The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning,9780387212401

The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning

by ;
ISBN13: 9780387212401
ISBN10: 038721240X
Format: Hardcover
Pub. Date: 2004-08-30
Publisher(s): Springer Verlag
List Price: $199.00

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Summary

The cross-entropy (CE) method is one of the most significant developments in stochastic optimization and simulation in recent years. This book explains in detail how and why the CE method works. The CE method involves an iterative procedure where each iteration can be broken down into two phases: (a) generate a random data sample (trajectories, vectors, etc.) according to a specified mechanism; (b) update the parameters of the random mechanism based on this data in order to produce a "better" sample in the next iteration. The simplicity and versatility of the method is illustrated via a diverse collection of optimization and estimation problems. The book is aimed at a broad audience of engineers, computer scientists, mathematicians, statisticians and in general anyone, theorist or practitioner, who is interested in fast simulation, including rare-event probability estimation, efficient combinatorial and continuous multi-extremal optimization, and machine learning algorithms. Reuven Y. Rubinstein is the Milford Bohm Professor of Management at the Faculty of Industrial Engineering and Management at the Technion (Israel Institute of Technology). His primary areas of interest are stochastic modelling, applied probability, and simulation. He has written over 100 articles and has published five books. He is the pioneer of the well-known score-function and cross-entropy methods. Dirk P. Kroese is an expert on the cross-entropy method. He has published close to 40 papers in a wide range of subjects in applied probability and simulation. He is on the editorial board of Methodology and Computing in Applied Probability and is Guest Editor of the Annals of Operations Research. He has held research and teaching positions at Princeton University and The University of Melbourne, and is currently working at the Department of Mathematics of The University of Queensland. From the reviews: "Rarely have I seen such a dense and straight to the point pedagogical monograph on such a modern subject. This excellent book, on the simulated cross-entropy method (CEM) pioneered by one of the authors (Rubinstein), is very well written..." Computing Reviews, Stochastic Programming November, 2004 "...I wholeheartedly recommend this book to anybody who is interested in stochastic optimization or simulation-based performance analysis of stochastic systems." Gazette of the Australian Mathematical Society, vol. 32 (3) 2005 "This book describes the cross-entropy method for a range of optimization problems. a? It is a substantial contribution to stochastic optimization and more generally to the stochastic numerical methods theory." (V.V.Fedorov, Short Book Reviews, Vol. 25 (1), 2005) "Since the CE method is a young and developing field, there is no book available in this area where the two authors are the pioneers. Therefore, it is quite a unique book and it may become a classic reference in the CE method literature." Technometrics, February 2005

Author Biography

Dirk P. Kroese is currently working at the Department of Mathematics of The University of Queensland.

