Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators

TL;DR

This paper analyzes the convergence of a stochastic forward-backward splitting algorithm for monotone operators, accommodating stochastic approximations, perturbations, relaxations, and non-vanishing parameters, with applications to primal-dual methods.

Contribution

It introduces a stochastic forward-backward splitting framework with relaxed conditions and proves convergence results, extending existing algorithms to stochastic and more general settings.

Findings

Proves almost sure convergence of stochastic forward-backward splitting.

Establishes convergence for stochastic primal-dual proximal methods.

Handles non-vanishing proximal parameters and stochastic perturbations.

Abstract

We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems.