On block Gaussian sketching for the Kaczmarz method

TL;DR

This paper provides theoretical convergence guarantees for the block Gaussian Kaczmarz method, a variant of the Kaczmarz algorithm using Gaussian sketches, and discusses its practical efficiency and potential advantages in noise reduction scenarios.

Contribution

It offers the first theoretical analysis of the block Gaussian Kaczmarz method, establishing exponential convergence, and compares its practical performance with other methods.

Findings

Proves exponential convergence of the block Gaussian Kaczmarz method.

Shows the method's numerical complexity often makes it less efficient than alternatives.

Identifies specific scenarios where the method's regularization benefits are advantageous.

Abstract

The Kaczmarz algorithm is one of the most popular methods for solving large-scale over-determined linear systems due to its simplicity and computational efficiency. This method can be viewed as a special instance of a more general class of sketch and project methods. Recently, a block Gaussian version was proposed that uses a block Gaussian sketch, enjoying the regularization properties of Gaussian sketching, combined with the acceleration of the block variants. Theoretical analysis was only provided for the non-block version of the Gaussian sketch method. Here, we provide theoretical guarantees for the block Gaussian Kaczmarz method, proving a number of convergence results showing convergence to the solution exponentially fast in expectation. On the flip side, with this theory and extensive experimental support, we observe that the numerical complexity of each iteration typically makes this method inferior to other iterative projection methods. We highlight only one setting in which it may be advantageous, namely when the regularizing effect is used to reduce variance in the iterates under certain noise models and convergence for some particular matrix constructions.