Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusions

TL;DR

This paper introduces novel stochastic Halpern iteration algorithms with variance reduction for solving stochastic monotone inclusion problems, achieving improved convergence rates over existing methods in machine learning applications.

Contribution

The paper develops new stochastic Halpern iteration variants with variance reduction, improving convergence rates for stochastic monotone inclusions under Lipschitz and strong monotonicity assumptions.

Findings

Achieves $rac{1}{ ext{epsilon}^3}$ complexity for Lipschitz-monotone cases.

Improves over the previous $rac{1}{ ext{epsilon}^4}$ complexity.

Provides a restart scheme reducing complexity to $rac{ ext{log}(1/ ext{epsilon})}{ ext{epsilon}^2}$ under certain conditions.

Abstract

We study stochastic monotone inclusion problems, which widely appear in machine learning applications, including robust regression and adversarial learning. We propose novel variants of stochastic Halpern iteration with recursive variance reduction. In the cocoercive -- and more generally Lipschitz-monotone -- setup, our algorithm attains $\epsilon$ norm of the operator with $\mathcal{O}(\frac{1}{\epsilon^3})$ stochastic operator evaluations, which significantly improves over state of the art $\mathcal{O}(\frac{1}{\epsilon^4})$ stochastic operator evaluations required for existing monotone inclusion solvers applied to the same problem classes. We further show how to couple one of the proposed variants of stochastic Halpern iteration with a scheduled restart scheme to solve stochastic monotone inclusion problems with ${\mathcal{O}}(\frac{\log(1/\epsilon)}{\epsilon^2})$ stochastic operator evaluations under additional sharpness or strong monotonicity assumptions.