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Asymptotic Analysis of Stochastic Splitting Methods for Multivariate Monotone Inclusions

Patrick L. Combettes, Javier I. Madariaga

TL;DR

This work develops a general stochastic framework for convergence of monotone operator splitting methods in Hilbert spaces, accommodating stochastic operator approximations, random coordinate/block updates, and random relaxations. It proves almost surely and in $L^2$ convergence for stochastic proximal point variants and randomized block-iterative saddle projective splitting applied to multivariate inclusions that combine set-valued, cocoercive, and Lipschitzian operators. By leveraging a two-operator abstract model and a robust probabilistic toolkit, the paper unifies and extends deterministic splitting theories to stochastic settings and provides convergence guarantees for complex multivariate minimization problems and randomized Kuhn–Tucker schemes. The results tolerate summable stochastic errors and even allow super-relaxation parameters, broadening applicability to large-scale stochastic optimization and domain-decomposed systems. Overall, the framework significantly enhances the toolkit for solving stochastic monotone inclusions with structured, high-dimensional block architectures.

Abstract

We propose an abstract framework to establish the convergence of the iterates of stochastic versions of a broad range of monotone operator splitting methods in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The proposed analysis involves a reduced inclusion model with two operators. At each iteration, 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.

Asymptotic Analysis of Stochastic Splitting Methods for Multivariate Monotone Inclusions

TL;DR

This work develops a general stochastic framework for convergence of monotone operator splitting methods in Hilbert spaces, accommodating stochastic operator approximations, random coordinate/block updates, and random relaxations. It proves almost surely and in convergence for stochastic proximal point variants and randomized block-iterative saddle projective splitting applied to multivariate inclusions that combine set-valued, cocoercive, and Lipschitzian operators. By leveraging a two-operator abstract model and a robust probabilistic toolkit, the paper unifies and extends deterministic splitting theories to stochastic settings and provides convergence guarantees for complex multivariate minimization problems and randomized Kuhn–Tucker schemes. The results tolerate summable stochastic errors and even allow super-relaxation parameters, broadening applicability to large-scale stochastic optimization and domain-decomposed systems. Overall, the framework significantly enhances the toolkit for solving stochastic monotone inclusions with structured, high-dimensional block architectures.

Abstract

We propose an abstract framework to establish the convergence of the iterates of stochastic versions of a broad range of monotone operator splitting methods in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The proposed analysis involves a reduced inclusion model with two operators. At each iteration, 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 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.

Paper Structure

This paper contains 13 sections, 14 theorems, 102 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. General notation
  4. Operators
  5. Probabilistic setting
  6. Preliminary results
  7. Convergence analysis
  8. Stochastic proximal point algorithm
  9. Stochastic block-iterative saddle projective splitting
  10. Problem setting
  11. Algorithm and convergence
  12. Application to multivariate minimization
  13. Stochastic block-iterative Kuhn--Tucker projective splitting

Key Result

Lemma 2.1

Let $(x_{{\mathsf{n}}})_{\mathsf{n}\in\mathbb{N}}$ and $x$ be $\mathsf{H}$-valued random variables and let $\mathsf{p}\in\left[1,{+}\infty\right[$ be such that $(x_{{\mathsf{n}}})_{\mathsf{n}\in\mathbb{N}}$ converges strongly in $L^{\mathsf{p}}(\upOmega,\EuScript{F},\mathsf{P};\mathsf{H})$ to $x$. T

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Lemma 2.7: Sadd22
  • ...and 18 more