{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:ZLEJPH2P2E5PK6CHQWSHDM4KX2","short_pith_number":"pith:ZLEJPH2P","canonical_record":{"source":{"id":"1909.07278","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-09-16T15:31:25Z","cross_cats_sorted":["cs.LG","math.OC"],"title_canon_sha256":"a751e926c9d054f0aff54071c16c0d26feb5bb0421025ff9f961ad9c2975e75d","abstract_canon_sha256":"8fd3edd54773995fb0b04a843e73469982c47655d3890f1e7203e2cf38b532e7"},"schema_version":"1.0"},"canonical_sha256":"cac8979f4fd13af5784785a471b38abe9ef6b12bc9f76f6b0120803d119180fa","source":{"kind":"arxiv","id":"1909.07278","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.07278","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"arxiv_version","alias_value":"1909.07278v2","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.07278","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"pith_short_12","alias_value":"ZLEJPH2P2E5P","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"pith_short_16","alias_value":"ZLEJPH2P2E5PK6CH","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"pith_short_8","alias_value":"ZLEJPH2P","created_at":"2026-07-05T00:09:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:ZLEJPH2P2E5PK6CHQWSHDM4KX2","target":"record","payload":{"canonical_record":{"source":{"id":"1909.07278","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-09-16T15:31:25Z","cross_cats_sorted":["cs.LG","math.OC"],"title_canon_sha256":"a751e926c9d054f0aff54071c16c0d26feb5bb0421025ff9f961ad9c2975e75d","abstract_canon_sha256":"8fd3edd54773995fb0b04a843e73469982c47655d3890f1e7203e2cf38b532e7"},"schema_version":"1.0"},"canonical_sha256":"cac8979f4fd13af5784785a471b38abe9ef6b12bc9f76f6b0120803d119180fa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:09:32.163504Z","signature_b64":"eu93ShPP4Aeq5ZUz/jVdrrOx0qz1uicehaNMCsLw5O4zSDJfIS1stZcEEl2rmSA8vW+YAEr5CcJ+AlLPyR98DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cac8979f4fd13af5784785a471b38abe9ef6b12bc9f76f6b0120803d119180fa","last_reissued_at":"2026-07-05T00:09:32.163090Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:09:32.163090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1909.07278","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:09:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r0MVBS5urY/9/P7dAQhkKu2WvJZWccQGS0sBH45Ox8UUIl6npqlRLoypH2emKvLMnNdJD5YtUEcMYJjhdOlqAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T08:42:01.371138Z"},"content_sha256":"1d5a83dd66c0c2728741d519f8de5bb4817a9841ddc4c687a27004237e41bb4c","schema_version":"1.0","event_id":"sha256:1d5a83dd66c0c2728741d519f8de5bb4817a9841ddc4c687a27004237e41bb4c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:ZLEJPH2P2E5PK6CHQWSHDM4KX2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mean-field Langevin System, Optimal Control and Deep Neural Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.OC"],"primary_cat":"math.PR","authors_text":"Anna Kazeykina, Kaitong Hu, Zhenjie Ren","submitted_at":"2019-09-16T15:31:25Z","abstract_excerpt":"In this paper, we study a regularised relaxed optimal control problem and, in particular, we are concerned with the case where the control variable is of large dimension. We introduce a system of mean-field Langevin equations, the invariant measure of which is shown to be the optimal control of the initial problem under mild conditions. Therefore, this system of processes can be viewed as a continuous-time numerical algorithm for computing the optimal control. As an application, this result endorses the solvability of the stochastic gradient descent algorithm for a wide class of deep neural ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.07278","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/1909.07278/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":"2026-07-05T00:09:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eEvDmDvS2aaKID1qwD34Ci/HIE+0SZ6U1YmJbWBNu0aFRNRLl8eUnymVMDL5l7K4Kv47XX0uwFKgOziFDJ+9CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T08:42:01.371947Z"},"content_sha256":"53ef121723d1cd7d7e3342a367cbc6c61055ee5eb1e367abb3deffbcb5bed429","schema_version":"1.0","event_id":"sha256:53ef121723d1cd7d7e3342a367cbc6c61055ee5eb1e367abb3deffbcb5bed429"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2/bundle.json","state_url":