{"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"}