary problem 260

12.2.1 Setup 261

12.2.2 Solution 263

12.3 Bibliographic notes 266

Problems 266

13 Active learning in statistics 269

13.1 Deterministic policies 270

13.2 Sequential policies for classification 274

13.2.1 Uncertainty sampling 274

13.2.2 Query by committee 275

13.2.3 Expected error reduction 276

13.3 A variance minimizing policy 277

13.4 Mixtures of Gaussians 279

13.4.1 Estimating parameters 280

13.4.2 Active learning 281

13.5 Bibliographic notes 283

14 Simulation optimization 285

14.1 Indifference zone selection 287

14.1.1 Batch procedures 288

14.1.2 Sequential procedures 290

14.1.3 The 0-1 procedure: connection to linear loss 291

14.2 Optimal computing budget allocation 292

14.2.1 Indifference-zone version 293

14.2.2 Linear loss version 294

14.2.3 When does it work? 295

14.3 Model-based simulated annealing 296

14.4 Other areas of simulation optimization 298

14.5 Bibliographic notes 299

15 Learning in mathematical programming 301

15.1 Applications 303

15.1.1 Piloting a hot air balloon 303

15.1.2 Optimizing a portfolio 308

15.1.3 Network problems 309

15.1.4 Discussion 313

15.2 Learning on graphs 313

15.3 Alternative edge selection policies 316

15.4 Learning costs for linear programs* 317

15.5 Bibliographic notes 324

16 Optimizing over continuous measurements 325

16.1 The belief model 327

16.1.1 Updating equations 328

16.1.2 Parameter estimation 330

16.2 Sequential kriging optimization 332

16.3 The knowledge gradient for continuous parameters* 334

16.3.1 Maximizing the knowledge gradient 334

16.3.2 Approximating the knowledge gradient 335

16.3.3 The gradient of the knowledge gradient 336

16.3.4 Maximizing the knowledge gradient 338

16.3.5 The KGCP policy 339

16.4 Efficient global optimization 340

16.5 Experiments 341

16.6 Extension to higher dimensional problems 342

16.7 Bibliographic notes 343

17 Learning with a physical state 345

17.1 Introduction to dynamic programming 347

17.1.1 Approximate dynamic programming 348

17.1.2 The exploration vs. exploitation problem 350

17.1.3 Discussion 351

17.2 Some heuristic learning policies 352

17.3 The local bandit approximation 353

17.4 The knowledge gradient in dynamic programming 355

17.4.1 Generalized learning using basis functions 355

17.4.2 The knowledge gradient 358

17.4.3 Experiments 361

17.5 An expected improvement policy 363

17.6 Bibliographic notes 364

Index 379

WARREN B. POWELL, PhD, is Professor of Operations Research and Financial Engineering at Princeton University, where he is founder and Director of CASTLE Laboratory, a research unit that works with industrial partners to test new ideas found in operations research. The recipient of the 2004 INFORMS Fellow Award, Dr. Powell is the author of Approximate Dynamic Programming: Solving the Curses of Dimensionality, Second Edition (Wiley).

ILYA O. RYZHOV, PhD, is Assistant Professor in the Department of Decision, Operations, and Information Technologies at the Robert H. Smith School of Business at the University of Maryland. He has made fundamental contributions to bridge the fields of ranking and selection with multiarmed bandits and optimal learning with mathematical programming.