Rates of convergence of adaptive step-size of stochastic approximation algorithms
type: Item
instance of: scholarly article
title: Rates of convergence of adaptive step-size of stochastic approximation algorithms
zbMATH Open document ID: 0960.65016
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-12-14T00:00:00Z
review text: The stochastic approximation of the model \(y_n=x_n^T\) \(\theta +v_n\) is considered, where \(y_n\), \(x_n\), \(v_n\) are the observations, the random inputs, the noise, respectively, and \(\theta\) is the estimated parameter. The constant step-size \(\varepsilon\) of the classical adaptive algorithm \(\widehat{\theta}_{n+1}=\widehat{\theta}_n+\varepsilon x_n(y_n-x_n^T\widehat{\theta}_n)\) is replaced by an adaptive step-size \(\overline{\varepsilon}_n\), \(\widehat{\theta}_{n+1}=\widehat{\theta}_n+\overline{\varepsilon}_nx_n(y_n-x_n^T\widehat{\theta}_n)\), defined recursively by \(\overline{\varepsilon}_n=\overline{\varepsilon}_{n-1}+\frac 1n(\varepsilon_n-\overline{\varepsilon}_ n)=\frac 1n\sum_{i=1}^n \varepsilon_i\) with \(\varepsilon_{n+1}=\varepsilon_n+\gamma_nx_n(y_n-x_n^T\widehat{\theta}_n)\), \(\sum\varepsilon_n<\infty\) and \(\sum\gamma_n=\infty\) and \(\gamma_n\to 0\) slower than \(\frac 1n\). The convergence of \(\widehat{\theta}_n\) and \(\overline{\varepsilon}_n\) are proven and the rate of the convergence is analyzed by using the Gaussian approximation theorem.
reviewed by: Gong Guanglu
Mathematics Subject Classification ID: 65C60
Mathematics Subject Classification ID: 62L20
zbMATH DE Number: 1449211
zbMATH Keywords: adaptive algorithm
zbMATH Keywords: stochastic approximation
zbMATH Keywords: convergence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1449211