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We show that the Monotonic Value Propagation (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$\\tilde{O}\\left(\\left(\\sum_{\\Delta_h(s,a)>0} \\frac{H^2 \\log K \\land \\mathtt{Var}_{\\max}^{\\text{c}}}{\\Delta_h(s,a)} +\\sum_{\\Delta_h(s,a)=0}\\frac{ H^2 \\land \\mathtt{Var}_{\\max}^{\\text{c}}}{\\Delta_{\\mathrm{min}}} + SAH^4 (S \\lor H) \\right) \\log K\\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Here, $\\Delta_h(s,a) =V_h^* (a) - Q_"},"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":"2506.06521","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-06-06T20:33:57Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"56e09b0bc9898e601c2f9db4a3b7bfd6a1f6ce50f9681a84c718a5fbdaa817f3","abstract_canon_sha256":"03649888726816eee673573c7b83b77fe6ec0e1ac80a11e618820cec44fbb53f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:17:51.544790Z","signature_b64":"4tXMxdTdxRed2XAHb5sE/825lMIXf9yE9SE8fFwGSa9RNRo3gE4y1HR2y73ohlN51FsWxQr3XyXxWw0rAbhMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fb5d79040bdbbf7ecdf7608ad7cbe0faf73eef9a3cb18db5b835148ac0021f3","last_reissued_at":"2026-07-05T11:17:51.544267Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:17:51.544267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Maryam Fazel, Runlong Zhou, Shulun Chen, Simon S. Du, Zihan Zhang","submitted_at":"2025-06-06T20:33:57Z","abstract_excerpt":"We consider the gap-dependent regret bounds for episodic MDPs. We show that the Monotonic Value Propagation (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$\\tilde{O}\\left(\\left(\\sum_{\\Delta_h(s,a)>0} \\frac{H^2 \\log K \\land \\mathtt{Var}_{\\max}^{\\text{c}}}{\\Delta_h(s,a)} +\\sum_{\\Delta_h(s,a)=0}\\frac{ H^2 \\land \\mathtt{Var}_{\\max}^{\\text{c}}}{\\Delta_{\\mathrm{min}}} + SAH^4 (S \\lor H) \\right) \\log K\\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Here, $\\Delta_h(s,a) =V_h^* (a) - Q_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.06521","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/2506.06521/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":"2506.06521","created_at":"2026-07-05T11:17:51.544334+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.06521v1","created_at":"2026-07-05T11:17:51.544334+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.06521","created_at":"2026-07-05T11:17:51.544334+00:00"},{"alias_kind":"pith_short_12","alias_value":"L625PECAXW57","created_at":"2026-07-05T11:17:51.544334+00:00"},{"alias_kind":"pith_short_16","alias_value":"L625PECAXW57P3G7","created_at":"2026-07-05T11:17:51.544334+00:00"},{"alias_kind":"pith_short_8","alias_value":"L625PECA","created_at":"2026-07-05T11:17:51.544334+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.04979","citing_title":"On-line Learning in Tree MDPs by Treating Policies as Bandit Arms","ref_index":12,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6","json":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6.json","graph_json":"https://pith.science/api/pith-number/L625PECAXW57P3G7OYEK27F6B6/graph.json","events_json":"https://pith.science/api/pith-number/L625PECAXW57P3G7OYEK27F6B6/events.json","paper":"https://pith.science/paper/L625PECA"},"agent_actions":{"view_html":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6","download_json":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6.json","view_paper":"https://pith.science/paper/L625PECA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.06521&json=true","fetch_graph":"https://pith.science/api/pith-number/L625PECAXW57P3G7OYEK27F6B6/graph.json","fetch_events":"https://pith.science/api/pith-number/L625PECAXW57P3G7OYEK27F6B6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6/action/storage_attestation","attest_author":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6/action/author_attestation","sign_citation":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6/action/citation_signature","submit_replication":"https://pith.science/pith/L625PECAXW57P3G7OYEK27F6B6/action/replication_record"}},"created_at":"2026-07-05T11:17:51.544334+00:00","updated_at":"2026-07-05T11:17:51.544334+00:00"}