{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:F66G75C2S47X7TVQMY5YDAN5BC","short_pith_number":"pith:F66G75C2","canonical_record":{"source":{"id":"2407.02322","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-07-02T14:52:21Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"8c6c742577a57c9f62b013505d4b6e421a4566c9e97be96f2139fb37da2825b3","abstract_canon_sha256":"523f45ec29b15ea182f1e66bbcefd0f2fa0bbe4cefe32b960e26e71e592aa156"},"schema_version":"1.0"},"canonical_sha256":"2fbc6ff45a973f7fceb0663b8181bd089706abb76b0ce13b860ccda45f824820","source":{"kind":"arxiv","id":"2407.02322","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.02322","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"arxiv_version","alias_value":"2407.02322v1","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.02322","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"pith_short_12","alias_value":"F66G75C2S47X","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"pith_short_16","alias_value":"F66G75C2S47X7TVQ","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"pith_short_8","alias_value":"F66G75C2","created_at":"2026-07-05T08:39:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:F66G75C2S47X7TVQMY5YDAN5BC","target":"record","payload":{"canonical_record":{"source":{"id":"2407.02322","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-07-02T14:52:21Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"8c6c742577a57c9f62b013505d4b6e421a4566c9e97be96f2139fb37da2825b3","abstract_canon_sha256":"523f45ec29b15ea182f1e66bbcefd0f2fa0bbe4cefe32b960e26e71e592aa156"},"schema_version":"1.0"},"canonical_sha256":"2fbc6ff45a973f7fceb0663b8181bd089706abb76b0ce13b860ccda45f824820","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:39:22.657878Z","signature_b64":"H0n1uSJ1y6FadvK0jXbCEjG2jAJ8iiLk1Qz+YRb1V5edCYgrz6z9shrTzsrSKlY+gZAA7Z9VzYbmrbAnRAXZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fbc6ff45a973f7fceb0663b8181bd089706abb76b0ce13b860ccda45f824820","last_reissued_at":"2026-07-05T08:39:22.657403Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:39:22.657403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.02322","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:39:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4lFNtlEMcO+JhZGlKoVUHWBFr5qebtnDOXAhTS49PEPGek5W9VCKBtKMNFaugOafHwng+AGCFGqFvESo09mDDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:58:43.139560Z"},"content_sha256":"447eeebb0d2a9d2a301dee701581bd46774c1c7e555b73de3b0f58733016bff8","schema_version":"1.0","event_id":"sha256:447eeebb0d2a9d2a301dee701581bd46774c1c7e555b73de3b0f58733016bff8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:F66G75C2S47X7TVQMY5YDAN5BC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stochastic Differential Equations models for Least-Squares Stochastic Gradient Descent","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cs.LG","authors_text":"Adrien Schertzer, Loucas Pillaud-Vivien","submitted_at":"2024-07-02T14:52:21Z","abstract_excerpt":"We study the dynamics of a continuous-time model of the Stochastic Gradient Descent (SGD) for the least-square problem. Indeed, pursuing the work of Li et al. (2019), we analyze Stochastic Differential Equations (SDEs) that model SGD either in the case of the training loss (finite samples) or the population one (online setting). A key qualitative feature of the dynamics is the existence of a perfect interpolator of the data, irrespective of the sample size. In both scenarios, we provide precise, non-asymptotic rates of convergence to the (possibly degenerate) stationary distribution. Additiona"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.02322","kind":"arxiv","version":1},"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/2407.02322/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":"2026-07-05T08:39:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2fs3x7s4am1ZVDF3687nFLOSysrLjZj7Q9+peXKBlpClMU0SmhWgf1mjIkWfwZex10kLklsVE5pqSynqlNLZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T06:58:43.140053Z"},"content_sha256":"d8d8b0160ccea43eee48443c4c113076ac6d407ef5c3351362a10f9b69355073","schema_version":"1.0","event_id":"sha256:d8d8b0160ccea43eee48443c4c113076ac6d407ef5c3351362a10f9b69355073"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F66G75C2S47X7TVQMY5YDAN5BC/bundle.json","state_url":"https