# Incremental constraint projection methods for monotone stochastic variational inequalities

## TL;DR

This paper introduces an incremental constraint projection method for stochastic variational inequalities with monotone operators, achieving convergence and rate guarantees suitable for large-scale, online, and distributed applications.

## Contribution

It proposes a novel incremental projection approach combining stochastic approximation with constraint sampling, extending to weak-sharp and monotone cases with convergence rates.

## Key findings

- Achieves $O(1/k)$ feasibility rate in mean squared distance.
- Provides $O(1/\sqrt{k})$ solvability rate for weak-sharp cases.
- Extends to distributed stochastic Nash games with near-optimal convergence.

## Abstract

We consider stochastic variational inequalities with monotone operators defined as the expected value of a random operator. We assume the feasible set is the intersection of a large family of convex sets. We propose a method that combines stochastic approximation with incremental constraint projections meaning that at each iteration, a step similar to some variant of a deterministic projection method is taken after the random operator is sampled and a component of the intersection defining the feasible set is chosen at random. Such sequential scheme is well suited for applications involving large data sets, online optimization and distributed learning. First, we assume that the variational inequality is weak-sharp. We provide asymptotic convergence, feasibility rate of $O(1/k)$ in terms of the mean squared distance to the feasible set and solvability rate of $O(1/\sqrt{k})$ (up to first order logarithmic terms) in terms of the mean distance to the solution set for a bounded or unbounded feasible set. Then, we assume just monotonicity of the operator and introduce an explicit iterative Tykhonov regularization to the method. We consider Cartesian variational inequalities so as to encompass the distributed solution of stochastic Nash games or multi-agent optimization problems under a limited coordination. We provide asymptotic convergence, feasibility rate of $O(1/k)$ in terms of the mean squared distance to the feasible set and, in the case of a compact set, we provide a near-optimal solvability convergence rate of $O\left(\frac{k^\delta\ln k}{\sqrt{k}}\right)$ in terms of the mean dual gap-function of the SVI for arbitrarily small $\delta>0$.