# Ergodic convergence of a stochastic proximal point algorithm

arXiv: https://arxiv.org/abs/1504.05400 · HTML: https://mlsanity.com/paper/1504.05400/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Apr 2015.

## Built on by (most cited first, 12 of 12)

- [Coordinate Linear Variance Reduction for Generalized Linear Programming](https://mlsanity.com/paper/2111.01842/) (2021)
- [Nonconvex Stochastic Bregman Proximal Gradient Method with Application to Deep Learning](https://mlsanity.com/paper/2306.14522/) (2023)
- [Variance reduction techniques for stochastic proximal point algorithms](https://mlsanity.com/paper/2308.09310/) (2023)
- [On the Convergence of FedProx with Extrapolation and Inexact Prox](https://mlsanity.com/paper/2410.01410/) (2024)
- [Statistical inference with implicit SGD: proximal Robbins-Monro vs. Polyak-Ruppert](https://mlsanity.com/paper/2206.12663/) (2022)
- [Finding a stationary point of a stochastic convex problem](https://mlsanity.com/paper/2607.06883/) (2026)
- [The Stochastic Proximal Distance Algorithm](https://mlsanity.com/paper/2210.12277/) (2022)
- [Tighter Performance Theory of FedExProx](https://mlsanity.com/paper/2410.15368/) (2024)
- [An abstract effective convergence theorem for stochastic processes, with applications to stochastic approximation](https://mlsanity.com/paper/2504.12922/) (2025)
- [Mean-square and sublinear convergence of a stochastic proximal point algorithm in metric spaces of nonpositive curvature](https://mlsanity.com/paper/2510.10697/) (2025)
- [Convergence Rate of the Last Iterate of Stochastic Proximal Algorithms](https://mlsanity.com/paper/2602.05489/) (2026)
- [Dual-Anchor Acceleration Is Near-Optimal for Stochastic Monotone Root-Finding](https://mlsanity.com/paper/2609.36033/) (2026)

## Abstract (arXiv)

The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates (xn) given by xn+1 = (I+λn A(xin+1,,.,))-1(xn) where (A(s,,.,):s∈ E) is a collection of maximal monotone operators on a separable Hilbert space, (xin) is an independent identically distributed sequence of random variables on E and (λn) is a positive sequence in ℓ2backslash ℓ1. The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation mathbb E(A(xi1,,.,)) under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form min mathbb E(f(xi1,x)) w.r.t. x∈ bigcapi=1m Xi where f is a normal convex integrand and (Xi) 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.