Variance reduction in stochastic methods for large-scale regularised least-squares problems

TL;DR

This paper introduces a variance reduction technique for stochastic estimators in large-scale regularised least-squares problems, combining deterministic and stochastic methods to improve efficiency without losing unbiasedness.

Contribution

It proposes a novel variance reduction method that merges stochastic estimators based on DPPs with deterministic gradient steps for regularised least-squares problems.

Findings

Variance can be significantly reduced with minimal additional cost.

The method maintains unbiasedness of the estimator.

Effective for Tikhonov regularization on graphs.

Abstract

Large dimensional least-squares and regularised least-squares problems are expensive to solve. There exist many approximate techniques, some deterministic (like conjugate gradient), some stochastic (like stochastic gradient descent). Among the latter, a new class of techniques uses Determinantal Point Processes (DPPs) to produce unbiased estimators of the solution. In particular, they can be used to perform Tikhonov regularization on graphs using random spanning forests, a specific DPP. While the unbiasedness of these algorithms is attractive, their variance can be high. We show here that variance can be reduced by combining the stochastic estimator with a deterministic gradient-descent step, while keeping the property of unbiasedness. We apply this technique to Tikhonov regularization on graphs, where the reduction in variance is found to be substantial at very small extra cost.