A simple linear convergence analysis of the randomized reshuffling Kaczmarz method

TL;DR

This paper provides a straightforward linear convergence analysis of the randomized reshuffling Kaczmarz method, demonstrating its efficiency and tight bounds independent of matrix dimension.

Contribution

It offers the first comprehensive linear convergence proof for RRK and establishes a tight upper bound that is dimension-independent.

Findings

RRK converges linearly to the least-norm solution.

The convergence upper bound is tight and dimension-independent.

The analysis enhances understanding of RRK's efficiency in solving linear systems.

Abstract

The random reshuffling Kaczmarz (RRK) method enjoys the simplicity and efficiency in solving linear systems as a Kaczmarz-type method, whereas it also inherits the practical improvements of the stochastic gradient descent (SGD) with random reshuffling (RR) over original SGD. However, the current studies on RRK do not characterize its convergence comprehensively. In this paper, we present a novel analysis of the RRK method and prove its linear convergence towards the unique least-norm solution of the linear system. Furthermore, the convergence upper bound is tight and does not depend on the dimension of the coefficient matrix.