{
  "abstract": "We present a tightened convergence analysis of clipped gradient descent on $(L_0, L_1)$-smooth functions, with quantitative constants. Building on the ideas of Koloskova et al (2023), we refactor several case disjunctions to reveal the central role of a control of the bias derived from fundamental properties of $\\ell_2$-projection, simplifying proofs. We also extend the domain of validity from $η\\leq 1 / (9 β)$ to $η< 1 /β$ where $β= L_0 + c L_1$ for clipping constant $c$, which matches the more traditional analysis of smooth functions. We strengthen the convergence criterion from $\\left( \\min_{t < T} \\mathbb{E}[\\lVert \\nabla f(x_t) \\rVert_2] \\right)$ to $\\left( \\frac{1}{T} \\sum_{t < T} \\mathbb{E}[\\lVert \\nabla f(x_t) \\rVert_2] \\right)$ with matching speed, and lower the final achievable loss from $\\mathcal{O}(\\min(σ^2/c, σ))$ to the more precise $6 \\min(σ^2 /c, 3 σ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29458",
  "author_total": 1,
  "authors": [
    {
      "name": "David A. R. Robin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/89xrcm52p1c9k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29458",
  "fulltext_url": null,
  "id": "89xrcm52p1c9k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29458",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7214344355",
  "pdf_url": "https://arxiv.org/pdf/2609.29458",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/89xrcm52p1c9k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29458",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29458"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Precise Convergence Speed of Clipped SGD",
  "title_text": "Precise Convergence Speed of Clipped SGD",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/89xrcm52p1c9k",
  "venue": "arXiv",
  "versions": []
}