{
  "abstract": "Normalized gradient descent is a widely studied adaptive optimization method. Most existing analyses focus on the best iterate or a weighted average of the iterates, whereas practical implementations typically return the last iterate. In this paper, we study the last-iterate convergence of normalized gradient descent for convex, $(ν,M_ν)$-Hölder-smooth objectives. For a constant stepsize, we establish an upper bound of $\\mathcal{O}\\bigl((\\log^2(T)/T)^{(1+ν)/2}\\bigr)$, which contains a logarithmic overhead relative to the known $\\mathcal{O}\\bigl(T^{-(1+ν)/2}\\bigr)$ guarantees for the best and weighted-average iterates. For $ν= 0$, this overhead is known to be unavoidable. We complement this analysis with numerical results based on the performance estimation problem (PEP), investigating the finite-horizon worst-case behavior in the smooth setting and whether the logarithmic overhead reflects an intrinsic limitation of constant-step normalized gradient descent. We then show that a linearly decreasing stepsize yields a last-iterate guarantee of $\\mathcal{O}\\bigl(T^{-(1+ν)/2}\\bigr)$, matching the order of the best-iterate/weighted-average guarantees without requiring knowledge of $ν$ and $M_ν$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06070",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuki Takezawa"
    },
    {
      "name": "Eduard Gorbunov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/38vc96eb6490x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06070",
  "fulltext_url": null,
  "id": "38vc96eb6490x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06070",
  "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": "W7220678365",
  "pdf_url": "https://arxiv.org/pdf/2610.06070",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/38vc96eb6490x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06070",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06070"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Last-Iterate Convergence Rate of Normalized Gradient Descent under Hölder Smoothness",
  "title_text": "Last-Iterate Convergence Rate of Normalized Gradient Descent under Hölder Smoothness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/38vc96eb6490x",
  "venue": "arXiv",
  "versions": []
}