A Spectral Revisit of the Distributional Bellman Operator under the Cramér Metric

Distributional reinforcement learning (DRL) studies the evolution of full return distributions under Bellman updates rather than focusing on expected values. A classical result is that the distributional Bellman operator is contractive under the Cramér metric, which corresponds to an $L^2$ geometry on differences of cumulative distribution functions (CDFs). While this contraction ensures stability of policy evaluation, existing analyses remain largely metric, focusing on contraction properties without elucidating the structural action of the Bellman update on distributions. In this work, we analyse distributional Bellman dynamics directly at the level of CDFs, treating the Cramér geometry as the intrinsic analytical setting. At this level, the Bellman update acts affinely on CDFs and linearly on differences between CDFs, and its contraction property yields a uniform bound on this linear action. Building on this intrinsic formulation, we construct a family of regularised spectral Hilbert representations that realise the CDF-level geometry by exact conjugation, without modifying the underlying Bellman dynamics. The regularisation affects only the geometry and vanishes in the zero-regularisation limit, recovering the native Cramér metric. This framework clarifies the operator structure underlying distributional Bellman updates and provides a foundation for further functional and operator-theoretic analyses in DRL.

Dataset Clustering for Improved Offline Policy Learning

Offline policy learning aims to discover decision-making policies from previously-collected datasets without additional online interactions with the environment. As the training dataset is fixed, its quality becomes a crucial determining factor in the performance of the learned policy. This paper studies a dataset characteristic that we refer to as multi-behavior, indicating that the dataset is collected using multiple policies that exhibit distinct behaviors. In contrast, a uni-behavior dataset would be collected solely using one policy. We observed that policies learned from a uni-behavior dataset typically outperform those learned from multi-behavior datasets, despite the uni-behavior dataset having fewer examples and less diversity. Therefore, we propose a behavior-aware deep clustering approach that partitions multi-behavior datasets into several uni-behavior subsets, thereby benefiting downstream policy learning. Our approach is flexible and effective; it can adaptively estimate the number of clusters while demonstrating high clustering accuracy, achieving an average Adjusted Rand Index of 0.987 across various continuous control task datasets. Finally, we present improved policy learning examples using dataset clustering and discuss several potential scenarios where our approach might benefit the offline policy learning community.