Dynamical behavior of a stochastic forward-backward algorithm using random monotone operators

TL;DR

This paper analyzes the long-term behavior of a stochastic forward-backward algorithm with random monotone operators, showing convergence properties and applications in optimization under randomness.

Contribution

It introduces a framework for studying the dynamical behavior of stochastic forward-backward algorithms involving random maximal monotone operators, including convergence analysis.

Findings

Interpolated process is an asymptotic pseudo trajectory of a differential inclusion.

Empirical means of iterates converge to a zero of the sum of mean operators.

Sequence converges to a zero under demipositivity assumption.

Abstract

The purpose of this paper is to study the dynamical behavior of the sequence produced by a forward-backward algorithm involving two random maximal monotone operators and a sequence of decreasing step sizes. Defining a mean monotone operator as an Aumann integral, and assuming that the sum of the two mean operators is maximal (sufficient maximality conditions are provided), it is shown that with probability one, the interpolated process obtained from the iterates is an asymptotic pseudo trajectory in the sense of Bena\"{\i}m and Hirsch of the differential inclusion involving the sum of the mean operators. The convergence of the empirical means of the iterates towards a zero of the sum of the mean operators is shown, as well as the convergence of the sequence itself to such a zero under a demipositivity assumption. These results find applications in a wide range of optimization or variational inequality problems in random environments.