Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusions

Xufeng Cai, Chaobing Song, Cristóbal A Guzmán, Jelena Diakonikolas

Keywords: stochastic, monotone inclusion, Halpern iteration, last iterate convergence, variance reduction, min-max optimization

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.

TL;DR: We provide state-of-the art variance reduced guarantees for all standard classes of stochastic Lipschitz monotone inclusion problems, using variants of Halpern iteration.