{
  "abstract": "We study the online calibration of multidimensional forecasts over an arbitrary convex set $Y\\subseteq\\mathbb{R}^d$ relative to an arbitrary error norm $\\|\\cdot\\|_{L}$. For forecasting $d$ binary outcomes simultaneously ($Y=[0,1]^d$), we give the first algorithm that achieves $\\varepsilon$-calibration in a number of rounds that is polynomial in $d$ for every fixed accuracy. It requires $d^{O(1/\\varepsilon)}$ rounds, exponentially improving the dimension dependence of previous bounds. For multi-class forecasting ($Y=Δ_d$), we obtain the same $d^{O(1/\\varepsilon)}$ rate, improving the $d^{\\widetilde{O}(1/\\varepsilon^2)}$ bounds of Peng and Fishelson et al. Our algorithm is simple: on each round, it outputs a harmonically weighted distribution over harmonically smoothed past outcomes. The same algorithm works for every forecast set and norm. More generally, it achieves $\\varepsilon$-calibration after $\\exp(O(γ(Y,L)/\\varepsilon))$ rounds, where $γ(Y,L)$ is a geometric parameter defined by a matrix discrepancy problem. The harmonic weights are motivated by the fact that the discrete Hilbert transform matrix achieves the optimal discrepancy up to a universal constant, simultaneously for every $L$. This optimality result may be of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07740",
  "author_total": 2,
  "authors": [
    {
      "name": "Maxwell Fishelson"
    },
    {
      "name": "Mehryar Mohri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0c74x9b01rxqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07740",
  "fulltext_url": null,
  "id": "0c74x9b01rxqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07740",
  "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",
      "stat.ML"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7220954393",
  "pdf_url": "https://arxiv.org/pdf/2610.07740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0c74x9b01rxqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:32Z",
      "native_id": "2610.07740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "High-dimensional online calibration from harmonic weights",
  "title_text": "High-dimensional online calibration from harmonic weights",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:32Z",
  "url": "https://research.draconitic.com/paper/0c74x9b01rxqe",
  "venue": "arXiv",
  "versions": []
}