A Cram\'er Distance perspective on Quantile Regression based Distributional Reinforcement Learning
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 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.
Cite
@article{arxiv.2110.00535,
title = {A Cram\'er Distance perspective on Quantile Regression based Distributional Reinforcement Learning},
author = {Alix Lhéritier and Nicolas Bondoux},
journal= {arXiv preprint arXiv:2110.00535},
year = {2022}
}
Comments
Substantial changes in the experimental part, in particular in the architectures used for our results. Improvements in the presentation of the proof of Lemma 2. Added a Section to show soundness of TD-learning. To be published in AISTATS 2022