Faster Rates For Federated Variational Inequalities

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.

💡 Research Summary

This paper addresses a fundamental gap in federated learning theory: while federated optimization for smooth convex objectives enjoys optimal convergence rates (O(1/(K R) + σ√(M K R) + σ^{2/3}K^{1/3}R^{2/3})), the analogous results for stochastic variational inequalities (VIs) have been far weaker. Existing works on federated VIs either restrict themselves to strongly monotone problems or rely on the Local Extra‑SGD (LESGD) algorithm, which inherits the extra‑gradient step from the classic Mirror‑Prox method. The best known bound for LESGD is O(1/√(K R) + σ√(M K R) + σ√(K R) + σ²R), which contains a term σ√(K R) that does not decay with the number of local steps K or the number of clients M, thus questioning the benefit of local updates.

The authors first revisit LESGD and provide a refined analysis that shows only the squared client‑drift term matters for VI convergence. By carefully bounding this term, they prove Theorem 1: under L‑smoothness and monotonicity, LESGD attains
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