{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:M6FCWYQUQVFUE5G4QBKH2O5PYM","short_pith_number":"pith:M6FCWYQU","canonical_record":{"source":{"id":"1806.04781","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-12T21:56:48Z","cross_cats_sorted":[],"title_canon_sha256":"8891c2a2de1e785cca1d245406f1ea0ee898c840a1fe76c3e01f34013b53bb4a","abstract_canon_sha256":"6a845e12c366b00173ccde53fbfba5a26467331611f767f9993e0e82bbc887c9"},"schema_version":"1.0"},"canonical_sha256":"678a2b6214854b4274dc80547d3bafc309dffad81ef6a8bf499711887d7f7d87","source":{"kind":"arxiv","id":"1806.04781","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.04781","created_at":"2026-05-18T00:13:20Z"},{"alias_kind":"arxiv_version","alias_value":"1806.04781v1","created_at":"2026-05-18T00:13:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.04781","created_at":"2026-05-18T00:13:20Z"},{"alias_kind":"pith_short_12","alias_value":"M6FCWYQUQVFU","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"M6FCWYQUQVFUE5G4","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"M6FCWYQU","created_at":"2026-05-18T12:32:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:M6FCWYQUQVFUE5G4QBKH2O5PYM","target":"record","payload":{"canonical_record":{"source":{"id":"1806.04781","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-12T21:56:48Z","cross_cats_sorted":[],"title_canon_sha256":"8891c2a2de1e785cca1d245406f1ea0ee898c840a1fe76c3e01f34013b53bb4a","abstract_canon_sha256":"6a845e12c366b00173ccde53fbfba5a26467331611f767f9993e0e82bbc887c9"},"schema_version":"1.0"},"canonical_sha256":"678a2b6214854b4274dc80547d3bafc309dffad81ef6a8bf499711887d7f7d87","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:20.902226Z","signature_b64":"7lhKYgFAfJotQJb9s2qBDvwqopxzuGwz6n8uoSNneHwfZCe94GdEwczguB5vDqZ1iZ0YhPNpJpl/mjxyynaHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"678a2b6214854b4274dc80547d3bafc309dffad81ef6a8bf499711887d7f7d87","last_reissued_at":"2026-05-18T00:13:20.901737Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:20.901737Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.04781","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":"2026-05-18T00:13:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tgTWbIWoo9i3l6PXTtjeQw0yNrIXsDIfgy7xUEGhyRV+7VyCeY7TKJv08u2CbFUI8IfvzHxdCNCWop4uZtj4Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T15:01:10.251418Z"},"content_sha256":"e48eb6a189b36aae2b0e6a27811e46399e45c029cea53f913c0d4ef0cb91bcf1","schema_version":"1.0","event_id":"sha256:e48eb6a189b36aae2b0e6a27811e46399e45c029cea53f913c0d4ef0cb91bcf1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:M6FCWYQUQVFUE5G4QBKH2O5PYM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Convergence Rate of Stochastic Mirror Descent for Nonsmooth Nonconvex Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Niao He, Siqi Zhang","submitted_at":"2018-06-12T21:56:48Z","abstract_excerpt":"In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of $ \\mathcal{O}(1/\\sqrt{t}) $. The efficiency estimate matches with existing results for stochastic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04781","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":""},"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":"2026-05-18T00:13:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dlKnrCGX3O8EpPYthh3svJmmQv58LIcrETJ1NxH+xzCK3GY3O3smo+FgSW5vJ/wjsPI1iUqYxDYPchxgFEjIBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T15:01:10.251950Z"},"content_sha256":"5467d535c1fda477b14bb725d70ab62869254dae37584c12c02b35c8f3c52a61","schema_version":"1.0","event_id":"sha256:5467d535c1fda477b14bb725d70ab62869254dae37584c12c02b35c8f3c52a61"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM/bundle.json","state_url":"https://pith.science/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM/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":"2026-05-29T15:01:10Z","links":{"resolver":"https://pith.science/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM","bundle":"https://pith.science/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM/bundle.json","state":"https://pith.science/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M6FCWYQUQVFUE5G4QBKH2O5PYM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:M6FCWYQUQVFUE5G4QBKH2O5PYM","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":"6a845e12c366b00173ccde53fbfba5a26467331611f767f9993e0e82bbc887c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-12T21:56:48Z","title_canon_sha256":"8891c2a2de1e785cca1d245406f1ea0ee898c840a1fe76c3e01f34013b53bb4a"},"schema_version":"1.0","source":{"id":"1806.04781","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.04781","created_at":"2026-05-18T00:13:20Z"},{"alias_kind":"arxiv_version","alias_value":"1806.04781v1","created_at":"2026-05-18T00:13:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.04781","created_at":"2026-05-18T00:13:20Z"},{"alias_kind":"pith_short_12","alias_value":"M6FCWYQUQVFU","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"M6FCWYQUQVFUE5G4","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"M6FCWYQU","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:5467d535c1fda477b14bb725d70ab62869254dae37584c12c02b35c8f3c52a61","target":"graph","created_at":"2026-05-18T00:13:20Z","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"},"paper":{"abstract_excerpt":"In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of $ \\mathcal{O}(1/\\sqrt{t}) $. The efficiency estimate matches with existing results for stochastic","authors_text":"Niao He, Siqi Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-12T21:56:48Z","title":"On the Convergence Rate of Stochastic Mirror Descent for Nonsmooth Nonconvex Optimization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.04781","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:e48eb6a189b36aae2b0e6a27811e46399e45c029cea53f913c0d4ef0cb91bcf1","target":"record","created_at":"2026-05-18T00:13:20Z","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":"6a845e12c366b00173ccde53fbfba5a26467331611f767f9993e0e82bbc887c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-12T21:56:48Z","title_canon_sha256":"8891c2a2de1e785cca1d245406f1ea0ee898c840a1fe76c3e01f34013b53bb4a"},"schema_version":"1.0","source":{"id":"1806.04781","kind":"arxiv","version":1}},"canonical_sha256":"678a2b6214854b4274dc80547d3bafc309dffad81ef6a8bf499711887d7f7d87","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"678a2b6214854b4274dc80547d3bafc309dffad81ef6a8bf499711887d7f7d87","first_computed_at":"2026-05-18T00:13:20.901737Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:20.901737Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7lhKYgFAfJotQJb9s2qBDvwqopxzuGwz6n8uoSNneHwfZCe94GdEwczguB5vDqZ1iZ0YhPNpJpl/mjxyynaHAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:20.902226Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.04781","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e48eb6a189b36aae2b0e6a27811e46399e45c029cea53f913c0d4ef0cb91bcf1","sha256:5467d535c1fda477b14bb725d70ab62869254dae37584c12c02b35c8f3c52a61"],"state_sha256":"598ab7f2258cbbcf625ae13b5910712a1da91f0355dd3a46cb0097e9119e48f3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q1YiaFtCy8wBfKcGt1CFuAMOoSN9pOlCluUwVhemW/p9vVtOn5X5CaFT5N+T1CoD1+gpRpynhbLpXHwPY4p3DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T15:01:10.255477Z","bundle_sha256":"e3f30adc5775deea53e3cff7cd9afeb2a28955e593f16f100e4d57047301e4e3"}}