{
  "abstract": "We characterize the oracle complexity of finding first-order stationary points in smooth nonconvex--strongly-concave (NC--SC) minimax optimization when each query provides the full Hessian in addition to the function value and gradient. For $Φ(x)=\\max_y f(x,y)$, assume that $f$ has an $L$-Lipschitz gradient, a $ρ$-Lipschitz Hessian, and is $μ$-strongly concave in $y$. Let $κ=L/μ$, let $Δ$ denote the initial value gap, and define $r=ρκ^{3/2}\\varepsilon/L^2$. We establish matching deterministic upper bounds and randomized lower bounds, up to logarithmic factors and additive initialization costs. When the leading term dominates initialization, the resulting minimax complexity is \\[ \\widetildeΘ\\!\\left( \\frac{ΔL}{\\varepsilon^2}Ψ_κ(r) \\right), \\qquad Ψ_κ(r)= \\begin{cases} r^{1/2}, & 0<r\\le 1,\\\\ r^{2/7}, & 1<r<κ^{7/4},\\\\ \\sqrtκ, & r\\geκ^{7/4}. \\end{cases} \\] Despite the stronger oracle, the optimal complexity exhibits a nontrivial dependence on accuracy, conditioning, and Hessian smoothness. In particular, these regimes exhibit accuracy dependencies $\\varepsilon^{-3/2}$, $\\varepsilon^{-12/7}$, and $\\varepsilon^{-2}$, respectively. The upper bounds combine second-order regularity estimates for a saddle Moreau envelope with certified accelerated reduced-dual solves and amortized warm starts. The lower bounds combine an all-derivative lift with a serial composition of private-handle constructions, using hard instances chosen independently of the algorithm's realized random seed. The resulting characterization quantifies improvements over the classical first-order benchmark under gradient smoothness alone and identifies the regime in which that benchmark is recovered.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16771",
  "author_total": 8,
  "authors": [
    {
      "name": "Fangqing Zhang"
    },
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6p5x6b28m8wn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16771",
  "fulltext_url": null,
  "id": "6p5x6b28m8wn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16771",
  "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": "W7221803062",
  "pdf_url": "https://arxiv.org/pdf/2610.16771",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6p5x6b28m8wn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.16771",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16771"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Matching Full-Hessian Oracle Complexity for Nonconvex--Strongly-Concave Minimax Optimization",
  "title_text": "Matching Full-Hessian Oracle Complexity for Nonconvex--Strongly-Concave Minimax Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/6p5x6b28m8wn5",
  "venue": "arXiv",
  "versions": []
}