Distributional Reinforcement Learning with Dual Expectile-Quantile Regression

Distributional reinforcement learning (RL) has proven useful in multiple benchmarks as it enables approximating the full distribution of returns and makes a better use of environment samples. The commonly used quantile regression approach to distributional RL -- based on asymmetric $L1$ losses -- provides a flexible and effective way of learning arbitrary return distributions. In practice, it is often improved by using a more efficient, hybrid asymmetric $L1$-$L2$ Huber loss for quantile regression. However, by doing so, distributional estimation guarantees vanish, and we empirically observe that the estimated distribution rapidly collapses to its mean. Indeed, asymmetric $L2$ losses, corresponding to expectile regression, cannot be readily used for distributional temporal difference learning. Motivated by the efficiency of $L_2$-based learning, we propose to jointly learn expectiles and quantiles of the return distribution in a way that allows efficient learning while keeping an estimate of the full distribution of returns. We prove that our approach approximately learns the correct return distribution, and we benchmark a practical implementation on a toy example and at scale. On the Atari benchmark, our approach matches the performance of the Huber-based IQN-1 baseline after $200$M training frames but avoids distributional collapse and keeps estimates of the full distribution of returns.