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arXiv · 2610.12641

Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption

Kirill Novoselov·Yuriy Dorn·Darina Dvinskikh·Alexander Gasnikov
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Abstract

We study how interpolation affects local steps and parallel sampling in monotone variational inequalities (VIs), with $m$ workers, $K$ oracle calls per worker per round, and $N$ rounds. Under the strong growth condition (SGC), we determine the minimax rate for affine VIs, attained by local extragradient with worker averaging. For nonlinear monotone VIs and convex-concave saddle problems with Lipschitz operators satisfying SGC, we improve the stochastic terms of bounded-variance expected-gap guarantees from $(KN)^{-1/2}$ to $(KN)^{-1}$ with local steps and from $(mKN)^{-1/2}$ to $(mKN)^{-1}$ with minibatches. Under strong monotonicity, we improve the noise-dependent condition-number scaling of existing extragradient bounds from quadratic to linear. The resulting monotone-gap and strongly monotone accuracy bounds match lower bounds up to logarithmic factors in $m$. Under generalized smoothness and SGC, adaptive steps give polynomial dependence on the initial radius. For affine VIs with globally bounded oracle variance, we also attain the minimax rate: the deterministic rate over $KN$ sequential calls plus the statistical rate over all $mKN$ samples. For composite VIs with smooth convex sample interpolation and an exactly evaluated monotone Lipschitz component, accelerated mirror-prox sliding achieves an $N^{-2}$ gradient term, a $(KN)^{-1}$ monotone term, and a $(mKN)^{-1}$ sampling term, without assuming SGC.

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Kirill Novoselov, Yuriy Dorn, Darina Dvinskikh, Alexander Gasnikov. 2026-10-08. Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption. https://arxiv.org/abs/2610.12641

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