Schrödinger--Föllmer Actor--Critic: Diffusion Policy Improvement with Finite-Sample Analysis

Diffusion and flow policies can model complex behaviors in offline reinforcement learning (RL). However, penalizing their KL divergence from the behavior policy can discourage actions having high critic values with low behavior density. Directly refining behavior proposals may be an alternative, yet Gaussian or deterministic editors limit expressiveness to represent multiple separated modes for the same proposal. In this work, we introduce Proposal-Conditioned Refinement Flows (PReFlow), a policy extraction method combining critic-based proposal selection with a conditional refinement flow. To optimize proposal selection and refinement together, we formulate a KL-regularized objective whose optimum induces a Gibbs policy over final actions under a Gaussian-smoothed behavior prior. The refinement flow can represent multiple high value modes, while a proposal-centered Gaussian reference regulates large action changes. This Gaussian reference further enables us to make use of simulation-free, closed form adjoint matching targets from sampled endpoints and critic gradients, yielding a single velocity regression loss without a backward adjoint solve. On 50 OGBench tasks, PReFlow achieves competitive offline performance and the highest aggregate score among the compared methods after online fine-tuning, reaching 91% after 500K environment steps.

Online safe reinforcement learning (RL) seeks policies that maximize reward while satisfying safety constraints. Reward and safety can induce multimodal action distributions, challenging the prevailing primal-dual methods: Gaussian actors may collapse onto a single suboptimal mode, and optimization over the nonconvex Lagrangian landscape can be unstable. Diffusion and flow policies can represent such distributions, but recent work with a diffusion actor relies on estimating and matching the score of an augmented-Lagrangian target policy. Instead, we differentiate the augmented objective directly through the generation path of a flow policy, so no score needs to be estimated. Because a flow policy lacks a readily available action log-density for entropy regularization, we build on the density-free kinetic-energy regularizer of FLAC, a recent reward-only method, and propose Reparameterized Augmented-Lagrangian Flow Actor with Least Energy (RAFALE), an off-policy actor-critic method for safe RL. We formulate its update as a constrained one-ended generalized Schrödinger bridge and show that, for each source draw, this path-space problem is exactly an entropy-regularized problem in action space. At positive noise, its solution reweights the reward-only action distribution only where the estimated cost exceeds a threshold set by the Lagrange multiplier. As the noise vanishes, the optimal value converges to that of a least-energy map objective that the flow policy optimizes directly. Across seven Safety-Gymnasium tasks, RAFALE achieves competitive reward with mean final cost within budget on every task, whereas strong baselines trade one for the other; ablations support the necessity of both its augmented objective and its flow actor.

Several state-of-the-art methods for online reinforcement learning in continuous control improve policies using action gradients of a learned critic. However, critics are typically trained to predict returns, and accurate value predictions do not necessarily yield accurate action derivatives, potentially leading to unreliable policy updates. We propose Forward Entropy-Regularized Policy Optimization (FERPO), an on-policy maximum entropy reinforcement learning algorithm that performs policy improvement using critic values without differentiating the critic with respect to actions. FERPO derives an optimal target action distribution from a policy-improvement objective regularized by entropy and Kullback-Leibler (KL) divergence. We then fit the actor to this target by minimizing a forward-KL objective, estimated using self-normalized importance sampling (SNIS) with actions drawn from the rollout policy. By limiting the target distribution's deviation from the rollout policy, the KL regularization helps keep these importance weights well behaved. In contrast to reverse-KL objectives, which can favor a subset of the target distribution's modes, the forward-KL objective encourages coverage of multiple high-value modes and thereby promotes exploration. Experiments and ablations on MuJoCo Playground and ManiSkill show competitive performance and sample-efficiency gains. Computational benchmarks also demonstrate faster actor updates than Relative Entropy Pathwise Policy Optimization (REPPO).