{
  "abstract": "We establish tight randomized higher-order oracle complexity for finding first-order stationary points of nonconvex finite sums. Let $n$ be the number of components, $Δ>0$ the initial objective-gap bound, $L_p>0$ an individual $p$-th derivative Lipschitz bound, and $ε>0$ the target gradient norm. For every fixed integer $p\\ge 2$, the minimax number of exact component queries returning the value and all derivatives through order $p$, with success probability at least $2/3$, is \\[ Θ_p\\!\\left( n+ΔL_p^{1/p}n^{1-1/(2p)} ε^{-(p+1)/p} \\right), \\] where the constants depend only on $p$ and the worst case ranges over all finite dimensions. The lower bound holds for unrestricted randomized adaptive algorithms and closes the $\\sqrt{n}$ gap between the previously known general-order upper and lower bounds in their dependence on $n$. We extend dense weak hiding to complete higher-order replies while keeping each component's regularity independent of the chain length. The matching upper bound retains the known finite-sum exponent, requires only mean-squared $p$-th derivative increments, and removes the fixed-confidence logarithmic loss by verifying entire recursive-estimation epochs with exact function values. The characterization includes the additive $n$ term for every positive parameter regime; it counts oracle calls with unrestricted internal computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28202",
  "author_total": 8,
  "authors": [
    {
      "name": "Wendao Wu"
    },
    {
      "name": "Haihan Zhang"
    },
    {
      "name": "Chenheng Zhang"
    },
    {
      "name": "Yanyi Li"
    },
    {
      "name": "Chunyuan Zheng"
    },
    {
      "name": "Cong Fang"
    },
    {
      "name": "Haoxuan Li"
    },
    {
      "name": "Zhouchen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fxj5x56ddn6e1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28202",
  "fulltext_url": null,
  "id": "fxj5x56ddn6e1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28202",
  "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": "W7214155242",
  "pdf_url": "https://arxiv.org/pdf/2609.28202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fxj5x56ddn6e1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.28202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Matching Upper and Lower Bounds for Higher-Order Nonconvex Finite-Sum Optimization",
  "title_text": "Matching Upper and Lower Bounds for Higher-Order Nonconvex Finite-Sum Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/fxj5x56ddn6e1",
  "venue": "arXiv",
  "versions": []
}