Stochastic Approximation
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Stochastic Approximation and Recursive Algorithms and Applications
Author: Harold Kushner
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-05-04
The basic stochastic approximation algorithms introduced by Robbins and MonroandbyKieferandWolfowitzintheearly1950shavebeenthesubject of an enormous literature, both theoretical and applied. This is due to the large number of applications and the interesting theoretical issues in the analysis of “dynamically de?ned” stochastic processes. The basic paradigm is a stochastic di?erence equation such as ? = ? + Y , where ? takes n+1 n n n n its values in some Euclidean space, Y is a random variable, and the “step n size” > 0 is small and might go to zero as n??. In its simplest form, n ? is a parameter of a system, and the random vector Y is a function of n “noise-corrupted” observations taken on the system when the parameter is set to ? . One recursively adjusts the parameter so that some goal is met n asymptotically. Thisbookisconcernedwiththequalitativeandasymptotic properties of such recursive algorithms in the diverse forms in which they arise in applications. There are analogous continuous time algorithms, but the conditions and proofs are generally very close to those for the discrete time case. The original work was motivated by the problem of ?nding a root of a continuous function g ̄(?), where the function is not known but the - perimenter is able to take “noisy” measurements at any desired value of ?. Recursive methods for root ?nding are common in classical numerical analysis, and it is reasonable to expect that appropriate stochastic analogs would also perform well.
Stochastic Approximation
Author: Vivek S. Borkar
language: en
Publisher: Cambridge University Press
Release Date: 2008-09-01
This simple, compact toolkit for designing and analyzing stochastic approximation algorithms requires only a basic understanding of probability and differential equations. Although powerful, these algorithms have applications in control and communications engineering, artificial intelligence and economic modeling. Unique topics include finite-time behavior, multiple timescales and asynchronous implementation. There is a useful plethora of applications, each with concrete examples from engineering and economics. Notably it covers variants of stochastic gradient-based optimization schemes, fixed-point solvers, which are commonplace in learning algorithms for approximate dynamic programming, and some models of collective behavior.
Stochastic Approximation and Its Applications
Author: Han-Fu Chen
language: en
Publisher: Springer Science & Business Media
Release Date: 2005-12-30
Estimating unknown parameters based on observation data conta- ing information about the parameters is ubiquitous in diverse areas of both theory and application. For example, in system identification the unknown system coefficients are estimated on the basis of input-output data of the control system; in adaptive control systems the adaptive control gain should be defined based on observation data in such a way that the gain asymptotically tends to the optimal one; in blind ch- nel identification the channel coefficients are estimated using the output data obtained at the receiver; in signal processing the optimal weighting matrix is estimated on the basis of observations; in pattern classifi- tion the parameters specifying the partition hyperplane are searched by learning, and more examples may be added to this list. All these parameter estimation problems can be transformed to a root-seeking problem for an unknown function. To see this, let - note the observation at time i. e. , the information available about the unknown parameters at time It can be assumed that the parameter under estimation denoted by is a root of some unknown function This is not a restriction, because, for example, may serve as such a function.