This paper studies federated optimization for stochastic variational inequalities (VIs), a problem class whose convergence rates have trailed the known bounds for federated convex optimization. The authors narrow that gap with a series of improved guarantees for smooth and monotone VIs.

The first contribution is a refined analysis of the classical Local Extra SGD algorithm, yielding tighter convergence guarantees than previously established. The authors then identify an inherent limitation of Local Extra SGD: it can lead to excessive client drift. To address this, they propose the Local Inexact Proximal Point Algorithm with Extra Step (LIPPAX), which mitigates client drift and achieves improved guarantees in several regimes, including bounded Hessian, bounded operator, and low-variance settings. Finally, the results are extended to federated composite variational inequalities, where improved convergence guarantees are also established. The work was conducted at Georgia Institute of Technology and Apple.