{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:ON25ECWU7YVOFZ7KHJWRFYOHJS","short_pith_number":"pith:ON25ECWU","schema_version":"1.0","canonical_sha256":"7375d20ad4fe2ae2e7ea3a6d12e1c74c9ce0aeb71e2cad3b977bbf207b8942c1","source":{"kind":"arxiv","id":"2512.21075","version":2},"attestation_state":"computed","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 "},"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":"2512.21075","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-12-24T09:39:04Z","cross_cats_sorted":["cs.AI","math.PR","stat.ML"],"title_canon_sha256":"f681a678a515c588697b44471af8b50a099a038d207515149b5a0ea0d3534034","abstract_canon_sha256":"91734d4062d34a7c318692774f524346ae498c97a0a9ae90bc1e897a3636e805"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:32.417954Z","signature_b64":"smXP2zTB3L4AWZ8Um5iBUgTWVRLWltc/K1ghk+f3wd4pAdJEZ7U0FRDIRTdUh9xEW4J2HWY1ED8WrE2okKoGBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7375d20ad4fe2ae2e7ea3a6d12e1c74c9ce0aeb71e2cad3b977bbf207b8942c1","last_reissued_at":"2026-05-18T03:09:32.417066Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:32.417066Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2512.21075","created_at":"2026-05-18T03:09:32.417200+00:00"},{"alias_kind":"arxiv_version","alias_value":"2512.21075v2","created_at":"2026-05-18T03:09:32.417200+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.21075","created_at":"2026-05-18T03:09:32.417200+00:00"},{"alias_kind":"pith_short_12","alias_value":"ON25ECWU7YVO","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"ON25ECWU7YVOFZ7K","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"ON25ECWU","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2602.13298","citing_title":"The Effective Depth Paradox: Evaluating the Relationship between Architectural Topology and Trainability in Deep CNNs","ref_index":14,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS","json":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS.json","graph_json":"https://pith.science/api/pith-number/ON25ECWU7YVOFZ7KHJWRFYOHJS/graph.json","events_json":"https://pith.science/api/pith-number/ON25ECWU7YVOFZ7KHJWRFYOHJS/events.json","paper":"https://pith.science/paper/ON25ECWU"},"agent_actions":{"view_html":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS","download_json":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS.json","view_paper":"https://pith.science/paper/ON25ECWU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2512.21075&json=true","fetch_graph":"https://pith.science/api/pith-number/ON25ECWU7YVOFZ7KHJWRFYOHJS/graph.json","fetch_events":"https://pith.science/api/pith-number/ON25ECWU7YVOFZ7KHJWRFYOHJS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS/action/storage_attestation","attest_author":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS/action/author_attestation","sign_citation":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS/action/citation_signature","submit_replication":"https://pith.science/pith/ON25ECWU7YVOFZ7KHJWRFYOHJS/action/replication_record"}},"created_at":"2026-05-18T03:09:32.417200+00:00","updated_at":"2026-05-18T03:09:32.417200+00:00"}