{
  "abstract": "For smooth convex--concave minimax optimization, the higher-order lower bound of Chen et al. (2026) applies to a restricted tensor-algorithm class with prescribed regularized Taylor-model updates. We establish the same bound for arbitrary adaptive deterministic and randomized algorithms, matching, up to logarithmic factors, the upper bound of Zhang et al. (2026). Fix an integer $p\\ge 2$ and let $L_p>0$ bound the Lipschitz constant of the objective's $p$-th derivative on a compact convex product domain of diameter at most $D_Z>0$. Each feasible query returns the objective value and all derivatives through order $p$. For accuracy $ε>0$, set $Q_{\\mathrm{tan}}=L_pD_Z^p/ε$ for tangent residual and $Q_{\\mathrm{gap}}=L_pD_Z^{p+1}/ε$ for saddle gap. Let $T_E^{\\mathrm{det}}(ε)$ and $T_E^{\\mathrm{rand}}(ε)$ denote the high-dimensional minimax query complexities for criterion $E\\in\\{\\mathrm{tan},\\mathrm{gap}\\}$, with randomized success probability at least $2/3$ on every instance. Our lower bounds and the existing upper bound give $c_pQ_E^{2/(3p-1)}\\le T_E^{\\mathrm{rand}}(ε)\\le T_E^{\\mathrm{det}}(ε)\\le C_pQ_E^{2/(3p-1)}[1+\\log(3+Q_E)]^{6(p-1)}$ for sufficiently large $Q_E$, where $c_p,C_p>0$ depend only on $p$. Thus the same accuracy exponent holds beyond tensor update rules, even for randomized queries and arbitrary feasible outputs. The proof constructs a scalar convex--concave chain with exactly flat gates that hide complete derivative information. Direct product-domain error witnesses and adaptive transcript arguments establish the lower bounds for both criteria.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28246",
  "author_total": 8,
  "authors": [
    {
      "name": "Yanyi Li"
    },
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33x8mmkw29mbc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28246",
  "fulltext_url": null,
  "id": "33x8mmkw29mbc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28246",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214184962",
  "pdf_url": "https://arxiv.org/pdf/2609.28246",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33x8mmkw29mbc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2609.28246",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28246"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Near-Optimal Higher-Order Oracle Complexity for Convex--Concave Minimax Optimization",
  "title_text": "Near-Optimal Higher-Order Oracle Complexity for Convex--Concave Minimax Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/33x8mmkw29mbc",
  "venue": "arXiv",
  "versions": []
}