{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:VBJJJF7EXHAFBAIKHGHXXICVF2","short_pith_number":"pith:VBJJJF7E","schema_version":"1.0","canonical_sha256":"a8529497e4b9c050810a398f7ba0552e8953daf8bcf80f630236de216e586826","source":{"kind":"arxiv","id":"2307.11134","version":1},"attestation_state":"computed","paper":{"title":"Exact convergence rate of the last iterate in subgradient methods","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Fran\\c{c}ois Glineur, Moslem Zamani","submitted_at":"2023-07-20T16:27:12Z","abstract_excerpt":"We study the convergence of the last iterate in subgradient methods applied to the minimization of a nonsmooth convex function with bounded subgradients.\n  We first introduce a proof technique that generalizes the standard analysis of subgradient methods. It is based on tracking the distance between the current iterate and a different reference point at each iteration. Using this technique, we obtain the exact worst-case convergence rate for the objective accuracy of the last iterate of the projected subgradient method with either constant step sizes or constant step lengths. Tightness is show"},"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":"2307.11134","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.OC","submitted_at":"2023-07-20T16:27:12Z","cross_cats_sorted":[],"title_canon_sha256":"5d826d7c3974fe5c43451a9b247570cc8169a2a316fd110cb2d256a3941b76a4","abstract_canon_sha256":"895d5bf324f782c65c33b5d802a982a616e25714b5172cc9333df60c6f898509"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:33:23.988535Z","signature_b64":"FCtOER2c96VLiAcq7zu0qqUQcIlbS341Yq8MJoCMpzhc+nlj2nx4pq+kzFqVUZPwpOpY0dpqKq4VCCLZUqhdCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8529497e4b9c050810a398f7ba0552e8953daf8bcf80f630236de216e586826","last_reissued_at":"2026-07-05T06:33:23.988105Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:33:23.988105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact convergence rate of the last iterate in subgradient methods","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Fran\\c{c}ois Glineur, Moslem Zamani","submitted_at":"2023-07-20T16:27:12Z","abstract_excerpt":"We study the convergence of the last iterate in subgradient methods applied to the minimization of a nonsmooth convex function with bounded subgradients.\n  We first introduce a proof technique that generalizes the standard analysis of subgradient methods. It is based on tracking the distance between the current iterate and a different reference point at each iteration. Using this technique, we obtain the exact worst-case convergence rate for the objective accuracy of the last iterate of the projected subgradient method with either constant step sizes or constant step lengths. Tightness is show"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.11134","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/2307.11134/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":"2307.11134","created_at":"2026-07-05T06:33:23.988166+00:00"},{"alias_kind":"arxiv_version","alias_value":"2307.11134v1","created_at":"2026-07-05T06:33:23.988166+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.11134","created_at":"2026-07-05T06:33:23.988166+00:00"},{"alias_kind":"pith_short_12","alias_value":"VBJJJF7EXHAF","created_at":"2026-07-05T06:33:23.988166+00:00"},{"alias_kind":"pith_short_16","alias_value":"VBJJJF7EXHAFBAIK","created_at":"2026-07-05T06:33:23.988166+00:00"},{"alias_kind":"pith_short_8","alias_value":"VBJJJF7E","created_at":"2026-07-05T06:33:23.988166+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":4,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.11431","citing_title":"Mirror Descent Beyond Euclidean Stability: An Exponential Separation in Initialization Sensitivity","ref_index":16,"is_internal_anchor":false},{"citing_arxiv_id":"2606.30559","citing_title":"Convergence of Continual Learning in Homogeneous Deep Networks","ref_index":27,"is_internal_anchor":false},{"citing_arxiv_id":"2502.07529","citing_title":"Training Deep Learning Models with Norm-Constrained LMOs","ref_index":219,"is_internal_anchor":false},{"citing_arxiv_id":"2604.13870","citing_title":"Gradient Descent's Last Iterate is Often (slightly) Suboptimal","ref_index":29,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2","json":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2.json","graph_json":"https://pith.science/api/pith-number/VBJJJF7EXHAFBAIKHGHXXICVF2/graph.json","events_json":"https://pith.science/api/pith-number/VBJJJF7EXHAFBAIKHGHXXICVF2/events.json","paper":"https://pith.science/paper/VBJJJF7E"},"agent_actions":{"view_html":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2","download_json":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2.json","view_paper":"https://pith.science/paper/VBJJJF7E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2307.11134&json=true","fetch_graph":"https://pith.science/api/pith-number/VBJJJF7EXHAFBAIKHGHXXICVF2/graph.json","fetch_events":"https://pith.science/api/pith-number/VBJJJF7EXHAFBAIKHGHXXICVF2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2/action/storage_attestation","attest_author":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2/action/author_attestation","sign_citation":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2/action/citation_signature","submit_replication":"https://pith.science/pith/VBJJJF7EXHAFBAIKHGHXXICVF2/action/replication_record"}},"created_at":"2026-07-05T06:33:23.988166+00:00","updated_at":"2026-07-05T06:33:23.988166+00:00"}