{
  "abstract": "Convex optimization with Hölder-continuous gradients has attracted considerable attention as a framework bridging smooth and nonsmooth optimization. For this class, Nesterov's universal fast gradient method attains the optimal black-box complexity, and we study whether exploiting further problem structure can further improve this bound. In this paper, we study an important class of Hölder-smooth problems, namely regularized $\\ell_p$ regression with $p\\in(1,2)$, whose loss has a $(p-1)$-Hölder-continuous gradient and for which the universal method requires $\\mathcal{O}(\\varepsilon^{-2/(3p-2)})$ iterations. Exploiting the problem structure, we characterize a family of smooth approximations based on $p$-Huber smoothing and prove that, for any prescribed smoothness level, it minimizes the worst-case approximation error among smooth lower approximations of the scalar loss. Combined with accelerated proximal gradients, the smoothing approach attains an $\\varepsilon$-optimal solution in $\\mathcal{O}(\\varepsilon^{-1/p})$ matrix--vector products, which significantly improves the complexity guarantee for $1<p<2$. We show that this bound is optimal by providing a matching lower bound. Numerical experiments on synthetic and real data demonstrate the practical performance of the proposed methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21595",
  "author_total": 2,
  "authors": [
    {
      "name": "Baiying Xin"
    },
    {
      "name": "Qi Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dwv590nzxfrjq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21595",
  "fulltext_url": null,
  "id": "dwv590nzxfrjq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21595",
  "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": "W7222425260",
  "pdf_url": "https://arxiv.org/pdf/2610.21595",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dwv590nzxfrjq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21595",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21595"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal First-Order Methods for a Class of Structured Hölder-Smooth Problems",
  "title_text": "Optimal First-Order Methods for a Class of Structured Hölder-Smooth Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/dwv590nzxfrjq",
  "venue": "arXiv",
  "versions": []
}