{
  "abstract": "Can the classical Heavy-Ball method, with arbitrary horizon-dependent parameters chosen in advance, achieve Nesterov's $O(T^{-2})$ last-iterate rate on every smooth convex objective? We provide a negative answer. For every horizon $T\\ge2$ and every predetermined schedule with nonnegative step sizes and momenta in $[0,1)$, there exists a convex $1$-smooth objective, with initialization distance at most one and zero initial velocity, for which the last iterate of the Heavy-Ball method satisfies \\[ f(x_T)-f^\\star=Ω\\!\\left(\\frac{1}{T^α\\log T}\\right), \\qquad α=\\frac{1+\\sqrt5}{2}. \\] Thus even fully nonstationary, horizon-dependent tuning cannot give the classical Heavy-Ball method a Nesterov-rate guarantee on the smooth convex class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08656",
  "author_total": 2,
  "authors": [
    {
      "name": "Jianhao Ma"
    },
    {
      "name": "Jingzhao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22hbgz0j6qp5w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08656",
  "fulltext_url": null,
  "id": "22hbgz0j6qp5w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08656",
  "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"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7212029601",
  "pdf_url": "https://arxiv.org/pdf/2609.08656",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22hbgz0j6qp5w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08656",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08656"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions",
  "title_text": "A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/22hbgz0j6qp5w",
  "venue": "arXiv",
  "versions": []
}