{
  "abstract": "We study full-information online linear optimisation on the unit $\\ell_p$ ball in $\\mathbb{R}^d$, for fixed $2\\le p<\\infty$ and losses of dual norm at most one. Let $Δ_T$ denote the cumulative loss of the best fixed action above each round's minimum. A follow-the-regularised-leader (FTRL) algorithm attains, on every loss sequence and without knowing $T$ or $Δ_T$, regret $O_p(1+\\min\\{Δ_T^{1-1/p},\\, d^{1/2-1/p}\\sqrt{Δ_T}\\})$. Matching lower bounds show that this rate is minimax for each fixed $p$, and its constant $(2+o(1))p/\\log p$ is optimal as $p\\to\\infty$ among guarantees uniform in $d$ and $T$. The algorithm uses a bounded radial regulariser and switches at most once to its quadratic augmentation. On some paths where follow-the-leader (FTL) has regret one, its regret grows with $T$. A switching extension keeps the sharp constant and also guarantees $\\mathrm{Reg}_T\\le 2\\,\\mathrm{Reg}_T^{\\mathrm{FTL}}$. For i.i.d. losses whose mean has no zero coordinates, we compute the exact $\\log T$ coefficient of the expected regret, which separates FTL's variance term from the bias of the regulariser.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36097",
  "author_total": 1,
  "authors": [
    {
      "name": "Jack J. Mayo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18njvdb4ga37c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36097",
  "fulltext_url": null,
  "id": "18njvdb4ga37c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36097",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7224083285",
  "pdf_url": "https://arxiv.org/pdf/2610.36097",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18njvdb4ga37c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36097",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36097"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Minimax-Optimal Small-Loss Bounds for Online Linear Optimisation on $\\ell_p$ Balls",
  "title_text": "Minimax-Optimal Small-Loss Bounds for Online Linear Optimisation on ℓ_p Balls",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:59Z",
  "url": "https://research.draconitic.com/paper/18njvdb4ga37c",
  "venue": "arXiv",
  "versions": []
}