{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:Y5VVQRJ27EHDSTDVRFSWWNC4VI","short_pith_number":"pith:Y5VVQRJ2","schema_version":"1.0","canonical_sha256":"c76b58453af90e394c7589656b345caa1447f6be81b1b38a57220f8da413fb63","source":{"kind":"arxiv","id":"2605.13684","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.13684","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-13T15:41:30Z","cross_cats_sorted":["cs.IT","math.IT"],"title_canon_sha256":"a24f49898933fac590f36df73ac706625010550d4db5b3a6713d6d6a14053f7c","abstract_canon_sha256":"738bd5a200a235f069c4876d6e7662332e05dc641469a3eee3c104736d18e6c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:17.010053Z","signature_b64":"ImN5BLPFbx9HWL/X7l1FBeSr1CgXvLUc3d8nytdMUxr37+xDY7Ev4sYSY+U5itqMxRXGaDNWvborAI1f1jFpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c76b58453af90e394c7589656b345caa1447f6be81b1b38a57220f8da413fb63","last_reissued_at":"2026-05-18T02:44:17.009589Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:17.009589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.13684","created_at":"2026-05-18T02:44:17.009668+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.13684v1","created_at":"2026-05-18T02:44:17.009668+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13684","created_at":"2026-05-18T02:44:17.009668+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y5VVQRJ27EHD","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y5VVQRJ27EHDSTDV","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y5VVQRJ2","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI","json":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI.json","graph_json":"https://pith.science/api/pith-number/Y5VVQRJ27EHDSTDVRFSWWNC4VI/graph.json","events_json":"https://pith.science/api/pith-number/Y5VVQRJ27EHDSTDVRFSWWNC4VI/events.json","paper":"https://pith.science/paper/Y5VVQRJ2"},"agent_actions":{"view_html":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI","download_json":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI.json","view_paper":"https://pith.science/paper/Y5VVQRJ2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.13684&json=true","fetch_graph":"https://pith.science/api/pith-number/Y5VVQRJ27EHDSTDVRFSWWNC4VI/graph.json","fetch_events":"https://pith.science/api/pith-number/Y5VVQRJ27EHDSTDVRFSWWNC4VI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI/action/storage_attestation","attest_author":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI/action/author_attestation","sign_citation":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI/action/citation_signature","submit_replication":"https://pith.science/pith/Y5VVQRJ27EHDSTDVRFSWWNC4VI/action/replication_record"}},"created_at":"2026-05-18T02:44:17.009668+00:00","updated_at":"2026-05-18T02:44:17.009668+00:00"}