{"paper":{"title":"Path-Coupled Bellman Flows for Distributional Reinforcement Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Path-Coupled Bellman Flows learn return distributions by matching flows along source-consistent paths that couple current and successor distributions through shared noise.","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Boyang Xu, Hao Yan, Qing Zou, Siqin Yang","submitted_at":"2026-05-07T19:05:01Z","abstract_excerpt":"Distributional reinforcement learning (DRL) models the full return distribution, but existing finite-support or quantile-based methods rely on projections, while recent flow-based approaches can suffer from \\emph{boundary mismatch} at the flow source or from \\emph{high-variance} bootstrapping when current and successor noises are independent. We propose Path-Coupled Bellman Flows (PCBF), a continuous-time DRL method that learns return distributions with flow matching using \\textbf{source-consistent Bellman-coupled paths}: the current path starts from the required base prior at $t{=}0$, reaches"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"PCBF learns return distributions with flow matching using source-consistent Bellman-coupled paths: the current path starts from the required base prior at t=0, reaches the Bellman target at t=1, and maintains a pathwise affine relation to the successor flow at intermediate times, with a lambda-parameterized control-variate target that trades controlled bias for variance reduction.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the pathwise affine relation and coupling via shared base noise preserve the necessary distributional properties for correct Bellman updates without requiring time-t marginals to satisfy a distributional Bellman fixed point for all t, and that this coupling does not introduce new inconsistencies in the continuous-time flow.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Path-Coupled Bellman Flows use source-consistent Bellman-coupled paths and a lambda-parameterized control-variate to learn return distributions via flow matching, improving fidelity and stability over prior DRL approaches.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Path-Coupled Bellman Flows learn return distributions by matching flows along source-consistent paths that couple current and successor distributions through shared noise.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c999209a3a003948ddfefab264abec217fd4b1b4105355e3ddf829567aa73d63"},"source":{"id":"2605.08253","kind":"arxiv","version":2},"verdict":{"id":"bfbb9445-33a8-44e6-a159-7e3da3345d02","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T02:03:35.025384Z","strongest_claim":"PCBF learns return distributions with flow matching using source-consistent Bellman-coupled paths: the current path starts from the required base prior at t=0, reaches the Bellman target at t=1, and maintains a pathwise affine relation to the successor flow at intermediate times, with a lambda-parameterized control-variate target that trades controlled bias for variance reduction.","one_line_summary":"Path-Coupled Bellman Flows use source-consistent Bellman-coupled paths and a lambda-parameterized control-variate to learn return distributions via flow matching, improving fidelity and stability over prior DRL approaches.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the pathwise affine relation and coupling via shared base noise preserve the necessary distributional properties for correct Bellman updates without requiring time-t marginals to satisfy a distributional Bellman fixed point for all t, and that this coupling does not introduce new inconsistencies in the continuous-time flow.","pith_extraction_headline":"Path-Coupled Bellman Flows learn return distributions by matching flows along source-consistent paths that couple current and successor distributions through shared noise."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.08253/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T12:02:03.721352Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-20T07:33:40.011882Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T17:31:19.623104Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T12:20:49.106433Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"a542f4b50cbea51e8d3eaa106e7723854b849f88c5e79bd802d127aaff35983c"},"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"}