{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:4TWACHJL35DWDN4FKE3B2F7RUN","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":"4b3d83789b7467b53cc2efb62e217a79a020ed1ccfeb52a2134fc397e477a4d2","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"stat.ML","submitted_at":"2024-05-23T08:00:45Z","title_canon_sha256":"e79fc20c2e212876f6656888edf6a7c156293190de9c8da8d0396784c6706741"},"schema_version":"1.0","source":{"id":"2405.14285","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.14285","created_at":"2026-07-05T09:25:43Z"},{"alias_kind":"arxiv_version","alias_value":"2405.14285v2","created_at":"2026-07-05T09:25:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.14285","created_at":"2026-07-05T09:25:43Z"},{"alias_kind":"pith_short_12","alias_value":"4TWACHJL35DW","created_at":"2026-07-05T09:25:43Z"},{"alias_kind":"pith_short_16","alias_value":"4TWACHJL35DWDN4F","created_at":"2026-07-05T09:25:43Z"},{"alias_kind":"pith_short_8","alias_value":"4TWACHJL","created_at":"2026-07-05T09:25:43Z"}],"graph_snapshots":[{"event_id":"sha256:254fcf16de0b3f75576506fecb93104141c3a25642ec34e38ce5b7ae69a75cdc","target":"graph","created_at":"2026-07-05T09:25:43Z","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/2405.14285/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study stochastic approximation algorithms with Markovian noise and constant step-size $\\alpha$. We develop a method based on infinitesimal generator comparisons to study the bias of the algorithm, which is the expected difference between $\\theta_n$ -- the value at iteration $n$ -- and $\\theta^*$ -- the unique equilibrium of the corresponding ODE. We show that, under some smoothness conditions, this bias is of order $O(\\alpha)$. Furthermore, we show that the time-averaged bias is equal to $\\alpha V + O(\\alpha^2)$, where $V$ is a constant characterized by a Lyapunov equation, showing that $\\m","authors_text":"Nicolas Gast, Sebastian Allmeier","cross_cats":["cs.LG","math.OC"],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"stat.ML","submitted_at":"2024-05-23T08:00:45Z","title":"Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.14285","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":"sha256:77e9a26ef7365adcdeda5b3bb41cc681e2403ebb260d9b1a02c216454ce8ecaa","target":"record","created_at":"2026-07-05T09:25:43Z","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":"4b3d83789b7467b53cc2efb62e217a79a020ed1ccfeb52a2134fc397e477a4d2","cross_cats_sorted":["cs.LG","math.OC"],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"stat.ML","submitted_at":"2024-05-23T08:00:45Z","title_canon_sha256":"e79fc20c2e212876f6656888edf6a7c156293190de9c8da8d0396784c6706741"},"schema_version":"1.0","source":{"id":"2405.14285","kind":"arxiv","version":2}},"canonical_sha256":"e4ec011d2bdf4761b78551361d17f1a34083fb4b3ca289e7aa81834c904b61fa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e4ec011d2bdf4761b78551361d17f1a34083fb4b3ca289e7aa81834c904b61fa","first_computed_at":"2026-07-05T09:25:43.496508Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:25:43.496508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tpJwF7EAdjxgSb17lp9ArffRLf8VEapHrUHM3DHzSWba+sy5Vy52DKcR8aV+Vsp834M4+d4kai0TA1coSslwDg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:25:43.497001Z","signed_message":"canonical_sha256_bytes"},"source_id":"2405.14285","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:77e9a26ef7365adcdeda5b3bb41cc681e2403ebb260d9b1a02c216454ce8ecaa","sha256:254fcf16de0b3f75576506fecb93104141c3a25642ec34e38ce5b7ae69a75cdc"],"state_sha256":"da39bc6491adb55097faf33906328d43de68a37672d6ef91398ebb71707694e2"}