Stochastic Krasnosel skii-Mann Iterations in Banach Spaces with Bregman Distances

TL;DR

This paper introduces a stochastic Krasnoselskii-Mann algorithm in Banach spaces with Bregman distances, proving convergence and residual bounds, and demonstrating applications in reinforcement learning and mirror descent.

Contribution

It generalizes the SKM algorithm to Bregman geometries in Banach spaces, providing convergence analysis and practical extensions.

Findings

Almost-sure convergence to fixed points

Residual bounds depending on convexity modulus

Effective in entropy-regularized reinforcement learning

Abstract

We propose a generalization of the stochastic Krasnoselskil-Mann $(SKM)$ algorithm to reflexive Banach spaces endowed with Bregman distances. Under standard martingale-difference noise assumptions in the dual space and mild conditions on the distance-generating function, we establish almost-sure convergence to a fixed point and derive non-asymptotic residual bounds that depend on the uniform convexity modulus of the generating function. Extensions to adaptive Bregman geometries and robust noise models are also discussed. Numerical experiments on entropy-regularized reinforcement learning and mirror-descent illustrate the theoretical findings.