{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:SRQTNCAEFH4S7DQGYLT2VC6HNV","short_pith_number":"pith:SRQTNCAE","canonical_record":{"source":{"id":"2605.20999","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-20T10:38:08Z","cross_cats_sorted":["cs.LG","math.OC","stat.ML"],"title_canon_sha256":"3d2cd5a58dfa168b317a787ef0c9e2a7767679ddd64bb864269f0c79254d20a4","abstract_canon_sha256":"5a8834395c235cbafe246ec45a4cae4a0e3a696a1b213c7f1204e715354bea9a"},"schema_version":"1.0"},"canonical_sha256":"946136880429f92f8e06c2e7aa8bc76d74cc88d65c91aad21cd6ef52156a3ea4","source":{"kind":"arxiv","id":"2605.20999","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20999","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20999v1","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20999","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"pith_short_12","alias_value":"SRQTNCAEFH4S","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"pith_short_16","alias_value":"SRQTNCAEFH4S7DQG","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"pith_short_8","alias_value":"SRQTNCAE","created_at":"2026-05-21T01:05:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:SRQTNCAEFH4S7DQGYLT2VC6HNV","target":"record","payload":{"canonical_record":{"source":{"id":"2605.20999","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-20T10:38:08Z","cross_cats_sorted":["cs.LG","math.OC","stat.ML"],"title_canon_sha256":"3d2cd5a58dfa168b317a787ef0c9e2a7767679ddd64bb864269f0c79254d20a4","abstract_canon_sha256":"5a8834395c235cbafe246ec45a4cae4a0e3a696a1b213c7f1204e715354bea9a"},"schema_version":"1.0"},"canonical_sha256":"946136880429f92f8e06c2e7aa8bc76d74cc88d65c91aad21cd6ef52156a3ea4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:05:31.740522Z","signature_b64":"nYUhLfl1IeiNFzj2KAakFq0mMD23Z8OElQJC2gRVc7LBbfKXOKXKJSOO4tlBMsYMmFJjwBN82qdwwSpF1X7aCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"946136880429f92f8e06c2e7aa8bc76d74cc88d65c91aad21cd6ef52156a3ea4","last_reissued_at":"2026-05-21T01:05:31.739983Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:05:31.739983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.20999","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":"2026-05-21T01:05:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kq3O+nXVrlZmtAHxzRhihJEfSYfw8+MVmgbsve+apyLWXQ8Nulzc5sd1pkxfdxfCPELORHUVPws1mVp5u8miBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T18:32:30.472777Z"},"content_sha256":"e0d2baf9a52793644195515b7ce60e38d19722da853dce46030debd8e5f305d7","schema_version":"1.0","event_id":"sha256:e0d2baf9a52793644195515b7ce60e38d19722da853dce46030debd8e5f305d7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:SRQTNCAEFH4S7DQGYLT2VC6HNV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Concentration of General Stochastic Approximation Under Heavy-Tailed Markovian Noise","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.OC","stat.ML"],"primary_cat":"math.PR","authors_text":"Martin Zubeldia, Shubhada Agrawal, Siva Theja Maguluri","submitted_at":"2026-05-20T10:38:08Z","abstract_excerpt":"We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the Martingale-difference noise is bounded, we show that the tail of the error can be sub-Gaussian, sub-Weibull, or something lighter than any Pareto but heavier than any Weibull, depending on the step size sequence and on whether the random operator is almost surely contractive, almost surely non-expansive, or expansive with positive probability. Our analysis reli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20999","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/2605.20999/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":"2026-05-21T01:05:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RDTSamDtvquc8MFTAgCfz7zdHpCP7HvMOfeZFJxqzPRV+mniPXhImGI8zyUBUuQlIgIbK7ozsxmYe0EcoucPCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T18:32:30.473469Z"},"content_sha256":"f34a1a5877ae04e13f7e2114032263a29c3e2ffa04433563c99bcebf7ad53b41","schema_version":"1.0","event_id":"sha256:f34a1a5877ae04e13f7e2114032263a29c3e2ffa04433563c99bcebf7ad53b41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV/bundle.json