{
  "abstract": "We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class and every $γ>0$, uniform convergence at scale $γ$, agnostic learnability at scale $γ/2$, and finiteness of the fat-shattering dimension at every scale $γ'>γ$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS 1998), which incur such a loss. The key technical ingredient is a direct bound on empirical $\\ell_\\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $γ$: an $O(\\log^2 n)$ bound holds already at scale $γ/2$, while an $O(\\log n)$ bound holds at scale $2γ$. We further show that the $O(\\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM 1997) and Rudelson and Vershynin (Ann. of Math. 2006). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML 2026). We also highlight several open questions on quantitative sample complexity and evaluability.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.13684",
  "author_total": 5,
  "authors": [
    {
      "name": "Shashaank Aiyer"
    },
    {
      "name": "Yishay Mansour"
    },
    {
      "name": "Shay Moran"
    },
    {
      "name": "Han Shao"
    },
    {
      "name": "Tom Waknine"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq6zv3wjq6s6b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.13684",
  "fulltext_url": null,
  "id": "dq6zv3wjq6s6b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.13684",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Machine Learning and Algorithms"
    ]
  ],
  "openalex": "W7161152370",
  "pdf_url": "https://arxiv.org/pdf/2605.13684",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq6zv3wjq6s6b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2605.13684",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.13684"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale",
  "title_text": "Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/dq6zv3wjq6s6b",
  "venue": "arXiv",
  "versions": []
}