{"paper":{"title":"Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For any bounded real-valued function class, uniform convergence at scale γ is equivalent to agnostic learnability at scale γ/2 and finite fat-shattering dimension at all scales above γ.","cross_cats":["cs.IT","math.IT"],"primary_cat":"cs.LG","authors_text":"Han Shao, Shashaank Aiyer, Shay Moran, Tom Waknine, Yishay Mansour","submitted_at":"2026-05-13T15:41:30Z","abstract_excerpt":"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 $\\gamma>0$, uniform convergence at scale $\\gamma$, agnostic learnability at scale $\\gamma/2$, and finiteness of the fat-shattering dimension at every scale $\\gamma'>\\gamma$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture att"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The real-valued function class must be bounded, which is required for the scale-sensitive notions of uniform convergence and learnability to be well-defined.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For bounded real-valued function classes, uniform convergence at scale γ, agnostic learnability at γ/2, and finite fat-shattering dimension above γ are equivalent.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any bounded real-valued function class, uniform convergence at scale γ is equivalent to agnostic learnability at scale γ/2 and finite fat-shattering dimension at all scales above γ.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3d88f0f825a391714058079e5f78537eae8ce344881ecd10b649a750fae3c162"},"source":{"id":"2605.13684","kind":"arxiv","version":1},"verdict":{"id":"457b5c4c-e90b-4b9e-9ca4-998a0badabdc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:29:49.960172Z","strongest_claim":"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","one_line_summary":"For bounded real-valued function classes, uniform convergence at scale γ, agnostic learnability at γ/2, and finite fat-shattering dimension above γ are equivalent.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The real-valued function class must be bounded, which is required for the scale-sensitive notions of uniform convergence and learnability to be well-defined.","pith_extraction_headline":"For any bounded real-valued function class, uniform convergence at scale γ is equivalent to agnostic learnability at scale γ/2 and finite fat-shattering dimension at all scales above γ."},"references":{"count":41,"sample":[{"doi":"10.1007/3-540-44581-1","year":2001,"title":"Long , editor =","work_id":"db13ffc6-669b-4732-8852-7c9d48c9afa7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Inventiones Mathematicae , volume =","work_id":"f10998e9-b7f0-41cc-9f7c-dcedd5fbd9c8","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"IEEE Transactions on Information Theory , volume =","work_id":"8de30bea-6784-48d2-8f96-1d57bb38534a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"Bartlett and Sanjeev R","work_id":"360d4056-43c6-4880-b633-8e821dc5ccc0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"International Conference on Computational Learning Theory , pages=","work_id":"c2ad8b6d-0fbb-4c85-8ff3-0cf526d23569","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":41,"snapshot_sha256":"8cabf195106c1985666950c6c7b0b716c8e4f6718fc7a4deb9723f4a1e5a0927","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"52df45566254febcbd7e44449da6a732c97a5edc4b739f22cd7fbc618c64832e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}