{
  "abstract": "We establish tight polynomial tolerance bounds for stochastic second-order stationarity when each fresh oracle response is a derivative of one common random scalar loss. For a population objective $F$ with Lipschitz gradient and Hessian, the target is $\\|\\nabla F(x)\\|\\le ε$ and $λ_{\\min}(\\nabla^2 F(x))\\ge -γ$, with independent tolerances $ε,γ>0$. Under bounded gradient variance and almost-surely bounded Hessian error, the minimax number of fresh gradient or Hessian-vector-product calls is $\\widetildeΘ\\!\\left(ε^{-3}+γ^{-5}\\right)$. The characterization fixes positive gap, smoothness, and noise parameters, suppresses logarithmic factors, and allows dimension to grow within an explicit polynomial envelope. The upper bound removes the mixed term $ε^{-2}γ^{-2}$ from the earlier fresh-HVP guarantee. Direct random-line Hessian estimates and a dyadic gradient tracker separate gradient drift from randomly signed curvature motion. The lower bound realizes the endpoint costs through globally defined smooth random losses: a smooth partition localizes scalar noise without a chain-length penalty, while exact cancellation limits the information in the entire response. Consequently, the same tolerance exponents hold even for joint value, gradient, and full-Hessian responses with bounded value variance. For a population Hessian with Hölder exponent $ν\\in(0,1]$, fresh gradient/HVP complexity becomes $\\widetildeΘ\\!\\left(ε^{-3}+γ^{-(3+2/ν)}\\right)$ under the corresponding dimension envelope.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28238",
  "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/cdt8q7srb8xda.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28238",
  "fulltext_url": null,
  "id": "cdt8q7srb8xda",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28238",
  "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": "W7214203671",
  "pdf_url": "https://arxiv.org/pdf/2609.28238",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cdt8q7srb8xda.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2609.28238",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28238"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Second-Order Stationarity with Common Random Losses: Matching Tolerance Bounds",
  "title_text": "Second-Order Stationarity with Common Random Losses: Matching Tolerance Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/cdt8q7srb8xda",
  "venue": "arXiv",
  "versions": []
}