A Tseng type stochastic forward-backward algorithm for monotone inclusions

TL;DR

This paper introduces a stochastic Tseng-type forward-backward algorithm with inertial terms for monotone inclusions, proving convergence and rate results in both general and specific cases, including saddle point problems.

Contribution

It develops a stochastic version of Tseng's forward-backward-forward method with inertial terms, providing convergence proofs and rate analysis for monotone inclusion problems.

Findings

Almost sure convergence in the general case

Rate of (1/n) in expectation for strong monotonicity

(1/n) rate for primal-dual gap in saddle point problems

Abstract

In this paper, we propose a stochastic version of the classical Tseng's forward-backward-forward method with inertial term for solving monotone inclusions given by the sum of a maximal monotone operator and a single-valued monotone operator in real Hilbert spaces. We obtain the almost sure convergence for the general case and the rate $\mathcal{O}(1/n)$ in expectation for the strong monotone case. Furthermore, we derive $\mathcal{O}(1/n)$ rate convergence of the primal-dual gap for saddle point problems.