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# A Cramér Distance perspective on Quantile Regression based Distributional Reinforcement Learning

*Proceedings of The 25th International Conference on Artificial Intelligence and Statistics*, PMLR 151:5774-5789, 2022.

#### Abstract

Distributional reinforcement learning (DRL) extends the value-based approach by approximating the full distribution over future returns instead of the mean only, providing a richer signal that leads to improved performances. Quantile Regression (QR)-based methods like QR-DQN project arbitrary distributions into a parametric subset of staircase distributions by minimizing the 1-Wasserstein distance. However, due to biases in the gradients, the quantile regression loss is used instead for training, guaranteeing the same minimizer and enjoying unbiased gradients. Non-crossing constraints on the quantiles have been shown to improve the performance of QR-DQN for uncertainty-based exploration strategies. The contribution of this work is in the setting of fixed quantile levels and is twofold. First, we prove that the Cramer distance yields a projection that coincides with the 1-Wasserstein one and that, under non-crossing constraints, the squared Cramer and the quantile regression losses yield collinear gradients, shedding light on the connection between these important elements of DRL. Second, we propose a low complexity algorithm to compute the Cramer distance.