Mini-Batch Stochastic Krasnosel'ski\u\i-Mann Algorithm for Nonexpansive Fixed Point Problems
Optimization and Control
2026-04-09 v1
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.
Cite
@article{arxiv.2604.06909,
title = {Mini-Batch Stochastic Krasnosel'ski\u\i-Mann Algorithm for Nonexpansive Fixed Point Problems},
author = {Hideaki Iiduka},
journal= {arXiv preprint arXiv:2604.06909},
year = {2026}
}