{
  "abstract": "We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Writing $p_{\\mathrm{sil}}=\\log_2(1+\\sqrt{2})$, we prove an $Ω\\left(n^{-p_{\\mathrm{sil}}-O(\\sqrt{\\log\\log n/\\log n})}\\right)$ non-anytime lower bound. In the anytime setting, every infinite schedule has infinitely many horizons with error $Ω\\left(n^{-\\frac{2p_{\\mathrm{sil}}}{1+p_{\\mathrm{sil}}}-O(\\sqrt{\\log\\log n/\\log n})}\\right)$. Together with the silver-schedule upper bound [Altschuler and Parrilo, 2025] and the anytime upper bound [Zhang et al., 2025], our results determine the optimal polynomial convergence exponents in both settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09152",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuhan Ye"
    },
    {
      "name": "Kaizhao Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7pm4gn8gax7c4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09152",
  "fulltext_url": null,
  "id": "7pm4gn8gax7c4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09152",
  "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": "W7212060869",
  "pdf_url": "https://arxiv.org/pdf/2609.09152",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7pm4gn8gax7c4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.09152",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09152"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Silver Rate Is (Almost) Optimal for Gradient Descent",
  "title_text": "Silver Rate Is (Almost) Optimal for Gradient Descent",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/7pm4gn8gax7c4",
  "venue": "arXiv",
  "versions": []
}