Random Function Iterations for Consistent Stochastic Feasibility

TL;DR

This paper investigates the convergence behavior of stochastic fixed point iterations, establishing necessary and sufficient conditions for geometric convergence in expectation across various settings.

Contribution

It generalizes previous results by showing that conditions for linear convergence in stochastic projection algorithms are also necessary for broader classes of iterated random functions.

Findings

Necessary and sufficient conditions for geometric convergence.

Convergence analyzed under decreasing regularity assumptions.

Results apply to a wide range of stochastic fixed point iterations.

Abstract

We study the convergence of stochastic fixed point iterations in the consistent case (in the sense of Butnariu and Fl{\aa}m (1995)) in several different settings, under decreasingly restrictive regularity assumptions of the fixed point mappings. The iterations are Markov chains and, for the purposes of this study, convergence is understood in very restrictive terms. We show that sufficient conditions for geometric (linear) convergence in expectation of stochastic projection algorithms presented in Nedi\'c (2011), are in fact necessary for geometric (linear) convergence in expectation more generally of iterated random functions.