Asymptotic Analysis of an Abstract Stochastic Scheme for Solving Monotone Inclusions

TL;DR

This paper introduces an abstract stochastic framework for solving monotone operator inclusion problems in Hilbert spaces, analyzing convergence of stochastic algorithms with multiple sources of randomness.

Contribution

It develops a unified stochastic scheme for monotone inclusions, allowing stochasticity in operator approximation, coordinate selection, and relaxation, with rigorous convergence analysis.

Findings

Proves almost sure convergence of stochastic proximal point algorithms.

Establishes $L^2$ convergence for randomized block-iterative methods.

Applies framework to systems with mixed monotone operators.

Abstract

We propose an abstract stochastic scheme for solving a broad range of monotone operator inclusion problems in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels in monotone operator splitting methods: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The analysis involves an abstract reduced inclusion model with two operators. At each iteration of the proposed scheme, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and $L^2$ convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations.