Selectable Set Randomized Kaczmarz

TL;DR

This paper introduces Selectable Set Randomized Kaczmarz (SSRK), an adaptive variant of RK that improves convergence by using problem-specific information, supported by theoretical guarantees and numerical experiments.

Contribution

The paper proposes a general framework for selectable set approaches in RK, introduces two new sampling strategies, and provides convergence analysis and empirical validation.

Findings

Selectable set strategies have competitive convergence guarantees.

Information about previous iterates improves convergence.

Orthogonality-based sampling leverages Gramian structure effectively.

Abstract

The Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz (SSRK) is an variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive "selectable set" and thus yields an improved convergence guarantee. In this paper, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.