Stochastic Approximation and Its Application
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Optimization by Stochastic Approximation. Applications To Signal Processing. Application to Systems and Control. View via Publisher. Alternate Sources. Save to Library. Create Alert. Convergence and convergence rates of the Kiefer-Wolfowitz KW algorithm with expanding truncations and randomized differences are established.
A global optimization method consisting in combination of the KW algorithms with search methods is defined, and its a. Finally, the global optimization method is applied to solving the model reduction problem. In Chapter 5 the general theory is applied to the problems arising from signal processing.
Applying TS method to principal component analysis results in improving conditions for convergence. Stochastic approximation algorithms with expanding truncations with TS method are also applied to adaptive filters with and without constraints.
As a result, conditions required for convergence have been considerably improved in comparison with the existing results. Finally, the expanding truncation technique and TS method are applied to the asynchronous stochastic approximation. In the last chapter, the general theory is applied to problems arising from systems and control. The ideal parameter for operation is identified for stochastic systems by using the methods developed in this book.
Then the obtained results are applied to the adaptive quadratic control problem. Adaptive regulation for a nonlinear nonparametric system and learning pole assignment are also solved by the stochastic approximation method. The book is self-contained in the sense that there are only a few points using knowledge for which we refer to other sources, and these points can be ignored when reading the main body of the book.
The basic mathematical tools used in the book are calculus and linear algebra based on which one will have no difficulty to read the fundamental convergence Theorems 2.
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To understand other material, probability concept, especially the convergence theorems for martingale difference sequences are needed. Necessary concept of probability theory is given in Appendix A. Some facts from probability that are used at a few specific points are listed in Appendix A but without proof, because omitting the corresponding parts still makes the rest of the book readable. The book is written for students, engineers and researchers working in the areas of systems and control, communication and signal processing, optimization and operation research, and mathematical statistics.
The author would like to express his gratitude to Dr.
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Haitao Fang for his helpful suggestions and useful discussions. The author would also like to thank Ms. Jinling Chang for her skilled typing and to thank my wife Shujun Wang for her constant support. It is quite often that an optimization problem can be reduced to finding zeros roots of an unknown function which can be observed but the observation may be corrupted by errors. This is the topic of stochastic approximation SA. The error source may be observation noise, but may also come from structural inaccuracy of the observed function.
For example, one wants to find zeros of but he actually observes functions which are different from Let us denote by the observation at time the observation noise: Here, is the additional error caused by the structural inaccuracy. It is worth noting that the structural error normally depends on and it is hard to require it to have a certain probabilistic property such as independence, stationarity or martingale property.
We call this kind of noises as state-dependent noise. The basic recursive algorithm for finding roots of an unknown function on the basis of noisy observations is the Robbins-Monro RM algorithm, which is characterized by its simplicity in computation.
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This chapter serves as an introduction to SA, describing various methods for analyzing convergence of the RM algorithm. In Section 1. The ODE method is demonstrated in Section 1. So, we call this method as Trajectory-Subsequence TS method, which is the basic tool used in the subsequent chapters. In this book our main concern is the path-wise convergence of the algorithm.
However, there is another approach to convergence analysis called the weak convergence method, which is briefly introduced in Section 1. Notes and references are given in the last section. This chapter introduces main methods used in literature for convergence analysis, but restricted to the single root case. Extension to more general cases in various aspects is given in later chapters. Finding Zeros of a Function. Many theoretical and practical problems in diverse areas can be reduced to finding zeros of a function. To see this it suffices to notice that solving many problems finally consists in optimizing some function i.
If is differentiable, then the optimization problem reduces to finding the roots of where the derivative of In the case where the function or its derivatives can be observed without errors, there are many numerical methods for solving the problem. For example, the gradient method, by which the estimate for the root of is recursively generated by the following algorithm where denotes the derivative of This kind of problems belongs to the topics of optimization theory, which considers general cases where may be nonconvex, nonsmooth, and with constraints. In contrast to the optimization theory, SA is devoted to finding zeros of an unknown function which can be observed, but the observations are corrupted by errors.
Since is not exactly known and even may not exist, 1. Consider the following simple example. Let be a linear function If the derivative of is available, i. Assume the derivative of is unavailable but can exactly be observed. Let us replace by in 1. Then we derive or This is a linear difference equation, which can inductively be solved, and the solution of 1. Let us consider the case where is observed with errors: where denotes the observation at time the corresponding observation error and the estimate for the root of at time It is natural to ask, how will behave if the exact value of in 1.
It is worth noting that in lieu of 1. This simple example demonstrates the basic features of the algorithm 1. From 1. In the case where is a sequence of independent and identically distributed random variables with zero mean and bounded variance, then by the iterated logarithm law. This means that convergence rate for algorithms 1. Probabilistic Method We have just shown how to find the root of an unknown linear function based on noisy observations. We now formulate the general problem. In the pioneer work of this area Robbins and Monro proposed the following algorithm to estimate where step size is decreasing and satisfies the fol- lowing conditions and They proved We explain the meaning of conditions required for step size Condition aims at reducing the effect of observation noises.
To see this, consider the case where is close to and is close to zero, say, with small. Throughout the book, always means the Euclidean norm of a vector and denotes the square root of the maximum eigenvalue of the matrix where means the transpose of the matrix A. Therefore, in order to have the desired consistency, i.
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Therefore, in this case if the initial value is far from the true root and hence will never converge to The algorithm 1. We now present a typical convergence theorem by this approach. Related concept and results from probability theory are given in Appendices A and B. In fact, we will use the martingale convergence theorem to prove the path-wise convergence of i. Prior to formulating the theorem we need some auxiliary results. Let be an adapted sequence, i. The following lemma concerning convergence of an adapted sequence will be used in the proof for convergence of the RM algorithm, but the lemma is of interest by itself.
Lemma 1. For proving i set Then we have By the convergence theorem for nonnegative supermartingales, verges a. Theorem 1.
Stochastic Approximation and Its Applications / Edition 1
Then for any initial value, given by the RM algorithm 1. Let be the Lyapunov function given in A1. Expanding to the Taylor series, we obtain where and denote the gradient and Hessian of respectively, is a vector with components located in-between the corresponding components of and and denotes the constant such that by A1. Noticing that is and taking conditional expectation for 1.
Since also converges a. For any Let denote be the first exit time of from and let where denotes the complement to This means that is the first exit time from after Since is nonpositive, from 1.
Otherwise, we would have a contradiction to A1. Therefore, a. By A1. Remark 1. As noted in Section 1. The noise condition A1. As to be shown in the subsequent chapters, may be composed of not only the random noise but also structural errors which hardly have nice probabilistic properties such as martingale difference, stationarity or with bounded variances etc. As in many cases, one can take to serve as Then from 1.
This is a major restriction to apply Theorem 1. However, if we a priori assume that generated by the algorithm 1. We explain the idea of the method. The estimate generated by the RM algorithm is interpolated to a continuous function with interpolating length equal to the step size used in the algorithm.