Mini-Batch Stochastic Krasnosel'ski\u\i-Mann Algorithm for Nonexpansive Fixed Point Problems

TL;DR

This paper introduces a mini-batch stochastic version of the Krasnosel'ski-Mann algorithm for large-scale fixed point problems, with proven convergence and rate analysis.

Contribution

It proposes a novel stochastic mini-batch approach that enables practical application of the Krasnosel'ski-Mann algorithm to large-scale problems.

Findings

The stochastic algorithm converges almost surely to a fixed point.

Convergence rate analysis is provided for the proposed method.

Abstract

The Krasnosel'ski\u\i-Mann algorithm is a well-known method for finding fixed points of a nonexpansive mapping with strong theoretical guarantees. However, there are practical large-scale problems to which this algorithm cannot be applied. Here, to resolve the issue caused by the computational difficulty of the mapping, we define a computable mini-batch stochastic mapping, which is a unbiased estimator of the nonexpansive mapping, and implement it in the Krasnosel'ski\u\i-Mann algorithm. We show that the algorithm with increasing batch sizes converges almost surely to a fixed point of the nonexpansive mapping. We also perform a convergence rate analysis on the algorithm.