# Faster randomized block Kaczmarz algorithms

## TL;DR

This paper introduces a family of randomized block Kaczmarz algorithms optimized for distributed systems, with proven linear convergence and novel strategies for block selection and stepsize extrapolation.

## Contribution

It develops a new framework for randomized block Kaczmarz algorithms with convergence guarantees and practical strategies for distributed implementation.

## Key findings

- Algorithms converge linearly in expectation.
- Performance depends on matrix conditioning and block sampling.
- Resolves open problem on practical efficiency of extrapolated methods.

## Abstract

The Kaczmarz algorithm is a simple iterative scheme for solving consistent linear systems. At each step, the method projects the current iterate onto the solution space of a single constraint. Hence, it requires very low cost per iteration and storage, and it has a linear rate of convergence. Distributed implementations of Kaczmarz have become, in recent years, the de facto architectural choice for large-scale linear systems. Therefore, in this paper we develop a family of randomized block Kaczmarz algorithms that uses at each step a subset of the constraints and extrapolated stepsizes, and can be deployed on distributed computing units. Our approach is based on several new ideas and tools, including stochastic selection rule for the blocks of rows, stochastic conditioning of the linear system, and novel strategies for designing extrapolated stepsizes. We prove that randomized block Kaczmarz algorithm converges linearly in expectation, with a rate depending on the geometric properties of the matrix and its submatrices and on the size of the blocks. Our convergence analysis reveals that the algorithm is most effective when it is given a good sampling of the rows into well-conditioned blocks. Besides providing a general framework for the design and analysis of randomized block Kaczmarz methods, our results resolve an open problem in the literature related to the theoretical understanding of observed practical efficiency of extrapolated block Kaczmarz methods.