{
  "abstract": "Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice. In contrast, we study nonconvex stochastic optimization under the joint relaxation of these assumptions: objectives with $(L,s)$-Hölder continuous gradients, $s\\in(0,1]$, and gradient noise satisfying only a bounded $α$-th moment condition for $α\\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $α\\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and Hölder smoothness simultaneously. Secondly, we analyze $δ$-regularized gradient clipping ($δ$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(α-1)/[(1+s)(2α-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $α\\ge1+s$ while in the very heavy-tailed regime $α<1+s$, it has a convergence rate $O(T^{-2s(α-1)/[(α-1)+s(2α-1)]})$ --- the first convergence guarantee in this regime for any stochastic gradient based method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12785",
  "author_total": 2,
  "authors": [
    {
      "name": "Misbah Uz Zaman"
    },
    {
      "name": "Anirbit Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9pfq791jybrb7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12785",
  "fulltext_url": null,
  "id": "9pfq791jybrb7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12785",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7212645724",
  "pdf_url": "https://arxiv.org/pdf/2609.12785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9pfq791jybrb7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Convergence of Stochastic Gradient Methods under Heavy-Tailed Noise and Hölder Smoothness",
  "title_text": "Convergence of Stochastic Gradient Methods under Heavy-Tailed Noise and Hölder Smoothness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:17Z",
  "url": "https://research.draconitic.com/paper/9pfq791jybrb7",
  "venue": "arXiv",
  "versions": []
}