Stein Variational Distributional Reinforcement Learning: Score-Guided Particles for Multivariate Return Distributions

Distributional reinforcement learning (DRL) models the full return distribution rather than expectations, but extending it to multivariate settings remains challenging. Many common metrics do not naturally generalize beyond one dimension or lose computational tractability, and the multivariate case introduces additional difficulties such as general matrix discounting, for which no contraction results are available. We introduce Sliced Distributional Reinforcement Learning (SDRL), which lifts tractable one-dimensional divergences to multivariate return distributions via projections. We prove Bellman contraction for uniform slicing under shared scalar discounting, and introduce a maximum-slicing variant with contraction under general dense discount matrices. SDRL supports a broad class of base divergences; we analyze Wasserstein, Cramér, and Maximum Mean Discrepancy (MMD), and characterize which SDRL variants suit the standard single-sample Bellman update used in distributional RL. We evaluate SDRL on a toy chain problem and a gridworld image-based environment as well as a subset of Atari games.

Unlike standard expected-return Reinforcement Learning (RL), Distributional RL (DRL) models the full return distribution, making it better-suited for uncertainty-aware and risk-sensitive decision-making. Conditional Flow Matching (CFM) critics have recently attracted attention for modelling continuous, multi-modal return distributions. Despite this interest, there remains a substantial metric mismatch: DRL theory relies on the distributional Bellman operator being contractive in the $p$-Wasserstein distance, yet existing CFM critics are trained with arbitrary source-target couplings, so their flow-matching losses are not Wasserstein-aligned surrogates for matching Bellman target return distributions. In this work, we address this mismatch by proposing FlowIQN, a CFM critic that sorts source and Bellman target samples within each mini-batch to approximate the monotone optimal transport coupling, replacing arbitrary pairings with quantile-aligned flow paths. We prove that the loss of our quantile-coupled CFM critic yields a Wasserstein-aligned approximate projection compatible with the foundations of DRL. To our knowledge, FlowIQN is the first flow-matching distributional critic with an explicit Wasserstein-aligned projection guarantee. We further extend FlowIQN with shortcut models for efficient inference. Empirical results show that FlowIQN improves Wasserstein return-distribution accuracy over other CFM critics. It also yields competitive performance on offline RL benchmarks across multiple policy extraction methods, providing a theoretically grounded CFM critic that is readily compatible with DRL pipelines. Code: https://github.com/ori-goals/flowIQN.

Reinforcement learning with verifiable rewards spends much of its rollout budget on states that yield no learning signal. We recast this as a question about the sampling distribution itself: under a fixed rollout budget, which distribution over states minimizes the variance of the policy-gradient estimator? The answer admits a closed-form Neyman allocation, $μ_{\mathrm{var}}^π(s) \propto d^π(s)\,σ_π(s)$, which connects naturally to GRPO's group-relative advantage and induces an emergent reverse curriculum as the policy improves, with no externally imposed schedule. Building on this principle, we propose VSR (Variance-driven State Resampling), a plug-in modification to GRPO using a prefix replay buffer and a zero-cost variance proxy. On mathematical reasoning benchmarks and long-horizon agentic tasks (Sokoban, BabyAI), VSR outperforms GRPO and GiGPO on most benchmarks with fewer generated tokens per update.