{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:CNHEAXLGP3O6GPDGCTNSH6Z3AD","short_pith_number":"pith:CNHEAXLG","canonical_record":{"source":{"id":"2303.03944","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.OC","submitted_at":"2023-03-07T14:55:05Z","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"title_canon_sha256":"b2b25e642678b57750d3267e9892cb5add4df69609c2ad290c873338d5281e16","abstract_canon_sha256":"d72292d1e1002b0a726d7feeab9a43c96ccb230dd8145f63c06d73ad32378300"},"schema_version":"1.0"},"canonical_sha256":"134e405d667edde33c6614db23fb3b00f764648356071193a7a19ea3592fdea4","source":{"kind":"arxiv","id":"2303.03944","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.03944","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"arxiv_version","alias_value":"2303.03944v4","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.03944","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"pith_short_12","alias_value":"CNHEAXLGP3O6","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"pith_short_16","alias_value":"CNHEAXLGP3O6GPDG","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"pith_short_8","alias_value":"CNHEAXLG","created_at":"2026-07-05T07:14:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:CNHEAXLGP3O6GPDGCTNSH6Z3AD","target":"record","payload":{"canonical_record":{"source":{"id":"2303.03944","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.OC","submitted_at":"2023-03-07T14:55:05Z","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"title_canon_sha256":"b2b25e642678b57750d3267e9892cb5add4df69609c2ad290c873338d5281e16","abstract_canon_sha256":"d72292d1e1002b0a726d7feeab9a43c96ccb230dd8145f63c06d73ad32378300"},"schema_version":"1.0"},"canonical_sha256":"134e405d667edde33c6614db23fb3b00f764648356071193a7a19ea3592fdea4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:14:09.999186Z","signature_b64":"SO5tQ453I7fHRpLFN9jhH12F8b5+PNquM3fUGVVQKsvBfmdJNksrDtYIkHd9gWOwFuQhiYXrBYPKcLDgDyOJCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"134e405d667edde33c6614db23fb3b00f764648356071193a7a19ea3592fdea4","last_reissued_at":"2026-07-05T07:14:09.998801Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:14:09.998801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.03944","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:14:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R0o0hFLlEnVMFFhiNATRwjqe2nEnoFLcRWZiABWJmlZww1hH4B2qkcD8F3DVnxdeJw/S8RF6JqZag+UACTfyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T00:11:52.419077Z"},"content_sha256":"a006c9130bc16bbb9e95952e431949506b561bb0b2492b1546714ff3925a931f","schema_version":"1.0","event_id":"sha256:a006c9130bc16bbb9e95952e431949506b561bb0b2492b1546714ff3925a931f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:CNHEAXLGP3O6GPDGCTNSH6Z3AD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Momentum-Based Gradient Methods for Bilevel Optimization with Nonconvex Lower-Level","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.LG","cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Feihu Huang","submitted_at":"2023-03-07T14:55:05Z","abstract_excerpt":"Bilevel optimization is a popular two-level hierarchical optimization, which has been widely applied to many machine learning tasks such as hyperparameter learning, meta learning and continual learning. Although many bilevel optimization methods recently have been developed, the bilevel methods are not well studied when the lower-level problem is nonconvex. To fill this gap, in the paper, we study a class of nonconvex bilevel optimization problems, where both upper-level and lower-level problems are nonconvex, and the lower-level problem satisfies Polyak-{\\L}ojasiewicz (PL) condition. We propo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.03944","kind":"arxiv","version":4},"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/2303.03944/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":"2026-07-05T07:14:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KoIiE8eDOoq9zd4Xv5KlOR2Dkaaj2q8EB6XdbY3LiOVCouheUZ1e3s9DW0ROexSwmss/LWnkk6K4u8D6nrPBDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T00:11:52.420005Z"},"content_sha256":"15255686c609d6a595195d6de1463b4be306a885988acbd2838ae3737fb66de8","schema_version":"1.0","event_id":"sha256:15255686c609d6a595195d6de1463b4be306a885988acbd2838ae3737fb66de8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD/bundle.json","state_url":"https://pith.science/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD/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":"2026-08-07T00:11:52Z","links":{"resolver":"https://pith.science/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD","bundle":"https://pith.science/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD/bundle.json","state":"https://pith.science/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CNHEAXLGP3O6GPDGCTNSH6Z3AD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:CNHEAXLGP3O6GPDGCTNSH6Z3AD","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":"d72292d1e1002b0a726d7feeab9a43c96ccb230dd8145f63c06d73ad32378300","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.OC","submitted_at":"2023-03-07T14:55:05Z","title_canon_sha256":"b2b25e642678b57750d3267e9892cb5add4df69609c2ad290c873338d5281e16"},"schema_version":"1.0","source":{"id":"2303.03944","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.03944","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"arxiv_version","alias_value":"2303.03944v4","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.03944","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"pith_short_12","alias_value":"CNHEAXLGP3O6","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"pith_short_16","alias_value":"CNHEAXLGP3O6GPDG","created_at":"2026-07-05T07:14:09Z"},{"alias_kind":"pith_short_8","alias_value":"CNHEAXLG","created_at":"2026-07-05T07:14:09Z"}],"graph_snapshots":[{"event_id":"sha256:15255686c609d6a595195d6de1463b4be306a885988acbd2838ae3737fb66de8","target":"graph","created_at":"2026-07-05T07:14:09Z","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/2303.03944/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Bilevel optimization is a popular two-level hierarchical optimization, which has been widely applied to many machine learning tasks such as hyperparameter learning, meta learning and continual learning. Although many bilevel optimization methods recently have been developed, the bilevel methods are not well studied when the lower-level problem is nonconvex. To fill this gap, in the paper, we study a class of nonconvex bilevel optimization problems, where both upper-level and lower-level problems are nonconvex, and the lower-level problem satisfies Polyak-{\\L}ojasiewicz (PL) condition. We propo","authors_text":"Feihu Huang","cross_cats":["cs.LG","cs.NA","math.NA"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.OC","submitted_at":"2023-03-07T14:55:05Z","title":"On Momentum-Based Gradient Methods for Bilevel Optimization with Nonconvex Lower-Level"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.03944","kind":"arxiv","version":4},"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:a006c9130bc16bbb9e95952e431949506b561bb0b2492b1546714ff3925a931f","target":"record","created_at":"2026-07-05T07:14:09Z","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":"d72292d1e1002b0a726d7feeab9a43c96ccb230dd8145f63c06d73ad32378300","cross_cats_sorted":["cs.LG","cs.NA","math.NA"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.OC","submitted_at":"2023-03-07T14:55:05Z","title_canon_sha256":"b2b25e642678b57750d3267e9892cb5add4df69609c2ad290c873338d5281e16"},"schema_version":"1.0","source":{"id":"2303.03944","kind":"arxiv","version":4}},"canonical_sha256":"134e405d667edde33c6614db23fb3b00f764648356071193a7a19ea3592fdea4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"134e405d667edde33c6614db23fb3b00f764648356071193a7a19ea3592fdea4","first_computed_at":"2026-07-05T07:14:09.998801Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:14:09.998801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SO5tQ453I7fHRpLFN9jhH12F8b5+PNquM3fUGVVQKsvBfmdJNksrDtYIkHd9gWOwFuQhiYXrBYPKcLDgDyOJCA==","signature_status":"signed_v1","signed_at":"2026-07-05T07:14:09.999186Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.03944","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a006c9130bc16bbb9e95952e431949506b561bb0b2492b1546714ff3925a931f","sha256:15255686c609d6a595195d6de1463b4be306a885988acbd2838ae3737fb66de8"],"state_sha256":"d31cf562b13230ad391bdd1a903bf645b65a77410021e97bc4e9048f8cd39dc1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U7Rp3uU04qFaMckpve04t7siQcoAThZJdmFR2mEk8hePBcAlfh0Hyus+LhV10gn68P4M5oOKVcSLTNqj7US4Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T00:11:52.425963Z","bundle_sha256":"e05829bc04f0549bf601a51fd7c478aca436e90178f917e1d32f34d19ab78116"}}