://pith.science/pith/F66G75C2S47X7TVQMY5YDAN5BC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F66G75C2S47X7TVQMY5YDAN5BC/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":"2026-08-09T06:58:43Z","links":{"resolver":"https://pith.science/pith/F66G75C2S47X7TVQMY5YDAN5BC","bundle":"https://pith.science/pith/F66G75C2S47X7TVQMY5YDAN5BC/bundle.json","state":"https://pith.science/pith/F66G75C2S47X7TVQMY5YDAN5BC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F66G75C2S47X7TVQMY5YDAN5BC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:F66G75C2S47X7TVQMY5YDAN5BC","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":"523f45ec29b15ea182f1e66bbcefd0f2fa0bbe4cefe32b960e26e71e592aa156","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-07-02T14:52:21Z","title_canon_sha256":"8c6c742577a57c9f62b013505d4b6e421a4566c9e97be96f2139fb37da2825b3"},"schema_version":"1.0","source":{"id":"2407.02322","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.02322","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"arxiv_version","alias_value":"2407.02322v1","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.02322","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"pith_short_12","alias_value":"F66G75C2S47X","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"pith_short_16","alias_value":"F66G75C2S47X7TVQ","created_at":"2026-07-05T08:39:22Z"},{"alias_kind":"pith_short_8","alias_value":"F66G75C2","created_at":"2026-07-05T08:39:22Z"}],"graph_snapshots":[{"event_id":"sha256:d8d8b0160ccea43eee48443c4c113076ac6d407ef5c3351362a10f9b69355073","target":"graph","created_at":"2026-07-05T08:39:22Z","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/2407.02322/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the dynamics of a continuous-time model of the Stochastic Gradient Descent (SGD) for the least-square problem. Indeed, pursuing the work of Li et al. (2019), we analyze Stochastic Differential Equations (SDEs) that model SGD either in the case of the training loss (finite samples) or the population one (online setting). A key qualitative feature of the dynamics is the existence of a perfect interpolator of the data, irrespective of the sample size. In both scenarios, we provide precise, non-asymptotic rates of convergence to the (possibly degenerate) stationary distribution. Additiona","authors_text":"Adrien Schertzer, Loucas Pillaud-Vivien","cross_cats":["math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-07-02T14:52:21Z","title":"Stochastic Differential Equations models for Least-Squares Stochastic Gradient Descent"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.02322","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":"sha256:447eeebb0d2a9d2a301dee701581bd46774c1c7e555b73de3b0f58733016bff8","target":"record","created_at":"2026-07-05T08:39:22Z","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":"523f45ec29b15ea182f1e66bbcefd0f2fa0bbe4cefe32b960e26e71e592aa156","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-07-02T14:52:21Z","title_canon_sha256":"8c6c742577a57c9f62b013505d4b6e421a4566c9e97be96f2139fb37da2825b3"},"schema_version":"1.0","source":{"id":"2407.02322","kind":"arxiv","version":1}},"canonical_sha256":"2fbc6ff45a973f7fceb0663b8181bd089706abb76b0ce13b860ccda45f824820","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2fbc6ff45a973f7fceb0663b8181bd089706abb76b0ce13b860ccda45f824820","first_computed_at":"2026-07-05T08:39:22.657403Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:39:22.657403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"H0n1uSJ1y6FadvK0jXbCEjG2jAJ8iiLk1Qz+YRb1V5edCYgrz6z9shrTzsrSKlY+gZAA7Z9VzYbmrbAnRAXZCA==","signature_status":"signed_v1","signed_at":"2026-07-05T08:39:22.657878Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.02322","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:447eeebb0d2a9d2a301dee701581bd46774c1c7e555b73de3b0f58733016bff8","sha256:d8d8b0160ccea43eee48443c4c113076ac6d407ef5c3351362a10f9b69355073"],"state_sha256":"af20ea4c9f42bfded3ed7228c5605f11ce4ebe61f741ef1bb88c774e12ec1638"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c0eI5bdu5HGFhj3pIVTcPlrVaswzz18IHOn0T0/m+uNop2l5u250hULzj6ZC761YP7V5IvaTz8Ltb26ot3XlCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T06:58:43.144416Z","bundle_sha256":"4418a3a742c0f124d35390a06ae2d663bd95a3f0af344ab8cb6081b8b95baed9"}}