{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:45F5CUW3IMYQOQIDNSAHHNCXXO","short_pith_number":"pith:45F5CUW3","schema_version":"1.0","canonical_sha256":"e74bd152db43310741036c8073b457bbae4282571238749770d7854d1ea081f4","source":{"kind":"arxiv","id":"2510.02779","version":4},"attestation_state":"computed","paper":{"title":"Optimal Rates for Generalization of Gradient Descent for Deep ReLU Classification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Under NTK separability with margin γ, gradient descent on deep ReLU networks attains an excess risk of Õ(L^6/(n γ²)).","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Yiming Ying, Yuanfan Li, Yunwen Lei, Zheng-Chu Guo","submitted_at":"2025-10-03T07:22:36Z","abstract_excerpt":"Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates comparable to the minimax optimal rates established in the kernel setting. Existing results either yield suboptimal rates of $O(1/\\sqrt{n})$, or focus on networks with smooth activation functions, incurring exponential dependence on network depth $L$. In this work, we establish optimal generalization rates for GD with deep ReLU networks by carefully trading off "},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2510.02779","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-10-03T07:22:36Z","cross_cats_sorted":[],"title_canon_sha256":"4bc1d06d560a9fc43b13cd540ea8c56d0986ecce85e5c3fe2893cbb0477bcf0e","abstract_canon_sha256":"d9dbf981bef549d6867176946de85939839ae7d66dff89e44ec69cc0326de424"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:45.518159Z","signature_b64":"Ze/h3oof7o32FV0R8IUewD4+heKvY0ruVaa7V73Hw4kztoMZ4Ayg36tpBRiD/luRtzae6+dXQqT4ogYcVJtcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e74bd152db43310741036c8073b457bbae4282571238749770d7854d1ea081f4","last_reissued_at":"2026-06-03T01:05:45.517619Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:45.517619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal Rates for Generalization of Gradient Descent for Deep ReLU Classification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Under NTK separability with margin γ, gradient descent on deep ReLU networks attains an excess risk of Õ(L^6/(n γ²)).","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Yiming Ying, Yuanfan Li, Yunwen Lei, Zheng-Chu Guo","submitted_at":"2025-10-03T07:22:36Z","abstract_excerpt":"Recent advances have significantly improved our understanding of the generalization performance of gradient descent (GD) methods in deep neural networks. A natural and fundamental question is whether GD can achieve generalization rates comparable to the minimax optimal rates established in the kernel setting. Existing results either yield suboptimal rates of $O(1/\\sqrt{n})$, or focus on networks with smooth activation functions, incurring exponential dependence on network depth $L$. In this work, we establish optimal generalization rates for GD with deep ReLU networks by carefully trading off "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under the assumption that the data are NTK separable from the margin γ, we prove an excess risk rate of Õ(L^6 / (n γ²))","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"the assumption that the data are NTK separable from the margin γ (stated in the abstract as the condition under which the rate holds)","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Gradient descent on deep ReLU networks achieves excess risk rate of order L^6/(n gamma^2) under NTK margin separability via activation pattern control.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Under NTK separability with margin γ, gradient descent on deep ReLU networks attains an excess risk of Õ(L^6/(n γ²)).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4b9cf7645b6f3a321fff3d62f9b10a2d649b08264e9c9e27f7e46206ad4fab8b"},"source":{"id":"2510.02779","kind":"arxiv","version":4},"verdict":{"id":"b659c8cc-a3bc-4133-a4ca-b70e63599fbb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T10:21:36.489766Z","strongest_claim":"under the assumption that the data are NTK separable from the margin γ, we prove an excess risk rate of Õ(L^6 / (n γ²))","one_line_summary":"Gradient descent on deep ReLU networks achieves excess risk rate of order L^6/(n gamma^2) under NTK margin separability via activation pattern control.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"the assumption that the data are NTK separable from the margin γ (stated in the abstract as the condition under which the rate holds)","pith_extraction_headline":"Under NTK separability with margin γ, gradient descent on deep ReLU networks attains an excess risk of Õ(L^6/(n γ²))."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.02779/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"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":"2510.02779","created_at":"2026-06-03T01:05:45.517678+00:00"},{"alias_kind":"arxiv_version","alias_value":"2510.02779v4","created_at":"2026-06-03T01:05:45.517678+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.02779","created_at":"2026-06-03T01:05:45.517678+00:00"},{"alias_kind":"pith_short_12","alias_value":"45F5CUW3IMYQ","created_at":"2026-06-03T01:05:45.517678+00:00"},{"alias_kind":"pith_short_16","alias_value":"45F5CUW3IMYQOQID","created_at":"2026-06-03T01:05:45.517678+00:00"},{"alias_kind":"pith_short_8","alias_value":"45F5CUW3","created_at":"2026-06-03T01:05:45.517678+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO","json":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO.json","graph_json":"https://pith.science/api/pith-number/45F5CUW3IMYQOQIDNSAHHNCXXO/graph.json","events_json":"https://pith.science/api/pith-number/45F5CUW3IMYQOQIDNSAHHNCXXO/events.json","paper":"https://pith.science/paper/45F5CUW3"},"agent_actions":{"view_html":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO","download_json":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO.json","view_paper":"https://pith.science/paper/45F5CUW3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2510.02779&json=true","fetch_graph":"https://pith.science/api/pith-number/45F5CUW3IMYQOQIDNSAHHNCXXO/graph.json","fetch_events":"https://pith.science/api/pith-number/45F5CUW3IMYQOQIDNSAHHNCXXO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO/action/storage_attestation","attest_author":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO/action/author_attestation","sign_citation":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO/action/citation_signature","submit_replication":"https://pith.science/pith/45F5CUW3IMYQOQIDNSAHHNCXXO/action/replication_record"}},"created_at":"2026-06-03T01:05:45.517678+00:00","updated_at":"2026-06-03T01:05:45.517678+00:00"}