{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:6IJ3EQSZPDROL47QIMJOWUHXH3","short_pith_number":"pith:6IJ3EQSZ","canonical_record":{"source":{"id":"2311.15051","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-11-25T15:10:00Z","cross_cats_sorted":["math.OC","stat.ML"],"title_canon_sha256":"0bd018627a285d82c5ea37f6cce4827736a30ee919333c40202c1fbe24931736","abstract_canon_sha256":"8b95eb1c503afd7052566c8f366ef722241781a7c3139b6edb9da0e9e2a58de9"},"schema_version":"1.0"},"canonical_sha256":"f213b2425978e2e5f3f04312eb50f73efbc9e34a7b78081121342104e136a158","source":{"kind":"arxiv","id":"2311.15051","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.15051","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"arxiv_version","alias_value":"2311.15051v3","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.15051","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"pith_short_12","alias_value":"6IJ3EQSZPDRO","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"pith_short_16","alias_value":"6IJ3EQSZPDROL47Q","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"pith_short_8","alias_value":"6IJ3EQSZ","created_at":"2026-07-05T08:24:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:6IJ3EQSZPDROL47QIMJOWUHXH3","target":"record","payload":{"canonical_record":{"source":{"id":"2311.15051","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-11-25T15:10:00Z","cross_cats_sorted":["math.OC","stat.ML"],"title_canon_sha256":"0bd018627a285d82c5ea37f6cce4827736a30ee919333c40202c1fbe24931736","abstract_canon_sha256":"8b95eb1c503afd7052566c8f366ef722241781a7c3139b6edb9da0e9e2a58de9"},"schema_version":"1.0"},"canonical_sha256":"f213b2425978e2e5f3f04312eb50f73efbc9e34a7b78081121342104e136a158","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:24:23.151492Z","signature_b64":"q/skbQjviZl86AleseGvnNjrYzpue1Yq6wqDKHo3QVIQ3UbHUT89IK3bihScGOkZWG6tfpvcKxxJ/sWbZKRFDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f213b2425978e2e5f3f04312eb50f73efbc9e34a7b78081121342104e136a158","last_reissued_at":"2026-07-05T08:24:23.151005Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:24:23.151005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2311.15051","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:24:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sHctYa/PkBA/tUaDNubukZ79rC9ZKnuJi2LIWHPoAsK+/k1wi797NNjOmzHXXDcxbEUO29j0rGxwMM89AAKIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T06:32:32.054725Z"},"content_sha256":"22dc4ad37d5dae3bcdadb4b2daad000303578306e3324f046ca3f70a2799d171","schema_version":"1.0","event_id":"sha256:22dc4ad37d5dae3bcdadb4b2daad000303578306e3324f046ca3f70a2799d171"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:6IJ3EQSZPDROL47QIMJOWUHXH3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Bohan Wang, Chulhee Yun, Huishuai Zhang, Junghyun Lee, Prin Phunyaphibarn","submitted_at":"2023-11-25T15:10:00Z","abstract_excerpt":"Although gradient descent with Polyak's momentum is widely used in modern machine and deep learning, a concrete understanding of its effects on the training trajectory remains elusive. In this work, we empirically show that for linear diagonal networks and nonlinear neural networks, momentum gradient descent with a large learning rate displays large catapults, driving the iterates towards much flatter minima than those found by gradient descent. We hypothesize that the large catapult is caused by momentum \"prolonging\" the self-stabilization effect (Damian et al., 2023). We provide theoretical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.15051","kind":"arxiv","version":3},"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/2311.15051/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:24:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/qmtI5WTN+7hdxDiSliAuPA6ED42oEmV1MBZD7u/wdtCTnvfvAce94E32jjz5ss83cg3Mz8f9vmOjq80TRnlDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T06:32:32.055432Z"},"content_sha256":"94e62a04474535b08c86d34a7c0e41114545fe3527d844e2cd19eaefb03adf74","schema_version":"1.0","event_id":"sha256:94e62a04474535b08c86d34a7c0e41114545fe3527d844e2cd19eaefb03adf74"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6IJ3EQSZPDROL47QIMJOWUHXH3/bundle.json","state_url":"https://pith.science/pith/6IJ3EQSZPDROL47QIMJOWUHXH3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