Parallel random block-coordinate forward–backward algorithm: a unified convergence analysis

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