"https://pith.science/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2/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":"2026-08-09T08:42:01Z","links":{"resolver":"https://pith.science/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2","bundle":"https://pith.science/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2/bundle.json","state":"https://pith.science/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLEJPH2P2E5PK6CHQWSHDM4KX2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:ZLEJPH2P2E5PK6CHQWSHDM4KX2","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":"8fd3edd54773995fb0b04a843e73469982c47655d3890f1e7203e2cf38b532e7","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-09-16T15:31:25Z","title_canon_sha256":"a751e926c9d054f0aff54071c16c0d26feb5bb0421025ff9f961ad9c2975e75d"},"schema_version":"1.0","source":{"id":"1909.07278","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1909.07278","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"arxiv_version","alias_value":"1909.07278v2","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1909.07278","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"pith_short_12","alias_value":"ZLEJPH2P2E5P","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"pith_short_16","alias_value":"ZLEJPH2P2E5PK6CH","created_at":"2026-07-05T00:09:32Z"},{"alias_kind":"pith_short_8","alias_value":"ZLEJPH2P","created_at":"2026-07-05T00:09:32Z"}],"graph_snapshots":[{"event_id":"sha256:53ef121723d1cd7d7e3342a367cbc6c61055ee5eb1e367abb3deffbcb5bed429","target":"graph","created_at":"2026-07-05T00:09:32Z","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/1909.07278/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study a regularised relaxed optimal control problem and, in particular, we are concerned with the case where the control variable is of large dimension. We introduce a system of mean-field Langevin equations, the invariant measure of which is shown to be the optimal control of the initial problem under mild conditions. Therefore, this system of processes can be viewed as a continuous-time numerical algorithm for computing the optimal control. As an application, this result endorses the solvability of the stochastic gradient descent algorithm for a wide class of deep neural ne","authors_text":"Anna Kazeykina, Kaitong Hu, Zhenjie Ren","cross_cats":["cs.LG","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-09-16T15:31:25Z","title":"Mean-field Langevin System, Optimal Control and Deep Neural Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.07278","kind":"arxiv","version":2},"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":"sha256:1d5a83dd66c0c2728741d519f8de5bb4817a9841ddc4c687a27004237e41bb4c","target":"record","created_at":"2026-07-05T00:09:32Z","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":"8fd3edd54773995fb0b04a843e73469982c47655d3890f1e7203e2cf38b532e7","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-09-16T15:31:25Z","title_canon_sha256":"a751e926c9d054f0aff54071c16c0d26feb5bb0421025ff9f961ad9c2975e75d"},"schema_version":"1.0","source":{"id":"1909.07278","kind":"arxiv","version":2}},"canonical_sha256":"cac8979f4fd13af5784785a471b38abe9ef6b12bc9f76f6b0120803d119180fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cac8979f4fd13af5784785a471b38abe9ef6b12bc9f76f6b0120803d119180fa","first_computed_at":"2026-07-05T00:09:32.163090Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:09:32.163090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eu93ShPP4Aeq5ZUz/jVdrrOx0qz1uicehaNMCsLw5O4zSDJfIS1stZcEEl2rmSA8vW+YAEr5CcJ+AlLPyR98DA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:09:32.163504Z","signed_message":"canonical_sha256_bytes"},"source_id":"1909.07278","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1d5a83dd66c0c2728741d519f8de5bb4817a9841ddc4c687a27004237e41bb4c","sha256:53ef121723d1cd7d7e3342a367cbc6c61055ee5eb1e367abb3deffbcb5bed429"],"state_sha256":"a32a1fb6b3621a789863d8d2b7fde3a7065c16528ceb8c5484105d08eb34cf4f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L5w6ex6zKnJRS56N01jeDrnBu0xXT0kWTjXlR1amTLS/UtQtTcev5i202QrI0ValNOzTweGcz8+AUOtJ5VLwBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T08:42:01.377940Z","bundle_sha256":"09909f9035660768eaa6d9b9bc6e303bfc77bb34d67a51c9915d72ec71da1948"}}