{
  "abstract": "We study the query complexity of finding approximate stationary points under higher-order smoothness using only function values or only gradients. Our results cover different dimensions, oracle models, randomization, and constraints. For constant dimension $d$ and $C^{p,β_p}$ objectives, with $ν:=p+β_p$, we give deterministic function-value gradient trapping algorithms with query complexities $O(\\varepsilon^{-(d-1)})$ on $\\mathbb R^d$ and $O(\\varepsilon^{-(d-1)/ν})$ on $[0,1]^d$. High-order line integration enables gradient-only trapping. For two-dimensional $C^{1,1}$ objectives, randomized integration yields $O(\\varepsilon^{-4/3})$ gradient queries on $\\mathbb R^2$, breaking the deterministic gradient-only $Θ(\\varepsilon^{-2})$ barrier. For general dimension, we introduce a trapping-or-descent framework that lazily reuses boundary models. For $C^{p,β_p}$ objectives with $p\\ge2$, it achieves deterministic query complexities $O(d^p+d^{p-1+β_p/(ν-1)} \\varepsilon^{-ν/(ν-1)})$ using function values and $O(d^{p-1}+d^{p-2+β_p/(ν-1)} \\varepsilon^{-ν/(ν-1)})$ using gradients. These bounds apply to approximate stationary points on $\\mathbb R^d$ and approximate KKT points on $[0,1]^d$. Their dimension-independent accuracy exponent matches the classical AR$p$ exponent $(p+1)/p$ when $β_p=1$. For sufficiently high dimension, we prove randomized lower bounds for gradient-only methods. For $C^{1,1}\\cap C^{p,1}$ objectives, our lower bound is $Ω(\\varepsilon^{-(3/2+1/(2p))})$, matching known upper bounds for $p=2$ and, up to logarithmic factors, for $p=3$. Without the $C^{1,1}$ assumption, we obtain an $Ω(\\varepsilon^{-5/3})$ lower bound for every fixed $p\\ge3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02791",
  "author_total": 3,
  "authors": [
    {
      "name": "Huanjian Zhou"
    },
    {
      "name": "Masashi Sugiyama"
    },
    {
      "name": "Taiji Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2bn52r4qgw486.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02791",
  "fulltext_url": null,
  "id": "2bn52r4qgw486",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02791",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7220466417",
  "pdf_url": "https://arxiv.org/pdf/2610.02791",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2bn52r4qgw486.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02791",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02791"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Finding Stationary Points under Higher-Order Smoothness: Trapping Algorithms and First-Order Lower Bounds",
  "title_text": "Finding Stationary Points under Higher-Order Smoothness: Trapping Algorithms and First-Order Lower Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/2bn52r4qgw486",
  "venue": "arXiv",
  "versions": []
}