{
  "abstract": "Predetermined stepsize schedules featuring carefully chosen long steps have recently been shown to accelerate gradient descent (GD) on smooth convex functions. A prominent class of such schedules is built through recursive composition. In this paper, we characterize the convergence of these optimized recursive schedules, revealing a nonconstant log-periodic modulation across prescribed horizons. Specifically, we prove that for every $N \\geq 1$, the optimized ConPP/OBS-S schedules satisfy $$f(x_{N-1})-f_\\ast \\le \\frac{1}{2N^pΦ(\\log_2N)-1} \\frac{L}{2}\\|x_0-x_\\ast\\|^2, \\qquad p=\\log_2(1+\\sqrt2), $$ where $Φ$ is a positive, Lipschitz, nonconstant $1$-periodic function. We derive this by proving that balanced splitting is optimal at every horizon for these constructions, resolving a conjecture of Zhang and Jiang. Furthermore, for their asymmetric counterparts (ConPD/OBS-F schedules), we show that although optimal splits are not necessarily balanced, the same Silver exponent asymptotically persists alongside a distinct log-periodic modulation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11788",
  "author_total": 4,
  "authors": [
    {
      "name": "Yu Liu"
    },
    {
      "name": "Kang Chen"
    },
    {
      "name": "Rujun Jiang"
    },
    {
      "name": "Tianyu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1pjmn144v771a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11788",
  "fulltext_url": null,
  "id": "1pjmn144v771a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11788",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7212259628",
  "pdf_url": "https://arxiv.org/pdf/2609.11788",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1pjmn144v771a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.11788",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11788"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal Recursive Composition and Dyadic Phase Laws for Gradient Descent with Predetermined Stepsizes",
  "title_text": "Optimal Recursive Composition and Dyadic Phase Laws for Gradient Descent with Predetermined Stepsizes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/1pjmn144v771a",
  "venue": "arXiv",
  "versions": []
}