Optimal Learning

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.

Preface xv 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