{
  "abstract": "Despite having an asymptotically optimal rate of convergence on convex quadratics, the acceleration of Polyak's heavy-ball method and a characterization of its dynamics within more general settings have remained long standing open problems. Recently, the work of Goujaud et al. [Math. Program. (2025)] made progress regarding acceleration in the context of smooth, strongly convex minimization: for a small enough inverse condition number, either the method does not accelerate on quadratics, or there exists a smooth strongly convex function such that heavy-ball cycles over the roots of unity. The constructed functions have two variables, thus leaving the question of acceleration over univariate functions open. In this work, we use a Fourier-based analysis to i) provide a proof for the non-acceleration of the heavy-ball method over the class of univariate smooth strongly convex functions. We also ii) provide a strict enlargement of the previous root-of-unity cycling region, with insights regarding the \"fractional\" sets of Goujaud et al. [arXiv:2502.19916], and iii) give a higher dimensional cycling instance which is not covered by root-of-unity cycling.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25104",
  "author_total": 4,
  "authors": [
    {
      "name": "Konstantinos Fotopoulos"
    },
    {
      "name": "Michael Helcig"
    },
    {
      "name": "Karl Deck"
    },
    {
      "name": "Jannis Alsbach"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7rv0nvf773m1v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25104",
  "fulltext_url": null,
  "id": "7rv0nvf773m1v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25104",
  "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",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7223055651",
  "pdf_url": "https://arxiv.org/pdf/2610.25104",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7rv0nvf773m1v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25104",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25104"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Non-Acceleration of the Heavy-Ball Method in 1D",
  "title_text": "Non-Acceleration of the Heavy-Ball Method in 1D",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7rv0nvf773m1v",
  "venue": "arXiv",
  "versions": []
}