{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12641",
  "author_total": 4,
  "authors": [
    {
      "name": "Kirill Novoselov"
    },
    {
      "name": "Yuriy Dorn"
    },
    {
      "name": "Darina Dvinskikh"
    },
    {
      "name": "Alexander Gasnikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2f9j0maes61td.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12641",
  "fulltext_url": null,
  "id": "2f9j0maes61td",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12641",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221527663",
  "pdf_url": "https://arxiv.org/pdf/2610.12641",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2f9j0maes61td.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption",
  "title_text": "Optimal Rates for Federated Variational Inequalities under different relaxations of bounded variance assumption",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/2f9j0maes61td",
  "venue": "arXiv",
  "versions": []
}