TITLE:
Stochastic Approximation Method for Fixed Point Problems
AUTHORS:
Ya. I. Alber, C. E. Chidume, Jinlu Li
KEYWORDS:
Hilbert Spaces; Stochastic Approximation Algorithm; Weakly Contractive Operators; Nonexpansive Operators; Fixed Points; Convergence in Mean Square; Convergence Almost Sure (a.s.); Nonasymptotic Estimates of Convergence Rate
JOURNAL NAME:
Applied Mathematics,
Vol.3 No.12A,
December
31,
2012
ABSTRACT: We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.