{
  "abstract": "We study the component-oracle complexity of finding approximate local minima of nonconvex finite sums under mean-squared averaged smoothness with parameter $L$ and a $ρ$-Lipschitz average Hessian. Each query returns the value, gradient, and one Hessian-vector product of a component. We show that the complexity in the high-accuracy regime $\\varepsilon\\le c_0L^2/(nρ)$ is $\\tildeΘ(n^{3/4}L^{1/2}ρ^{1/4}Δ\\varepsilon^{-7/4})$ for a sufficiently large initial gap $Δ$. We attain the upper bound by adapting the inverse solver and eigenvector routine of FastCubic to averaged smoothness. Our lower bound assigns the bridge edges of a nested chain to random components; a replicated spectral construction gives the same bound for unrestricted randomized algorithms. Under individual smoothness, a factor of $n^{1/4}$ remains for $\\varepsilon\\le cL^2/(n^2ρ)$, up to logarithmic factors. In the complementary low-accuracy regime, we propose PathVR, which uses $\\tilde O(n+\\sqrt{n}LΔ\\varepsilon^{-2})$ queries and matches a lower bound from dense weak hiding.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05807",
  "author_total": 1,
  "authors": [
    {
      "name": "Qihao Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7a3sv4m20x1wq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05807",
  "fulltext_url": null,
  "id": "7a3sv4m20x1wq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05807",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05807",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7a3sv4m20x1wq.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05807",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05807"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Near-Optimal Complexity of Finding Approximate Local Minima of Nonconvex Finite Sums",
  "title_text": "Near-Optimal Complexity of Finding Approximate Local Minima of Nonconvex Finite Sums",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/7a3sv4m20x1wq",
  "venue": "arXiv",
  "versions": []
}