Stochastic Approximation and Its Application

by Chen, Han-Fu

• ISBN13: 9781402008061
• ISBN10: 1402008066
• Format: Hardcover
• Copyright: 2002-09-01
• Publisher: Springer Nature

Summary

This book presents the recent development of stochastic approximation algorithms with expanding truncations based on the TS (trajectory-subsequence) method, a newly developed method for convergence analysis. This approach is so powerful that conditions used for guaranteeing convergence have been considerably weakened in comparison with those applied in the classical probability and ODE methods. The general convergence theorem is presented for sample paths and is proved in a purely deterministic way. The sample-path description of theorems is particularly convenient for applications. Convergence theory takes both observation noise and structural error of the regression function into consideration. Convergence rates, asymptotic normality and other asymptotic properties are presented as well. Applications of the developed theory to global optimization, blind channel identification, adaptive filtering, system parameter identification, adaptive stabilization and other problems arising from engineering fields are demonstrated.Audience: Researchers and students of both graduate and undergraduate levels in systems and control, optimization, signal processing, communication and statistics.

Table of Contents

Preface	p. ix
Acknowledgments	p. xv
Robbins-Monro Algorithm	p. 1
Finding Zeros of a Function	p. 2
Probabilistic Method	p. 4
ODE Method	p. 10
Truncated RM Algorithm and TS Method	p. 16
Weak Convergence Method	p. 21
Notes and References	p. 23
Stochastic Approximation Algorithms with Expanding Truncations	p. 25
Motivation	p. 26
General Convergence Theorems by TS Method	p. 28
Convergence Under State-Independent Conditions	p. 41
Necessity of Noise Condition	p. 45
Non-Additive Noise	p. 49
Connection Between Trajectory Convergence and Property of Limit Points	p. 57
Robustness of Stochastic Approximation Algorithms	p. 67
Dynamic Stochastic Approximation	p. 82
Notes and References	p. 93
Asymptotic Properties of Stochastic Approximation Algorithms	p. 95
Convergence Rate: Nondegenerate Case	p. 96
Convergence Rate: Degenerate Case	p. 103
Asymptotic Normality	p. 113
Asymptotic Efficiency	p. 130
Notes and References	p. 149
Optimization by Stochastic Approximation	p. 151
Kiefer-Wolfowitz Algorithm with Randomized Differences	p. 153
Asymptotic Properties of KW Algorithm	p. 166
Global Optimization	p. 172
Asymptotic Behavior of Global Optimization Algorithm	p. 194
Application to Model Reduction	p. 210
Notes and References	p. 218
Application to Signal Processing	p. 219
Recursive Blind Identification	p. 220
Principal Component Analysis	p. 238
Recursive Blind Identification by PCA	p. 246
Constrained Adaptive Filtering	p. 265
Adaptive Filtering by Sign Algorithms	p. 273
Asynchronous Stochastic Approximation	p. 278
Notes and References	p. 288
Application to Systems and Control	p. 289
Application to Identification and Adaptive Control	p. 290
Application to Adaptive Stabilization	p. 305
Application to Pole Assignment for Systems with Unknown Coefficients	p. 316
Application to Adaptive Regulation	p. 321
Notes and References	p. 327
Appendices	p. 329
Probability Space	p. 329
Random Variable and Distribution Function	p. 330
Expectation	p. 330
Convergence Theorems and Inequalities	p. 331
Conditional Expectation	p. 332
Independence	p. 333
Ergodicity	p. 333
Convergence Theorems for Martingale	p. 335
Convergence Theorems for MDS I	p. 339
Borel-Cantelli-Levy Lemma	p. 340
Convergence Criteria for Adapted Sequences	p. 341
Convergence Theorems for MDS II	p. 343
Weighted Sum of MDS	p. 344
References	p. 347
Index	p. 355
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