{
  "abstract": "We establish a randomized first-order lower bound for finding an $ε$-stationary point, $\\|\\nabla f(x)\\|\\le ε$, of a nonconvex function with initial gap at most $Δ$, $L_1$-Lipschitz gradient, and $L_2$-Lipschitz Hessian. Each oracle call returns the exact function value and gradient. Let $Q_{\\mathrm{rand},\\mathrm{FO}}^{\\infty}(ε;Δ,L_1,L_2)$ denote the minimax number of calls, maximized over finite dimensions, for arbitrary adaptive randomized algorithms with per-instance success probability at least $2/3$. In the regime $ε\\lesssim L_1^2/L_2$ and $ΔL_2^{1/2}ε^{-3/2}\\gtrsim 1$, we prove $ Q_{\\mathrm{rand},\\mathrm{FO}}^{\\infty}(ε;Δ,L_1,L_2) \\ge c\\,ΔL_1^{1/2}L_2^{1/4}ε^{-7/4}, $ where $c>0$ is an absolute constant. This matches the deterministic restarted accelerated-gradient upper bound of Li and Lin (2023) and extends the sharp deterministic lower bound of Zhou (2026) to unrestricted randomized algorithms. Thus, randomization does not improve the high-dimensional worst-case query rate. The construction keeps quadratic curvature visible while revealing the forcing directions sequentially. Large eigenspaces limit preprocessing, and a scheduled-disclosure coupling makes each fresh direction incur a separate cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00245",
  "author_total": 8,
  "authors": [
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Yanyi Li"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7zg8wb2r1mm7n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00245",
  "fulltext_url": null,
  "id": "7zg8wb2r1mm7n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00245",
  "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": "W7216583717",
  "pdf_url": "https://arxiv.org/pdf/2610.00245",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7zg8wb2r1mm7n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.00245",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00245"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Matching Lower Bounds for Randomized First-Order Methods in Hessian-Lipschitz Nonconvex Optimization",
  "title_text": "Matching Lower Bounds for Randomized First-Order Methods in Hessian-Lipschitz Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/7zg8wb2r1mm7n",
  "venue": "arXiv",
  "versions": []
}