Stochastic Proximal Point Methods for Monotone Inclusions under Expected Similarity

TL;DR

This paper introduces stochastic proximal point algorithms for solving monotone inclusions with a new notion of similarity between operators, achieving linear convergence under strong monotonicity.

Contribution

It proposes novel stochastic algorithms with variance reduction for monotone inclusions, incorporating a new similarity measure to handle discontinuous operators.

Findings

Algorithms achieve linear convergence under strong monotonicity.

The new similarity notion applies even to discontinuous operators.

Variance reduction improves convergence efficiency.

Abstract

Monotone inclusions have a wide range of applications, including minimization, saddle-point, and equilibria problems. We introduce new stochastic algorithms, with or without variance reduction, to estimate a root of the expectation of possibly set-valued monotone operators, using at every iteration one call to the resolvent of a randomly sampled operator. We also introduce a notion of similarity between the operators, which holds even for discontinuous operators. We leverage it to derive linear convergence results in the strongly monotone setting.