Faster Randomized Block Sparse Kaczmarz by Averaging

TL;DR

This paper introduces a parallel mini-batch version of the randomized sparse Kaczmarz method that leverages averaging of multiple steps, enabling faster convergence and parallel computation for solving linear systems.

Contribution

The paper proposes a novel parallel (mini-batch) algorithm for RSK, demonstrating linear convergence and improved speed over the standard method, with theoretical and numerical validation.

Findings

The parallel RSK method converges linearly in expectation.

The method outperforms the standard RSK in convergence speed when parallelization is utilized.

Numerical experiments confirm the theoretical advantages of the proposed algorithm.

Abstract

The standard randomized sparse Kaczmarz (RSK) method is an algorithm to compute sparse solutions of linear systems of equations and uses sequential updates, and thus, does not take advantage of parallel computations. In this work, we introduce a parallel (mini batch) version of RSK based on averaging several Kaczmarz steps. Naturally, this method allows for parallelization and we show that it can also leverage large over-relaxation. We prove linear expected convergence and show that, given that parallel computations can be exploited, the method provably provides faster convergence than the standard method. This method can also be viewed as a variant of the linearized Bregman algorithm, a randomized dual block coordinate descent update, a stochastic mirror descent update, or a relaxed version of RSK and we recover the standard RSK method when the batch size is equal to one. We also provide estimates for inconsistent systems and show that the iterates convergence to an error in the order of the noise level. Finally, numerical examples illustrate the benefits of the new algorithm.