{"paper":{"title":"Feature Learning Dynamics in Infinite-Depth Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Finite ResNet training dynamics converge to a decoupled Neural Feature Dynamics limit with O(L^{-1}) error under depth-μP scaling.","cross_cats":["cs.AI","math.PR","stat.ML"],"primary_cat":"cs.LG","authors_text":"Ruoyu Wu, Tianxiang Gao, Zihan Yao","submitted_at":"2025-12-24T09:39:04Z","abstract_excerpt":"Deep neural networks have achieved remarkable success in practice, yet a mechanistic understanding of how features evolve during training remains incomplete, especially in the large-depth limit. For ResNets under depth-$\\mu$P scaling, prior work treats the layer index $\\ell$ as a continuous time $t_\\ell = \\ell/L$, yielding SDE descriptions of the training dynamics. A key unresolved issue is that backpropagation reuses each forward weight matrix $W_\\ell$ through its transpose $W_\\ell^\\top$, creating correlations between forward features and backward gradients whose behavior and role in feature "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Under nondegeneracy assumptions, we prove that the finite-network training dynamics converge to its NFD limit with an O(L^{-1}) depth-discretization error, while the reused-weight coupling term has a faster O(L^{-2}) decay.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The nondegeneracy assumptions on the feature-gradient covariance structure generated during training, which are required to ensure the SDE limit exists and that the coupling remains higher-order in depth under depth-μP scaling.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Under depth-μP scaling, the reused-weight forward-backward coupling in one-layer ResNets vanishes at O(L^{-2}), enabling convergence to a decoupled Neural Feature Dynamics SDE limit with O(L^{-1}) discretization error.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Finite ResNet training dynamics converge to a decoupled Neural Feature Dynamics limit with O(L^{-1}) error under depth-μP scaling.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"87b8d87b67f7500719b5b9f349a1b632e89dc4d4b0f9b67dd2b0300ba00d78f5"},"source":{"id":"2512.21075","kind":"arxiv","version":2},"verdict":{"id":"a4c93aa3-e0ce-4656-8417-7d9fe1a74f46","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T20:13:31.612520Z","strongest_claim":"Under nondegeneracy assumptions, we prove that the finite-network training dynamics converge to its NFD limit with an O(L^{-1}) depth-discretization error, while the reused-weight coupling term has a faster O(L^{-2}) decay.","one_line_summary":"Under depth-μP scaling, the reused-weight forward-backward coupling in one-layer ResNets vanishes at O(L^{-2}), enabling convergence to a decoupled Neural Feature Dynamics SDE limit with O(L^{-1}) discretization error.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The nondegeneracy assumptions on the feature-gradient covariance structure generated during training, which are required to ensure the SDE limit exists and that the coupling remains higher-order in depth under depth-μP scaling.","pith_extraction_headline":"Finite ResNet training dynamics converge to a decoupled Neural Feature Dynamics limit with O(L^{-1}) error under depth-μP scaling."},"references":{"count":4,"sample":[{"doi":"","year":null,"title":"write newline","work_id":"8e5fda61-e601-4df4-8204-015bee341570","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"\\@ifxundefined[1] #1\\@undefined \\@firstoftwo \\@secondoftwo \\@ifnum[1] #1 \\@firstoftwo \\@secondoftwo \\@ifx[1] #1 \\@firstoftwo \\@secondoftwo [2] @ #1 \\@temptokena #2 #1 @ \\@temptokena \\@ifclassloaded ag","work_id":"b058608d-98d0-4821-a4ae-403d2b7cd411","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"\\@lbibitem[] @bibitem@first@sw\\@secondoftwo \\@lbibitem[#1]#2 \\@extra@b@citeb \\@ifundefined br@#2\\@extra@b@citeb \\@namedef br@#2 \\@nameuse br@#2\\@extra@b@citeb \\@ifundefined b@#2\\@extra@b@citeb @num @p","work_id":"ea79bfb8-d434-45e9-8607-416d3839ec5c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"@open @close @open @close and [1] URL: #1 \\@ifundefined chapter * \\@mkboth \\@ifxundefined @sectionbib * \\@mkboth * \\@mkboth\\@gobbletwo \\@ifclassloaded amsart * \\@ifclassloaded amsbook * \\@ifxundefined","work_id":"d051d6aa-efca-49eb-a66d-8ea01bf21294","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":4,"snapshot_sha256":"8cd3c2443fe68f5b0b489cafcb4da447febf1db446e7328adbfe038ec9ee0c09","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"47532bf44977a180848007fd5de320cc7c0c3e8dcc6ba27e37503faae83f9a12"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}