Ergodic convergence of a stochastic proximal point algorithm

TL;DR

This paper proves the almost sure weak ergodic convergence of a stochastic proximal point algorithm involving maximal monotone operators, with applications to stochastic convex optimization problems.

Contribution

It establishes the weak ergodic convergence of a stochastic proximal point algorithm under broad conditions, extending previous results to a more general setting.

Findings

Weighted averaged iterates converge weakly to a zero of the Aumann expectation.

Convergence holds under the assumption that the Aumann expectation is maximal.

Applications to stochastic convex optimization problems are demonstrated.

Abstract

The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates $(x_n)$ given by $x_{n+1} = (I+\lambda_n A(\xi_{n+1},\,.\,))^{-1}(x_n)$ where $(A(s,\,.\,):s\in E)$ is a collection of maximal monotone operators on a separable Hilbert space, $(\xi_n)$ is an independent identically distributed sequence of random variables on $E$ and $(\lambda_n)$ is a positive sequence in $\ell^2\backslash \ell^1$. The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation ${\mathbb E}(A(\xi_1,\,.\,))$ under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form $\min {\mathbb E}(f(\xi_1,x))$ w.r.t. $x\in \bigcap_{i=1}^m X_i$ where $f$ is a normal convex integrand and $(X_i)$ is a collection of closed convex sets. In this case, the iterations are closely related to a stochastic proximal algorithm recently proposed by Wang and Bertsekas.