Worth Their Weight: Randomized and Regularized Block Kaczmarz Algorithms without Preprocessing

TL;DR

This paper introduces ReBlocK, a regularized version of the randomized block Kaczmarz algorithm, which converges reliably without preprocessing and outperforms existing methods on certain large-scale least-squares problems.

Contribution

It provides the first analysis of RBK with uniform sampling, introduces regularization to control bias and variance, and demonstrates improved performance over traditional methods.

Findings

ReBlocK converges to a weighted least-squares solution without preprocessing.

Regularization effectively controls bias and variance in RBK.

ReBlocK outperforms RBK and minibatch SGD on problems with rapidly decaying singular values.

Abstract

Due to the ever growing amounts of data leveraged for machine learning and scientific computing, it is increasingly important to develop algorithms that sample only a small portion of the data at a time. In the case of linear least-squares, the randomized block Kaczmarz method (RBK) is an appealing example of such an algorithm, but its convergence is only understood under sampling distributions that require potentially prohibitively expensive preprocessing steps. To address this limitation, we analyze RBK when the data is sampled uniformly, showing that its iterates converge in a Monte Carlo sense to a $\textit{weighted}$ least-squares solution. Unfortunately, for general problems the bias of the weighted least-squares solution and the variance of the iterates can become arbitrarily large. We show that these quantities can be rigorously controlled by incorporating regularization into the RBK iterations, yielding the regularized algorithm ReBlocK. Numerical experiments including examples arising from natural gradient optimization demonstrate that ReBlocK can outperform both RBK and minibatch stochastic gradient descent for inconsistent problems with rapidly decaying singular values.