Parameter-Efficient Distributional RL via Normalizing Flows and a Geometry-Aware Cram\'er Surrogate

Abstract

Distributional Reinforcement Learning (DistRL) improves upon expectation-based methods by modeling full return distributions, but standard approaches often remain far from parsimonious. Categorical methods (e.g., C51) rely on fixed supports where parameter counts scale linearly with resolution, while quantile methods approximate distributions as discrete mixtures whose piecewise-constant densities can be wasteful when modeling complex multi-modal or heavy-tailed returns. We introduce NFDRL, a parsimonious architecture that models return distributions using continuous normalizing flows. Unlike categorical baselines, our flow-based model maintains a compact parameter footprint that does not grow with the effective resolution of the distribution, while providing a dynamic, adaptive support for returns. To train this continuous representation, we propose a Cram\'er-inspired, geometry-aware distance defined over probability masses obtained from the flow. We show that this distance is a true probability metric, that the associated distributional Bellman operator is a sqrt(gamma)-contraction, and that the resulting objective admits unbiased sample gradients, properties that are typically not simultaneously guaranteed in prior PDF-based DistRL methods. Empirically, NFDRL recovers rich, multi-modal return landscapes on toy MDPs and achieves performance competitive with categorical baselines on the Atari-5 benchmark, while offering substantially better parameter efficiency.