","state_url":"https://pith.science/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV/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":"2026-05-21T18:32:30Z","links":{"resolver":"https://pith.science/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV","bundle":"https://pith.science/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV/bundle.json","state":"https://pith.science/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SRQTNCAEFH4S7DQGYLT2VC6HNV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:SRQTNCAEFH4S7DQGYLT2VC6HNV","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":"5a8834395c235cbafe246ec45a4cae4a0e3a696a1b213c7f1204e715354bea9a","cross_cats_sorted":["cs.LG","math.OC","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-20T10:38:08Z","title_canon_sha256":"3d2cd5a58dfa168b317a787ef0c9e2a7767679ddd64bb864269f0c79254d20a4"},"schema_version":"1.0","source":{"id":"2605.20999","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20999","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20999v1","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20999","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"pith_short_12","alias_value":"SRQTNCAEFH4S","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"pith_short_16","alias_value":"SRQTNCAEFH4S7DQG","created_at":"2026-05-21T01:05:31Z"},{"alias_kind":"pith_short_8","alias_value":"SRQTNCAE","created_at":"2026-05-21T01:05:31Z"}],"graph_snapshots":[{"event_id":"sha256:f34a1a5877ae04e13f7e2114032263a29c3e2ffa04433563c99bcebf7ad53b41","target":"graph","created_at":"2026-05-21T01:05:31Z","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/2605.20999/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the Martingale-difference noise is bounded, we show that the tail of the error can be sub-Gaussian, sub-Weibull, or something lighter than any Pareto but heavier than any Weibull, depending on the step size sequence and on whether the random operator is almost surely contractive, almost surely non-expansive, or expansive with positive probability. Our analysis reli","authors_text":"Martin Zubeldia, Shubhada Agrawal, Siva Theja Maguluri","cross_cats":["cs.LG","math.OC","stat.ML"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-20T10:38:08Z","title":"Concentration of General Stochastic Approximation Under Heavy-Tailed Markovian Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20999","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:e0d2baf9a52793644195515b7ce60e38d19722da853dce46030debd8e5f305d7","target":"record","created_at":"2026-05-21T01:05:31Z","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":"5a8834395c235cbafe246ec45a4cae4a0e3a696a1b213c7f1204e715354bea9a","cross_cats_sorted":["cs.LG","math.OC","stat.ML"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-05-20T10:38:08Z","title_canon_sha256":"3d2cd5a58dfa168b317a787ef0c9e2a7767679ddd64bb864269f0c79254d20a4"},"schema_version":"1.0","source":{"id":"2605.20999","kind":"arxiv","version":1}},"canonical_sha256":"946136880429f92f8e06c2e7aa8bc76d74cc88d65c91aad21cd6ef52156a3ea4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"946136880429f92f8e06c2e7aa8bc76d74cc88d65c91aad21cd6ef52156a3ea4","first_computed_at":"2026-05-21T01:05:31.739983Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:05:31.739983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nYUhLfl1IeiNFzj2KAakFq0mMD23Z8OElQJC2gRVc7LBbfKXOKXKJSOO4tlBMsYMmFJjwBN82qdwwSpF1X7aCw==","signature_status":"signed_v1","signed_at":"2026-05-21T01:05:31.740522Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.20999","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0d2baf9a52793644195515b7ce60e38d19722da853dce46030debd8e5f305d7","sha256:f34a1a5877ae04e13f7e2114032263a29c3e2ffa04433563c99bcebf7ad53b41"],"state_sha256":"a5bd42a2c1b8e88106b7cc4aec549fe523d5cb7277dec2a0c497f1bce42613bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"naI+/8gIZmeDmjLkYqf0HIObfSNgEqFYxdolZFs+02VzqOULo2H5fIXrE7FSkf/d15apaPNVeDRvag0Jyo49CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T18:32:30.478046Z","bundle_sha256":"ec15bed26ec802e7cf0bb918dc49ee3b0aecad2b56b23b375fed759f70ed3e3b"}}