6IJ3EQSZPDROL47QIMJOWUHXH3/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-05T06:32:32Z","links":{"resolver":"https://pith.science/pith/6IJ3EQSZPDROL47QIMJOWUHXH3","bundle":"https://pith.science/pith/6IJ3EQSZPDROL47QIMJOWUHXH3/bundle.json","state":"https://pith.science/pith/6IJ3EQSZPDROL47QIMJOWUHXH3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6IJ3EQSZPDROL47QIMJOWUHXH3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:6IJ3EQSZPDROL47QIMJOWUHXH3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8b95eb1c503afd7052566c8f366ef722241781a7c3139b6edb9da0e9e2a58de9","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-11-25T15:10:00Z","title_canon_sha256":"0bd018627a285d82c5ea37f6cce4827736a30ee919333c40202c1fbe24931736"},"schema_version":"1.0","source":{"id":"2311.15051","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2311.15051","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"arxiv_version","alias_value":"2311.15051v3","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.15051","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"pith_short_12","alias_value":"6IJ3EQSZPDRO","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"pith_short_16","alias_value":"6IJ3EQSZPDROL47Q","created_at":"2026-07-05T08:24:23Z"},{"alias_kind":"pith_short_8","alias_value":"6IJ3EQSZ","created_at":"2026-07-05T08:24:23Z"}],"graph_snapshots":[{"event_id":"sha256:94e62a04474535b08c86d34a7c0e41114545fe3527d844e2cd19eaefb03adf74","target":"graph","created_at":"2026-07-05T08:24:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2311.15051/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Although gradient descent with Polyak's momentum is widely used in modern machine and deep learning, a concrete understanding of its effects on the training trajectory remains elusive. In this work, we empirically show that for linear diagonal networks and nonlinear neural networks, momentum gradient descent with a large learning rate displays large catapults, driving the iterates towards much flatter minima than those found by gradient descent. We hypothesize that the large catapult is caused by momentum \"prolonging\" the self-stabilization effect (Damian et al., 2023). We provide theoretical ","authors_text":"Bohan Wang, Chulhee Yun, Huishuai Zhang, Junghyun Lee, Prin Phunyaphibarn","cross_cats":["math.OC","stat.ML"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-11-25T15:10:00Z","title":"Gradient Descent with Polyak's Momentum Finds Flatter Minima via Large Catapults"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.15051","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:22dc4ad37d5dae3bcdadb4b2daad000303578306e3324f046ca3f70a2799d171","target":"record","created_at":"2026-07-05T08:24:23Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8b95eb1c503afd7052566c8f366ef722241781a7c3139b6edb9da0e9e2a58de9","cross_cats_sorted":["math.OC","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2023-11-25T15:10:00Z","title_canon_sha256":"0bd018627a285d82c5ea37f6cce4827736a30ee919333c40202c1fbe24931736"},"schema_version":"1.0","source":{"id":"2311.15051","kind":"arxiv","version":3}},"canonical_sha256":"f213b2425978e2e5f3f04312eb50f73efbc9e34a7b78081121342104e136a158","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f213b2425978e2e5f3f04312eb50f73efbc9e34a7b78081121342104e136a158","first_computed_at":"2026-07-05T08:24:23.151005Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:24:23.151005Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q/skbQjviZl86AleseGvnNjrYzpue1Yq6wqDKHo3QVIQ3UbHUT89IK3bihScGOkZWG6tfpvcKxxJ/sWbZKRFDg==","signature_status":"signed_v1","signed_at":"2026-07-05T08:24:23.151492Z","signed_message":"canonical_sha256_bytes"},"source_id":"2311.15051","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22dc4ad37d5dae3bcdadb4b2daad000303578306e3324f046ca3f70a2799d171","sha256:94e62a04474535b08c86d34a7c0e41114545fe3527d844e2cd19eaefb03adf74"],"state_sha256":"94fc0b7270e0289202e105b82811ffadbddc0aef31dc20206eeaa2c4b4433be3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q5WFhIA2lMT+dzvNMc/pzNwOU2zm81+Irr5G3I4dclC4BRU2qLP0WqZ+VNHZHrEd9Izd4+brAkoCm4kfNz5rBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T06:32:32.064521Z","bundle_sha256":"8dcf7c84897d5e4ca867f6db314c66f15ad74ce2d476195f3a1c10d82d5fb864"}}