  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 Cram\'er distance yields a
projection that coincides with the 1-Wasserstein one and that, under
non-crossing constraints, the squared Cram\'er 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 Cram\'er distance.
