{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:HIA5T2HDMV7GCEDVHIOLWLK23I","short_pith_number":"pith:HIA5T2HD","schema_version":"1.0","canonical_sha256":"3a01d9e8e3657e6110753a1cbb2d5ada395917a4628535963cde2c90b8f79317","source":{"kind":"arxiv","id":"2405.18373","version":3},"attestation_state":"computed","paper":{"title":"A Hessian-Aware Stochastic Differential Equation for Modelling SGD","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.OC"],"primary_cat":"stat.ML","authors_text":"Liang Zhang, Niao He, Xiang Li, Zebang Shen","submitted_at":"2024-05-28T17:11:34Z","abstract_excerpt":"Continuous-time approximation of Stochastic Gradient Descent (SGD) is a crucial tool to study its escaping behaviors from stationary points. However, existing stochastic differential equation (SDE) models fail to fully capture these behaviors, even for simple quadratic objectives. Built on a novel stochastic backward error analysis framework, we derive the Hessian-Aware Stochastic Modified Equation (HA-SME), an SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. Our analysis shows that HA-SME achieves the order-best approximation error g"},"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":"2405.18373","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2024-05-28T17:11:34Z","cross_cats_sorted":["cs.LG","math.OC"],"title_canon_sha256":"5b24623617396f06d1f9b3f529312f92d9111e4408ce72e44ab443b4b032e17e","abstract_canon_sha256":"567fd6e25490cda9a1f56b8bbbf0cb885be7e1a5bb780bf3d53948855565b1d0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:14:47.575950Z","signature_b64":"rkRW/bUuhzvu4k4UzIxwWt1q/8+BlfF58K+xW60jORopxXo6/A7PxhVEk4Ibtm58Meb9p9SgjCiMaQCduYpeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a01d9e8e3657e6110753a1cbb2d5ada395917a4628535963cde2c90b8f79317","last_reissued_at":"2026-07-05T11:14:47.575512Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:14:47.575512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Hessian-Aware Stochastic Differential Equation for Modelling SGD","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.OC"],"primary_cat":"stat.ML","authors_text":"Liang Zhang, Niao He, Xiang Li, Zebang Shen","submitted_at":"2024-05-28T17:11:34Z","abstract_excerpt":"Continuous-time approximation of Stochastic Gradient Descent (SGD) is a crucial tool to study its escaping behaviors from stationary points. However, existing stochastic differential equation (SDE) models fail to fully capture these behaviors, even for simple quadratic objectives. Built on a novel stochastic backward error analysis framework, we derive the Hessian-Aware Stochastic Modified Equation (HA-SME), an SDE that incorporates Hessian information of the objective function into both its drift and diffusion terms. Our analysis shows that HA-SME achieves the order-best approximation error g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.18373","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/2405.18373/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":"2405.18373","created_at":"2026-07-05T11:14:47.575580+00:00"},{"alias_kind":"arxiv_version","alias_value":"2405.18373v3","created_at":"2026-07-05T11:14:47.575580+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.18373","created_at":"2026-07-05T11:14:47.575580+00:00"},{"alias_kind":"pith_short_12","alias_value":"HIA5T2HDMV7G","created_at":"2026-07-05T11:14:47.575580+00:00"},{"alias_kind":"pith_short_16","alias_value":"HIA5T2HDMV7GCEDV","created_at":"2026-07-05T11:14:47.575580+00:00"},{"alias_kind":"pith_short_8","alias_value":"HIA5T2HD","created_at":"2026-07-05T11:14:47.575580+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.26016","citing_title":"MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generation","ref_index":24,"is_internal_anchor":false},{"citing_arxiv_id":"2606.07495","citing_title":"Second-Order Path Kernel Interpolation Formulas in Machine Learning","ref_index":115,"is_internal_anchor":false},{"citing_arxiv_id":"2606.26016","citing_title":"MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generation","ref_index":26,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I","json":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I.json","graph_json":"https://pith.science/api/pith-number/HIA5T2HDMV7GCEDVHIOLWLK23I/graph.json","events_json":"https://pith.science/api/pith-number/HIA5T2HDMV7GCEDVHIOLWLK23I/events.json","paper":"https://pith.science/paper/HIA5T2HD"},"agent_actions":{"view_html":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I","download_json":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I.json","view_paper":"https://pith.science/paper/HIA5T2HD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2405.18373&json=true","fetch_graph":"https://pith.science/api/pith-number/HIA5T2HDMV7GCEDVHIOLWLK23I/graph.json","fetch_events":"https://pith.science/api/pith-number/HIA5T2HDMV7GCEDVHIOLWLK23I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I/action/storage_attestation","attest_author":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I/action/author_attestation","sign_citation":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I/action/citation_signature","submit_replication":"https://pith.science/pith/HIA5T2HDMV7GCEDVHIOLWLK23I/action/replication_record"}},"created_at":"2026-07-05T11:14:47.575580+00:00","updated_at":"2026-07-05T11:14:47.575580+00:00"}