{
  "abstract": "In this paper, we study the last-iterate convergence of clipped stochastic gradient descent under heavy-tailed noise. We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\\le2$. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(LR^2\\log(1/δ)/T+σR(\\log(1/δ)/T)^{(p-1)/p})$. We further establish last-iterate guarantees for nonsmooth composite objectives and smooth composite objectives with proximal updates. In addition, when a causal adversary replaces at most $q$ gradients, the same clipped update achieves a bound with $\\log(1/δ)$ replaced by $\\log(1/δ)+q$. The corresponding lower bounds for the clean and corrupted oracle models are also provided.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18555",
  "author_total": 3,
  "authors": [
    {
      "name": "Wei Jiang"
    },
    {
      "name": "Haomin Bai"
    },
    {
      "name": "Zechao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ee7k1c0y8pnpr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18555",
  "fulltext_url": null,
  "id": "ee7k1c0y8pnpr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18555",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7221794827",
  "pdf_url": "https://arxiv.org/pdf/2610.18555",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ee7k1c0y8pnpr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18555",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18555"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Last-Iterate Convergence of Clipped SGD with Linear Decay",
  "title_text": "Last-Iterate Convergence of Clipped SGD with Linear Decay",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/ee7k1c0y8pnpr",
  "venue": "arXiv",
  "versions": []
}