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

arXiv: https://arxiv.org/abs/1906.07392 · HTML: https://mlsanity.com/paper/1906.07392/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2019.

## Built on by (most cited first, 2 of 2)

- [Laplacian-based Semi-Supervised Learning in Multilayer Hypergraphs by Coordinate Descent](https://mlsanity.com/paper/2301.12184/) (2023)
- [A block-coordinate descent framework for non-convex composite optimization. Application to sparse precision matrix estimation](https://mlsanity.com/paper/2601.21467/) (2026)

## Abstract (arXiv)

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.