{
  "abstract": "We study the deterministic first-order oracle complexity of finding \\(ε\\)-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical \\(ε^{-2}\\) rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the \\(ε^{-7/4}\\) rate under Lipschitz Hessians and the \\(ε^{-5/3}\\) rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We resolve this gap by proving a new dimension-free first-order lower bound for higher-order smooth nonconvex functions, valid for every finite smoothness order. In particular, our construction gives a matching \\(Ω(ε^{-7/4})\\) lower bound in the Hessian-Lipschitz case and a matching \\(Ω(ε^{-5/3})\\) lower bound in the third-order-smooth regime. The hard instance is based on a \\emph{block-chain} mechanism that enforces blockwise oracle revelation while preserving the smoothness structure needed for the scalar hard instance. The lower-bound construction was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.05438",
  "author_total": 1,
  "authors": [
    {
      "name": "Dongruo Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1deajfads085k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.05438",
  "fulltext_url": null,
  "id": "1deajfads085k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.05438",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7163636643",
  "pdf_url": "https://arxiv.org/pdf/2606.05438",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1deajfads085k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2606.05438",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.05438"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization",
  "title_text": "Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/1deajfads085k",
  "venue": "arXiv",
  "versions": []
}