{
  "abstract": "We study stochastic gradient descent with random reshuffling for finite sums \\[ F(x)=\\frac1n\\sum_{i=1}^n f_i(x). \\] For fresh reshuffling with a constant component stepsize, if each $f_i$ has an $L$-Lipschitz gradient and the average $F$ is $μ$-strongly convex with a Lipschitz-continuous Hessian, we prove the last-epoch rate \\[ \\mathbb E[F(y_K)-F(x_\\star)] =\\widetilde O\\!\\left(T^{-2}+n^2T^{-3}\\right), \\qquad T=nK, \\] matching the known quadratic lower bound in its $(n,K)$-dependence. The components may be nonconvex, and no componentwise Hessian continuity or separate bounded-iterate assumption is required. More generally, a $ν$-Hölder-continuous average Hessian adds only $\\widetilde O(n^{1+ν}T^{-2-2ν})$, so every $ν\\ge 1/2$ preserves the quadratic rate. Under convex components, a decreasing-stepsize result removes the large-epoch requirement and recovers the same two-term scale once $nK$ exceeds the condition-number scale. We also analyze epoch-wise ProxRR for $\\mathcal P=F+ψ$. Writing $x^\\dagger$ for the composite minimizer and $β_\\star=\\|\\nabla F(x^\\dagger)\\|$, we prove \\[ \\mathbb E\\|y_K-x^\\dagger\\|^2 =\\widetilde O\\!\\left( \\frac{β_\\star^2}{K^2} +T^{-2}+n^2T^{-3} +n^{1+ν}T^{-2-2ν} \\right). \\] For $ν\\ge 1/2$, we show that the $β_\\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04578",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiaxiang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0b9tpsqbpdfjc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04578",
  "fulltext_url": null,
  "id": "0b9tpsqbpdfjc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04578",
  "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": "W7211920400",
  "pdf_url": "https://arxiv.org/pdf/2609.04578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0b9tpsqbpdfjc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Centered Permutation Prefixes for SGD with Random Reshuffling: Sharp Rates, Hölder Geometry, and Composite Proximal Extensions",
  "title_text": "Centered Permutation Prefixes for SGD with Random Reshuffling: Sharp Rates, Hölder Geometry, and Composite Proximal Extensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/0b9tpsqbpdfjc",
  "venue": "arXiv",
  "versions": []
}