{
  "abstract": "Fix a target accuracy $\\varepsilon$, a gradient-noise level $s$, and a horizon $N$. We wish to design algorithms which minimize the probability of observing a suboptimality gap which exceeds the target accuracy, i.e., $\\mathcal{E}_N = -\\log \\sup_{f,P} \\mathbb{P}_P(f(x_A) - f_\\star \\ge \\varepsilon)$, with the noise law $P$ known only to be sub-Gaussian. We single out a uniformly averaged schedule which is harmonic (of the form $h_k = R^2/(\\varepsilon (N+m-k))$) and prove, via an optimized exponential supermartingale argument, that it attains the optimal exponent $\\mathcal{E}_N^\\star = \\varepsilon^2 N (1+o(1))/(2R^2 s^2)$. Optimality is certified by a matching impossibility result: under Gaussian noise, a gradient-masking change of measure caps the exponent of every algorithm at the same leading order. In a small-noise limit, our setting degenerates into the adversarial-error model of Gösgens and van Parys (2025) for subgradient methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37425",
  "author_total": 1,
  "authors": [
    {
      "name": "Bart P. G. van Parys"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23j6bt6gt91n4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37425",
  "fulltext_url": null,
  "id": "23j6bt6gt91n4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37425",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214898803",
  "pdf_url": "https://arxiv.org/pdf/2609.37425",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23j6bt6gt91n4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37425",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37425"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A stochastic subgradient method with optimal failure exponent",
  "title_text": "A stochastic subgradient method with optimal failure exponent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:41Z",
  "url": "https://research.draconitic.com/paper/23j6bt6gt91n4",
  "venue": "arXiv",
  "versions": []
}