{
  "abstract": "We show that gradient descent (GD) with step size $η$ can achieve $\\varepsilon$ optimality gap on the multiclass logistic regression loss in a number of iterations $T$ that is independent of $\\varepsilon$. To do so, the step size $η$ must be chosen large enough as a function of $\\varepsilon$ (and properties of the linearly separable dataset), while $T$ is a constant that only depends on the data. The relevant properties of the data include the number of points and their margin (for $T$), and one additional quantity (for $η$) that turns out to be necessary in dimension $d \\geq 3$. The proofs build on the known fact that, as $η\\to \\infty$, suitably rescaled GD becomes the perceptron algorithm, and the latter finds a perfect classifier in $T$ iterations. A similar claim was shown for two classes in dimension $d = 2$ using other proof techniques. This result adds to previous works that showed GD on the logistic regression loss can be accelerated by taking large step sizes. Those provoke a wild, oscillatory initial phase, followed by a stable phase. We provide results of this (qualitatively different) nature, too.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26748",
  "author_total": 2,
  "authors": [
    {
      "name": "Andreea-Alexandra Musat"
    },
    {
      "name": "Nicolas Boumal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/01rg9xkz2jag0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26748",
  "fulltext_url": null,
  "id": "01rg9xkz2jag0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26748",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7223069161",
  "pdf_url": "https://arxiv.org/pdf/2610.26748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/01rg9xkz2jag0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.26748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Gradient descent reaches arbitrary accuracy in a fixed number of large steps on multiclass logistic regression",
  "title_text": "Gradient descent reaches arbitrary accuracy in a fixed number of large steps on multiclass logistic regression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/01rg9xkz2jag0",
  "venue": "arXiv",
  "versions": []
}