{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:BHSZSRL6KIEAOXWSH2WY6CJAEF","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":"d774b5ca31abdc627be8c0c2b3197936a16e6b92a7b7a649302790c5747c3080","cross_cats_sorted":["math.DS","math.OC","math.PR","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-10-02T22:59:17Z","title_canon_sha256":"3f0e8fd8e89711c92567fd530c0e621e67b18864fa0f89951623f1d76b92949d"},"schema_version":"1.0","source":{"id":"2310.01687","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.01687","created_at":"2026-07-05T06:56:44Z"},{"alias_kind":"arxiv_version","alias_value":"2310.01687v1","created_at":"2026-07-05T06:56:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.01687","created_at":"2026-07-05T06:56:44Z"},{"alias_kind":"pith_short_12","alias_value":"BHSZSRL6KIEA","created_at":"2026-07-05T06:56:44Z"},{"alias_kind":"pith_short_16","alias_value":"BHSZSRL6KIEAOXWS","created_at":"2026-07-05T06:56:44Z"},{"alias_kind":"pith_short_8","alias_value":"BHSZSRL6","created_at":"2026-07-05T06:56:44Z"}],"graph_snapshots":[{"event_id":"sha256:5b73e348efff080b56c2d45415c198fd7fc6aa31cad37be69eea7ca5c0277d3f","target":"graph","created_at":"2026-07-05T06:56:44Z","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/2310.01687/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We conduct a comprehensive investigation into the dynamics of gradient descent using large-order constant step-sizes in the context of quadratic regression models. Within this framework, we reveal that the dynamics can be encapsulated by a specific cubic map, naturally parameterized by the step-size. Through a fine-grained bifurcation analysis concerning the step-size parameter, we delineate five distinct training phases: (1) monotonic, (2) catapult, (3) periodic, (4) chaotic, and (5) divergent, precisely demarcating the boundaries of each phase. As illustrations, we provide examples involving","authors_text":"Bhavya Agrawalla, Krishnakumar Balasubramanian, Promit Ghosal, Xuxing Chen","cross_cats":["math.DS","math.OC","math.PR","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-10-02T22:59:17Z","title":"From Stability to Chaos: Analyzing Gradient Descent Dynamics in Quadratic Regression"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.01687","kind":"arxiv","version":1},"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:b98acd955491497fec34333aa4c8f7ee0bd36065d8e766b97e892366a4c0c346","target":"record","created_at":"2026-07-05T06:56:44Z","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":"d774b5ca31abdc627be8c0c2b3197936a16e6b92a7b7a649302790c5747c3080","cross_cats_sorted":["math.DS","math.OC","math.PR","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2023-10-02T22:59:17Z","title_canon_sha256":"3f0e8fd8e89711c92567fd530c0e621e67b18864fa0f89951623f1d76b92949d"},"schema_version":"1.0","source":{"id":"2310.01687","kind":"arxiv","version":1}},"canonical_sha256":"09e599457e5208075ed23ead8f092021744b6bbfe6c3011d9a910f797d879ad7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09e599457e5208075ed23ead8f092021744b6bbfe6c3011d9a910f797d879ad7","first_computed_at":"2026-07-05T06:56:44.825409Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:56:44.825409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"esFyLQojWaPFxSA7Qxgb7kDQpvNydkYPjz7fEF2E2swGSqK/2nTR4n+hzLthUP30Tj+tilNqOtP7N9VQpPl9Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T06:56:44.825872Z","signed_message":"canonical_sha256_bytes"},"source_id":"2310.01687","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b98acd955491497fec34333aa4c8f7ee0bd36065d8e766b97e892366a4c0c346","sha256:5b73e348efff080b56c2d45415c198fd7fc6aa31cad37be69eea7ca5c0277d3f"],"state_sha256":"2c20c7bac3d7abd620a927a1c44ddb1a493f695156582106acb68f7610eb1fde"}