Posterior-Averaged Bellman Flow Matching for Distributional Critics

Abstract

Distributional critics model the full return distribution, providing richer signals for exploration and risk-sensitive control. Recent work has shown that flow-based critics can model these distributions flexibly and provide effective supervision across flow time for temporal-difference learning. However, unlike standard flow matching with fixed data endpoints, a flow critic is trained on Bellman targets recursively generated by a learned target critic under the current policy. Existing objectives therefore either use noisy single-endpoint supervision or rely on the learned flow itself for additional guidance. Our key observation is that an intermediate bridge state may be compatible with several Bellman endpoints, so its velocity should average their directions under the corresponding posterior. We propose Posterior-Averaged Bellman Flow Matching (PA-BFM), which samples an endpoint pool, infers the closed-form posterior over the latent anchor, and regresses to the posterior-mean velocity. Uniform anchor selection makes this target an exact finite-pool Rao--Blackwellization of the single-endpoint target: for every \(K\), it preserves the population target, cannot increase conditional variance, and its conditional variance scales as \(O(K^{-1})\). The endpoint pool supports conservative updates, while the twin return flows provide critic-disagreement and unsupported-mass diagnostics. Controlled studies show that PA-BFM reduces fixed-query target mean-squared error by \(79.7\%\) and target-induced gradient variance by up to \(98.0\%\), while improving return-distribution fidelity. Across 50 OGBench tasks and four offline-to-online scenarios, PA-BFM achieves higher aggregate performance than Value Flows.