{
  "abstract": "Does randomization improve the worst-case first-order complexity of nonconvex optimization when the objective is very smooth? We prove that, for every fixed finite integer $p\\ge3$, finding an $\\eps$-stationary point requires $Ω_p(\\eps^{-5/3})$ exact value-and-gradient queries, even for $C^\\infty$ objectives with initial gap at most one and simultaneous global Lipschitz budgets of one for derivatives of orders $1,\\ldots,p$. Together with the known deterministic upper bound, this determines the accuracy exponent up to a single logarithmic factor. The lower bound holds for unrestricted adaptive randomized algorithms, unbounded query locations, and outputs that need not have been queried. One algorithm-independent hard distribution suffices in dimension $O_p(\\eps^{-47/3})$. The construction combines a block-chain potential with an exact quartic core, repeated quadratic spectra, and two static Gaussian tensor perturbations. A scheduled Haar argument controls the entire adaptive query span. An innovation-only Gaussian comparison then bounds exact transcript information without assuming finite precision or a minimum query separation. This establishes the $5/3$ randomized accuracy barrier under every prescribed finite smoothness order, extending the previously matched third-order case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17074",
  "author_total": 8,
  "authors": [
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Fangqing Zhang"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ed13r1ctfxj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17074",
  "fulltext_url": null,
  "id": "2ed13r1ctfxj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17074",
  "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": "W7221813324",
  "pdf_url": "https://arxiv.org/pdf/2610.17074",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ed13r1ctfxj1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.17074",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17074"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Matching Randomized First-Order Bounds for Higher-Order Smooth Nonconvex Optimization",
  "title_text": "Matching Randomized First-Order Bounds for Higher-Order Smooth Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/2ed13r1ctfxj1",
  "venue": "arXiv",
  "versions": []
}