Table of Contents

Acronyms xi
List of Symbols xiii
1 Preliminaries 1(28)
1.1 Motivation for the Cross-Entropy Method
1(1)
1.2 Random Experiments and Probability Distributions
2(4)
1.3 Exponential Families
6(2)
1.4 Efficiency of Estimators
8(2)
1.5 Information
10(6)
1.6 The Score Function Method *
16(3)
1.7 Generating Random Variables
19(5)
1.7.1 The Inverse-Transform Method
19(2)
1.7.2 Alias Method
21(1)
1.7.3 The Composition Method
21(1)
1.7.4 The Acceptance-Rejection Method
22(2)
1.8 Exercises
24(5)
2 A Tutorial Introduction to the Cross-Entropy Method 29(30)
2.1 Introduction
29(2)
2.2 Methodology: Two Examples
31(5)
2.2.1 A Rare-Event Simulation Example
31(3)
2.2.2 A Combinatorial Optimization Example
34(2)
2.3 The CE Method for Rare-Event Simulation
36(5)
2.4 The CE-Method for Combinatorial Optimization
41(4)
2.5 Two Applications
45(12)
2.5.1 The Max-Cut Problem
46(5)
2.5.2 The Travelling Salesman Problem
51(6)
2.6 Exercises
57(2)
3 Efficient Simulation via Cross-Entropy 59(70)
3.1 Introduction
59(3)
3.2 Importance Sampling
62(5)
3.3 Kullback-Leibler Cross-Entropy
67(5)
3.4 Estimation of Rare-Event Probabilities
72(3)
3.5 Examples
75(8)
3.6 Convergence Issues
83(7)
3.7 The Root-Finding Problem
90(1)
3.8 The TLR Method
91(5)
3.9 Equivalence Between SLR and TLR
96(4)
3.10 Stationary Waiting Time of the GI/G/1 Queue
100(4)
3.11 Big-Step CE Algorithm
104(1)
3.12 Numerical Results
105(13)
3.12.1 Sum of Weibull Random Variables
106(2)
3.12.2 Sum of Pareto Random Variables
108(1)
3.12.3 Stochastic Shortest Path
109(2)
3.12.4 GI/G/1 Queue
111(5)
3.12.5 Two Non-Markovian Queues with Feedback
116(2)
3.13 Appendix: The Sum of Two Weibulls
118(3)
3.14 Exercises
121(8)
4 Combinatorial Optimization via Cross-Entropy 129(58)
4.1 Introduction
129(3)
4.2 The Main CE Algorithm for Optimization
132(4)
4.3 Adaptive Parameter Updating in Stochastic Node Networks
136(2)
4.3.1 Conditional Sampling
137(1)
4.3.2 Degenerate Distribution
138(1)
4.4 Adaptive Parameter Updating in Stochastic Edge Networks
138(2)
4.4.1 Parameter Updating for Markov Chains
139(1)
4.4.2 Conditional Sampling
140(1)
4.4.3 Optimal Degenerate Transition Matrix
140(1)
4.5 The Max-Cut Problem
140(5)
4.6 The Partition Problem
145(2)
4.7 The Travelling Salesman Problem
147(7)
4.8 The Quadratic Assignment Problem
154(2)
4.9 Numerical Results for SNNs
156(12)
4.9.1 Synthetic Max-Cut Problem
157(3)
4.9.2 r-Partition
160(2)
4.9.3 Multiple Solutions
162(6)
4.9.4 Empirical Computational Complexity
168(1)
4.10 Numerical Results for SENs
168(6)
4.10.1 Synthetic TSP
169(1)
4.10.2 Multiple Solutions
170(1)
4.10.3 Experiments with Sparse Graphs
171(2)
4.10.4 Numerical Results for the QAP
173(1)
4.11 Appendices
174(11)
4.11.1 Two Tour Generation Algorithm for the TSP
174(3)
4.11.2 Speeding up Trajectory Generation
177(2)
4.11.3 An Analysis of Algorithm 4.5.2 for a Partition Problem
179(2)
4.11.4 Convergence of CE Using the Best Sample
181(4)
4.12 Exercises
185(2)
5 Continuous Optimization and Modifications 187(16)
5.1 Continuous Multi-Extremal Optimization
187(3)
5.2 Alternative Reward Functions
190(1)
5.3 Fully Adaptive CE Algorithm
191(3)
5.4 Numerical Results for Continuous Optimization
194(2)
5.5 Numerical Results for FACE
196(4)
5.6 Exercises
200(3)
6 Noisy Optimization with CE 203(24)
6.1 Introduction
203(1)
6.2 The CE Algorithm for Noisy Optimization
204(3)
6.3 Optimal Buffer Allocation in a Simulation-Based Environment [9]
207(6)
6.4 Numerical results for the BAP
213(3)
6.5 Numerical Results for the Noisy TSP
216(9)
6.6 Exercises
225(2)
7 Applications of CE to COPs 227(24)
7.1 Sequence Alignment
227(7)
7.2 Single Machine Total Weighted Tardiness Problem
234(3)
7.3 Single Machine Common Due Date Problem
237(1)
7.4 Capacitated Vehicle Routing Problem
238(2)
7.5 The Clique Problem
240(7)
7.6 Exercises
247(4)
8 Applications of CE to Machine Learning 251(20)
8.1 Mastermind Game
251(3)
8.1.1 Numerical Results
253(1)
8.2 The Markov Decision Process and Reinforcement Learning
254(6)
8.2.1 Policy Learning via the CE Method
256(3)
8.2.2 Numerical Results
259(1)
8.3 Clustering and Vector Quantization
260(8)
8.3.1 Numerical Results
263(5)
8.4 Exercises
268(3)
A Example Programs 271(16)
A.1 Rare Event Simulation
272(2)
A.2 The Max-Cut Problem
274(2)
A.3 Continuous Optimization via the Normal Distribution
276(1)
A.4 FACE
277(4)
A.5 Rosenbrock
281(2)
A.6 Beta Updating
283(2)
A.7 Banana Data
285(2)
References 287(10)
Index 297

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