Retimed Bellman Flows: Escaping the Impossible Triangle of Velocity Bootstrapping

Boyang Xu; Shengzhe Chen; Hao Yan

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arXiv · 2609.32828

Retimed Bellman Flows: Escaping the Impossible Triangle of Velocity Bootstrapping

Boyang Xu·Shengzhe Chen·Hao Yan

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Abstract

Flow critics learn return distributions by transporting Gaussian noise to Bellman endpoints via continuous velocity fields. While velocity bootstrapping stabilizes training by querying a successor teacher, existing methods face a structural dilemma: on straight paths, no residual-free same-time affine mapping can preserve Gaussian initial noise while maintaining an unbiased target. To overcome this limitation, we introduce Retimed Bellman Flows (ReBF). ReBF queries the teacher critic at a dynamically shifted earlier flow time, aligning intermediate student and teacher trajectories. By combining this retimed clock with fresh, decoupled noise generation, ReBF constructs a provably conditionally unbiased velocity target that preserves the Bellman fixed point and contracts under Wasserstein distances. Empirically, ReBF reduces $W_1$ distance to ground-truth return distributions by up to $7.7\times$ on synthetic MRPs and outperforms existing flow critics across 38 challenging OGBench and D4RL offline reinforcement learning tasks.

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BibTeX RIS

Boyang Xu, Shengzhe Chen, Hao Yan. 2026-09-26. Retimed Bellman Flows: Escaping the Impossible Triangle of Velocity Bootstrapping. https://arxiv.org/abs/2609.32828

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