On a Stochastic Differential Equation with Correction Term Governed by a Monotone and Lipschitz Continuous Operator

TL;DR

This paper studies a stochastic differential equation with a correction term for finding zeros of a monotone, Lipschitz operator, proving solution existence, convergence, and applying results to minimax problems and discretized algorithms.

Contribution

It establishes existence, uniqueness, and convergence of solutions for a stochastic differential equation with a correction term, and connects these results to discretized algorithms for minimax problems.

Findings

Solutions exist and are unique under given conditions.

Trajectories converge almost surely to zeros of the operator.

Discretized algorithms inherit convergence properties from the continuous model.

Abstract

In our pursuit of finding a zero for a monotone and Lipschitz continuous operator $M : \R^n \rightarrow \R^n$ amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction term. We present a result establishing the existence and uniqueness of solutions for the stochastic differential equations under examination. Additionally, assuming that the diffusion term is square-integrable, we demonstrate the almost sure convergence of the trajectory process $X(t)$ to a zero of $M$ and of $\|M(X(t))\|$ to $0$ as $t \rightarrow +\infty$. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for $\|M(X(t))\|^2$ and $\langle M(X(t), X(t)-x^*\rangle$, where $x^*$ is an arbitrary zero of the monotone operator. Subsequently, we apply these findings to a minimax problem. Finally, we analyze two temporal discretizations of the continuous-time models, resulting in stochastic variants of the Optimistic Gradient Descent Ascent and Extragradient methods, respectively, and assess their convergence properties.