Faster Rates For Federated Variational Inequalities

TL;DR

This paper advances federated optimization for stochastic variational inequalities by developing tighter convergence guarantees and introducing the LIPPAX algorithm to mitigate client drift, applicable across various regimes.

Contribution

It provides improved convergence rates for federated variational inequalities and introduces the LIPPAX algorithm to address client drift issues.

Findings

Tighter convergence guarantees for Local Extra SGD.

LIPPAX algorithm reduces client drift.

Enhanced results for federated composite variational inequalities.

Abstract

In this paper, we study federated optimization for solving stochastic variational inequalities (VIs), a problem that has attracted growing attention in recent years. Despite substantial progress, a significant gap remains between existing convergence rates and the state-of-the-art bounds known for federated convex optimization. In this work, we address this limitation by establishing a series of improved convergence rates. First, we show that, for general smooth and monotone variational inequalities, the classical Local Extra SGD algorithm admits tighter guarantees under a refined analysis. Next, we identify an inherent limitation of Local Extra SGD, which can lead to excessive client drift. Motivated by this observation, we propose a new algorithm, the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), and show that it mitigates client drift and achieves improved guarantees in several regimes, including bounded Hessian, bounded operator, and low-variance settings. Finally, we extend our results to federated composite variational inequalities and establish improved convergence guarantees.