{
  "abstract": "Uniform noise-moment bounds exclude stochastic gradients whose variability increases with the iterate. We study ordinary, single-sample stochastic gradient descent for smooth, lower-bounded, possibly nonconvex objectives under distance-dependent conditional moments. Under second moments alone, a direct descent--displacement argument yields $T^{-1/3}$ expected average squared-gradient stationarity with a horizon-dependent stepsize. An explicit oracle-complexity corollary matches the known smooth Blum--Gladyshev (BG-0) lower bound, including the $Lb_2Δ^3\\varepsilon^{-6}$ and $LΔσ^2\\varepsilon^{-4}$ stochastic terms, where $Δ$ is the initial objective gap and $σ^2+b_2\\|x-x_1\\|^2$ bounds the variance. Thus unchanged SGD attains the minimax stochastic complexity in this second-moment class. For $p>2$, predictable localization and a Hilbert-space Fuk--Nagaev inequality yield a high-probability bound separating logarithmic variance and polynomial rare-shock contributions. The localization radius is derived from the recursion: no bounded-iterate assumption, clipping, normalization, momentum, or increasing batch size is needed. We also give increasing-confidence rates, an objective-gap-growth refinement recovering root-$T$ stationarity, and stochastic $L^p$-Lipschitz examples. The broad BG-0 optimality statement is distinguished from the smaller mean-square-smooth class, in which additional oracle structure permits faster algorithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30499",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Biao Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drmv6ex5vfxs6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30499",
  "fulltext_url": null,
  "id": "drmv6ex5vfxs6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30499",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214658225",
  "pdf_url": "https://arxiv.org/pdf/2609.30499",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drmv6ex5vfxs6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:48Z",
      "native_id": "2609.30499",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30499"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Ordinary Nonconvex SGD under Distance-Dependent Moments: Finite-Horizon Stationarity and Nagaev Bounds",
  "title_text": "Ordinary Nonconvex SGD under Distance-Dependent Moments: Finite-Horizon Stationarity and Nagaev Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:48Z",
  "url": "https://research.draconitic.com/paper/drmv6ex5vfxs6",
  "venue": "arXiv",
  "versions": []
}