# Stochastic Quasi-Fej\'er Block-Coordinate Fixed Point Iterations With Random Sweeping II: Mean-Square and Linear Convergence

## TL;DR

This paper analyzes the convergence properties of a stochastic block-coordinate fixed point algorithm with random block selection and stochastic errors, establishing mean-square and linear convergence results for applications in optimization.

## Contribution

It introduces new convergence results for a stochastic block-coordinate fixed point method with random sweeping and stochastic errors, extending previous almost sure convergence analysis.

## Key findings

- Proves mean-square convergence of the algorithm.
- Establishes linear convergence under certain conditions.
- Applies results to monotone operator splitting and proximal algorithms.

## Abstract

Reference [11] investigated the almost sure weak convergence of block-coordinate fixed point algorithms and discussed their applications to nonlinear analysis and optimization. This algorithmic framework features random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and it allows for stochastic errors in the evaluation of the operators. The present paper establishes results on the mean-square and linear convergence of the iterates. Applications to monotone operator splitting and proximal optimization algorithms are presented.