Nonexpansive Markov Operators and Random Function Iterations for Stochastic Fixed Point Problems

TL;DR

This paper investigates the convergence of random function iterations in stochastic fixed point problems, establishing conditions for invariant measures and extending fixed point theory to stochastic algorithms with computational errors.

Contribution

It introduces a framework for analyzing convergence to invariant measures in stochastic fixed point problems, generalizing previous feasibility results and applicable to distributed algorithms and computational errors.

Findings

Convergence to invariant measures under certain conditions

Extension of fixed point theory to stochastic settings

Applicability to large-scale distributed algorithms

Abstract

We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes earlier work studying the stochastic feasibility problem, namely, to find points that are, with probability 1, fixed points of the random functions. When no such points exist, the stochastic feasibility problem is called inconsistent, but still under certain assumptions, the more general stochastic fixed point problem has a solution and the random function iterations converge to an invariant measure for the corresponding Markov operator. We show how common structures in deterministic fixed point theory can be exploited to establish existence of invariant measures and convergence in distribution of the Markov chain. This framework specializes to many applications of current interest including, for instance, stochastic algorithms for large-scale distributed computation, and deterministic iterative procedures with computational error. The theory developed in this study provides a solid basis for describing the convergence of simple computational methods without the assumption of infinite precision arithmetic or vanishing computational errors.