The book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. There is a complete development of both probability one and weak convergence methods for very general noise processes. The proofs of convergence use the ODE method, the most powerful to date, with which the asymptotic behavior is characterized by the limit behavior of a mean ODE. The assumptions and proof methods are designed to cover the needs of recent applications. The development proceeds from simple to complex…mehr
The book presents a thorough development of the modern theory of stochastic approximation or recursive stochastic algorithms for both constrained and unconstrained problems. There is a complete development of both probability one and weak convergence methods for very general noise processes. The proofs of convergence use the ODE method, the most powerful to date, with which the asymptotic behavior is characterized by the limit behavior of a mean ODE. The assumptions and proof methods are designed to cover the needs of recent applications. The development proceeds from simple to complex problems, allowing the underlying ideas to be more easily understood. Rate of convergence, iterate averaging, high-dimensional problems, stability-ODE methods, two time scale, asynchronous and decentralized algorithms, general correlated and state-dependent noise, perturbed test function methods, and large devitations methods, are covered. Many motivational examples from learning theory, ergodic cost problems for discrete event systems, wireless communications, adaptive control, signal processing, and elsewhere, illustrate the application of the theory. This second edition is a thorough revision, although the main features and the structure remain unchanged. It contains many additional applications and results, and more detailed discussion. Harold J. Kushner is a University Professor and Professor of Applied Mathematics at Brown University. He has written numerous books and articles on virtually all aspects of stochastic systems theory, and has received various awards including the IEEE Control Systems Field Award.
Introduction 1 Review of Continuous Time Models 1.1 Martingales and Martingale Inequalities 1.2 Stochastic Integration 1.3 Stochastic Differential Equations: Diffusions 1.4 Reflected Diffusions 1.5 Processes with Jumps 2 Controlled Markov Chains 2.1 Recursive Equations for the Cost 2.2 Optimal Stopping Problems 2.3 Discounted Cost 2.4 Control to a Target Set and Contraction Mappings 2.5 Finite Time Control Problems 3 Dynamic Programming Equations 3.1 Functionals of Uncontrolled Processes 3.2 The Optimal Stopping Problem 3.3 Control Until a Target Set Is Reached 3.4 A Discounted Problem with a Target Set and Reflection 3.5 Average Cost Per Unit Time 4 Markov Chain Approximation Method: Introduction 4.1 Markov Chain Approximation 4.2 Continuous Time Interpolation 4.3 A Markov Chain Interpolation 4.4 A Random Walk Approximation 4.5 A Deterministic Discounted Problem 4.6 Deterministic Relaxed Controls 5 Construction of the Approximating Markov Chains 5.1 One Dimensional Examples 5.2 Numerical Simplifications 5.3 The General Finite Difference Method 5.4 A Direct Construction 5.5 Variable Grids 5.6 Jump Diffusion Processes 5.7 Reflecting Boundaries 5.8 Dynamic Programming Equations 5.9 Controlled and State Dependent Variance 6 Computational Methods for Controlled Markov Chains 6.1 The Problem Formulation 6.2 Classical Iterative Methods 6.3 Error Bounds 6.4 Accelerated Jacobi and Gauss-Seidel Methods 6.5 Domain Decomposition 6.6 Coarse Grid-Fine Grid Solutions 6.7 A Multigrid Method 6.8 Linear Programming 7 The Ergodic Cost Problem: Formulation and Algorithms 7.1 Formulation of the Control Problem 7.2 A Jacobi Type Iteration 7.3 Approximation in Policy Space 7.4 Numerical Methods 7.5 The Control Problem 7.6 The Interpolated Process 7.7 Computations 7.8 Boundary Costs and Controls 8 Heavy Traffic and Singular Control 8.1 Motivating Examples&nb
From the reviews of the second edition: "This is the second edition of an excellent book on stochastic approximation, recursive algorithms and applications ... . Although the structure of the book has not been changed, the authors have thoroughly revised it and added additional material ... ." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1026, 2004) "The book attempts to convince that ... algorithms naturally arise in many application areas ... . I do not hesitate to conclude that this book is exceptionally well written. The literature citation is extensive, and pertinent to the topics at hand, throughout. This book could be well suited to those at the level of the graduate researcher and upwards." (A. C. Brooms, Journal of the Royal Statistical Society Series A: Statistics in Society, Vol. 169 (3), 2006)
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