{
  "abstract": "Haussler's covering lemma (JCTA 1995) gives the sharp $(C/δ)^d$ metric-entropy bound for binary VC classes under normalized $\\ell_1$ distance. We prove a sharp global dynamic analogue for bounded real-valued classes in normalized $\\ell_p$, for every $1\\le p<\\infty$. Writing $d=\\operatorname{sfat}_{\\varepsilon/32}(\\mathcal F)$, we obtain a global dynamic cover of size at most $(C/\\varepsilon^p)^d$ for every horizon and $0<\\varepsilon<1/2$. Here a single family of deterministic causal experts, fixed independently of the instance sequence, covers every target on every sequence. For $0<\\varepsilon<1/4$, a matching worst-case lower bound $(c/\\varepsilon)^{pd}$ holds already for binary classes on one fixed sequence, even with real-valued experts. The constants are universal, independent of $p$. For $p=2$, this resolves the horizon-independent sequential $\\ell_2$ covering problem of Rakhlin, Sridharan, and Tewari (PTRF 2015) in a stronger global form. For $p=1$, it gives the sharp $(C/δ)^{\\operatorname{Ldim}(\\mathcal H)}$ global dynamic covering bound for binary classes, resolving questions of Alon, Ben-Eliezer, Dagan, Moran, Naor, and Yogev (STOC 2021) and Malliaris and Moran (Trans. AMS 2024). Our proof combines rank-decreasing correction schedules and accurate sampled prefixes with exact continuations after target identification. Shared-parent routing distributes these continuations among eligible prefixes, eliminating the logarithmic overhead.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29898",
  "author_total": 2,
  "authors": [
    {
      "name": "Shihao Shao"
    },
    {
      "name": "Dacheng Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fh57ggndava6e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29898",
  "fulltext_url": null,
  "id": "fh57ggndava6e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29898",
  "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.DS"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7223209870",
  "pdf_url": "https://arxiv.org/pdf/2610.29898",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fh57ggndava6e.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2610.29898",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29898"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Sharp Dynamic Haussler Lemma",
  "title_text": "A Sharp Dynamic Haussler Lemma",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:28Z",
  "url": "https://research.draconitic.com/paper/fh57ggndava6e",
  "venue": "arXiv",
  "versions": []
}