{
  "abstract": "We study the stochastic first-order oracle (SFO) complexity of finding $ε$-stationary points of $(L_0,L_1)$-smooth nonconvex functions. Under bounded-variance noise, existing analyses of normalized and clipped stochastic gradient descent (SGD) establish optimal dependence on $ε$ and $L_0$, but the jointly optimal dependence on $L_0$, $L_1$, and $ε$ remains unclear. We resolve this question for an unbiased SFO whose gradient noise has a $p$th central moment bounded by $σ^p$ for some $σ\\geq 0$, where $p\\in(1,2]$.Specifically, we establish an SFO lower bound of $Ω\\big(\\big(σ^{\\frac{p}{p-1}}ε^{-\\frac{p}{p-1}}+1\\big) (L_0 Δε^{-2}+L_1Δε^{-1})\\big)$, where $Δ>0$ is the initial optimality gap.This bound identifies an unavoidable contribution proportional to $L_1ε^{-1}Δ$. We also prove that mini-batch (smoothed) normalized SGD can attain a matching upper bound, thereby establishing the optimal SFO complexity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21692",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanpeng Liang"
    },
    {
      "name": "Luo Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a88g45cscq928.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21692",
  "fulltext_url": null,
  "id": "a88g45cscq928",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21692",
  "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": "W7222750316",
  "pdf_url": "https://arxiv.org/pdf/2610.21692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a88g45cscq928.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the Complexity of Finding Stationary Points in Stochastic $(L_0, L_1)$-Smooth Nonconvex Optimization",
  "title_text": "On the Complexity of Finding Stationary Points in Stochastic (L_0, L_1)-Smooth Nonconvex Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/a88g45cscq928",
  "venue": "arXiv",
  "versions": []
}