# Parallel Random Block-Coordinate Forward-Backward Algorithm: A Unified Convergence Analysis

## TL;DR

This paper introduces a unified convergence analysis for a parallel, random block-coordinate forward-backward algorithm, demonstrating its convergence properties and rates under various conditions.

## Contribution

It provides a comprehensive convergence analysis for a flexible parallel block-coordinate algorithm with arbitrary update probabilities and stepsizes.

## Key findings

- Almost sure weak convergence in convex and infinite-dimensional settings
- O(1/n) convergence rate for mean function values
- Linear convergence under strong convexity and error bounds

## Abstract

We study the block-coordinate forward-backward algorithm in which the blocks are updated in a random and possibly parallel manner, according to arbitrary probabilities. The algorithm allows different stepsizes along the block-coordinates to fully exploit the smoothness properties of the objective function. In the convex case and in an infinite dimensional setting, we establish almost sure weak convergence of the iterates and the asymptotic rate o(1/n) for the mean of the function values. We derive linear rates under strong convexity and error bound conditions. Our analysis is based on an abstract convergence principle for stochastic descent algorithms which allows to extend and simplify existing results.