{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:YNOJFVNMUSPBGSXDYEHPH2CXA5","short_pith_number":"pith:YNOJFVNM","schema_version":"1.0","canonical_sha256":"c35c92d5aca49e134ae3c10ef3e857077d02ab40836f5e8e6af8bd0c9a681f8a","source":{"kind":"arxiv","id":"2603.05002","version":2},"attestation_state":"computed","paper":{"title":"Non-Euclidean Gradient Descent Operates at the Edge of Stability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Jeremy Cohen, Michael Crawshaw, Robert Gower, Rustem Islamov","submitted_at":"2026-03-05T09:49:33Z","abstract_excerpt":"The Edge of Stability (EoS) is a phenomenon where the sharpness (largest eigenvalue) of the Hessian approaches and then hovers near the stability threshold $2/\\eta$ during gradient descent (GD) with step size $\\eta$. Despite (apparently) violating classical smoothness assumptions, EoS has been widely observed in deep learning, but its theoretical foundations remain incomplete. We provide an interpretation of EoS through the lens of Directional Smoothness [Mishkin et al., 2024]. This interpretation naturally extends to non-Euclidean norms, which we use to define generalized sharpness under an a"},"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":"2603.05002","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-03-05T09:49:33Z","cross_cats_sorted":["math.OC","stat.ML"],"title_canon_sha256":"ff111847a471989a6cd7b951ae030d13dbcf591a13edb0cfff879e6fa20f9ddf","abstract_canon_sha256":"2286e7b45f48068b528c423ac3cb684d9a371e339a8827c45caa99f010d181ea"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T02:05:42.460246Z","signature_b64":"l6TntNOWMwl53rR6d3KhJkNtvLMAUcJKqYvriDOChI7xtrmNkoFEVsPEkleDWLxIWT7hm+5f6bXaxHun+rkuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c35c92d5aca49e134ae3c10ef3e857077d02ab40836f5e8e6af8bd0c9a681f8a","last_reissued_at":"2026-05-29T02:05:42.459399Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T02:05:42.459399Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-Euclidean Gradient Descent Operates at the Edge of Stability","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Jeremy Cohen, Michael Crawshaw, Robert Gower, Rustem Islamov","submitted_at":"2026-03-05T09:49:33Z","abstract_excerpt":"The Edge of Stability (EoS) is a phenomenon where the sharpness (largest eigenvalue) of the Hessian approaches and then hovers near the stability threshold $2/\\eta$ during gradient descent (GD) with step size $\\eta$. Despite (apparently) violating classical smoothness assumptions, EoS has been widely observed in deep learning, but its theoretical foundations remain incomplete. We provide an interpretation of EoS through the lens of Directional Smoothness [Mishkin et al., 2024]. This interpretation naturally extends to non-Euclidean norms, which we use to define generalized sharpness under an a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.05002","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.05002/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":"2603.05002","created_at":"2026-05-29T02:05:42.459526+00:00"},{"alias_kind":"arxiv_version","alias_value":"2603.05002v2","created_at":"2026-05-29T02:05:42.459526+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.05002","created_at":"2026-05-29T02:05:42.459526+00:00"},{"alias_kind":"pith_short_12","alias_value":"YNOJFVNMUSPB","created_at":"2026-05-29T02:05:42.459526+00:00"},{"alias_kind":"pith_short_16","alias_value":"YNOJFVNMUSPBGSXD","created_at":"2026-05-29T02:05:42.459526+00:00"},{"alias_kind":"pith_short_8","alias_value":"YNOJFVNM","created_at":"2026-05-29T02:05:42.459526+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2605.16622","citing_title":"Does Weight Decay Enhance Training Stability?","ref_index":22,"is_internal_anchor":true},{"citing_arxiv_id":"2605.11181","citing_title":"Muon is Not That Special: Random or Inverted Spectra Work Just as Well","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5","json":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5.json","graph_json":"https://pith.science/api/pith-number/YNOJFVNMUSPBGSXDYEHPH2CXA5/graph.json","events_json":"https://pith.science/api/pith-number/YNOJFVNMUSPBGSXDYEHPH2CXA5/events.json","paper":"https://pith.science/paper/YNOJFVNM"},"agent_actions":{"view_html":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5","download_json":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5.json","view_paper":"https://pith.science/paper/YNOJFVNM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2603.05002&json=true","fetch_graph":"https://pith.science/api/pith-number/YNOJFVNMUSPBGSXDYEHPH2CXA5/graph.json","fetch_events":"https://pith.science/api/pith-number/YNOJFVNMUSPBGSXDYEHPH2CXA5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5/action/storage_attestation","attest_author":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5/action/author_attestation","sign_citation":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5/action/citation_signature","submit_replication":"https://pith.science/pith/YNOJFVNMUSPBGSXDYEHPH2CXA5/action/replication_record"}},"created_at":"2026-05-29T02:05:42.459526+00:00","updated_at":"2026-05-29T02:05:42.459526+00:00"}