{
  "abstract": "We prove randomized matrix-vector query lower bounds for two normalized matrix-game geometries: a Euclidean unit ball against a probability simplex, with row norms at most one, and two probability simplices, with entries of absolute value at most one. Each query returns $(Ax,A^\\top y)$ for arbitrary real vectors. The algorithm must return a feasible pair with full saddle-point gap at most $\\varepsilon$, with probability at least $2/3$ on every admissible matrix. For sufficiently small $\\varepsilon$, the worst-case query complexities are $Ω(\\varepsilon^{-2/3}/(\\log^2(1/\\varepsilon)\\log\\log(1/\\varepsilon)))$ for ball-simplex games and $Ω(\\varepsilon^{-2/3}/(\\log^{7/3}(1/\\varepsilon)\\log\\log(1/\\varepsilon)))$ for simplex-simplex games. The hard instances have dimensions of order $\\varepsilon^{-2/3}$ and $\\varepsilon^{-2/3}/\\log^{1/3}(1/\\varepsilon)$, respectively, and the bounds extend to larger dimensions. These lower bounds match the deterministic upper bounds of Karmarkar, O'Carroll, and Sidford up to logarithmic factors. The proof extracts a fresh Gaussian core after adaptive two-sided queries and uses uncertainty in its smallest singular value to establish linear-system solve hardness. Two reductions transfer this hardness to matrix games by converting a small full gap into a small residual, with an additional logarithmic normalization loss only for simplex-simplex games.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02095",
  "author_total": 2,
  "authors": [
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Cong Fang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cj5dwjy807pde.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02095",
  "fulltext_url": null,
  "id": "cj5dwjy807pde",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02095",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7216342102",
  "pdf_url": "https://arxiv.org/pdf/2610.02095",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cj5dwjy807pde.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.02095",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02095"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Randomized Matvec Lower Bounds for Simplex-Based Matrix Games",
  "title_text": "Randomized Matvec Lower Bounds for Simplex-Based Matrix Games",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/cj5dwjy807pde",
  "venue": "arXiv",
  "versions": []
}