{
  "abstract": "What is the round complexity of Hessian-similar optimization when each round can contact at most $s$ of $m$ clients? Prior rates permit a free master or full participation and do not characterize strict caps. We count every one-vector query and reply. A hidden-client chain gives a lower bound for zero-respecting local algorithms with adaptive cohorts and random stopping: on strongly convex quadratics, the required rounds are $Ω((m/s)\\sqrt{\\min\\{δ_a,\\barδ/\\sqrt m\\}/μ}\\,\\log(1/ε))$ for $s\\leq(m-2)/2$ in the high-accuracy, high-dissimilarity regime. Here $μ$ is strong convexity, while $δ_a$ and $\\barδ$ bound anchor and average Hessian variation. To remove dependence on a poor anchor, virtual-master SVRS uses independent estimator and proximal cohorts and averages the proximal replies. With Catalyst, it reaches relative gap $ε$ in $\\tilde O(m/s+(\\sqrt{m/s}+m^{3/4}/s)\\sqrt{\\barδ/μ})$ expected rounds on quadratics with strongly convex clients, independent of $δ_a$. Combining this method with charged analyses of existing algorithms matches the lower bound's polynomial dependence for $s\\leq\\sqrt m$ and $s=Ω(m)$; the worst gap is $O(m^{1/8})$. A held-out 12-seed synthetic study finds fewer rounds than tuned SVRP in nine of ten dip-family incomplete-cohort cells, versus two of 30 other cells. The lower bounds concern zero-respecting algorithms and do not cover unrestricted messages.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33835",
  "author_total": 1,
  "authors": [
    {
      "name": "Rui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45hfw62h8ne2a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33835",
  "fulltext_url": null,
  "id": "45hfw62h8ne2a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33835",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7223787710",
  "pdf_url": "https://arxiv.org/pdf/2610.33835",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45hfw62h8ne2a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:49Z",
      "native_id": "2610.33835",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33835"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Hessian-Similar Optimization with Capped Participation: Lower Bounds and Virtual Masters",
  "title_text": "Hessian-Similar Optimization with Capped Participation: Lower Bounds and Virtual Masters",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:49Z",
  "url": "https://research.draconitic.com/paper/45hfw62h8ne2a",
  "venue": "arXiv",
  "versions": []
}