{
  "abstract": "Stochastic gradient descent (SGD) is widely used to train machine learning models, but subsampling the training data introduces noise into its updates. The strength and applicability of high-probability guarantees therefore depend critically on how the tails of gradient noise are modeled. Reports of heavy-tailed gradient noise in deep learning motivate relaxing the bounded-noise and sub-Gaussian assumptions commonly used in high-probability analyses of SGD. We use Young functions from Orlicz space theory to describe noise tails in a common framework. We model SGD gradient noise by adopting a Young function that preserves the finiteness of all polynomial moments while allowing tails heavier than sub-Weibull, including lognormal distributions. The resulting class is called $β$-heavy-tailed, with $β$ controlling the tail heaviness. We establish concentration inequalities for $β$-heavy-tailed noise and combine them with a uniform bound on the difference between empirical and population gradients along the SGD trajectory to obtain high-probability bounds on optimization and population-risk stationarity for smooth nonconvex losses under trajectory assumptions. The bounds are not restricted to a particular learning-rate decay rule and make explicit the effects of noise tails and learning-rate schedules. Under the Polyak-Łojasiewicz condition, we bound the risk at the last iterate. We also analyze SGD with gradient clipping under the $β$-heavy-tailed noise model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32195",
  "author_total": 3,
  "authors": [
    {
      "name": "Qijun Tong"
    },
    {
      "name": "Masahiro Ikeda"
    },
    {
      "name": "Ryota Kawasumi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9n06xf08d8d0h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32195",
  "fulltext_url": null,
  "id": "9n06xf08d8d0h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32195",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214791782",
  "pdf_url": "https://arxiv.org/pdf/2609.32195",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9n06xf08d8d0h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:26Z",
      "native_id": "2609.32195",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32195"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "High-Probability Guarantees for SGD under $β$-Heavy-Tailed Gradient Noise",
  "title_text": "High-Probability Guarantees for SGD under β-Heavy-Tailed Gradient Noise",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/9n06xf08d8d0h",
  "venue": "arXiv",
  "versions": []
}