  Robust Markov decision processes (MDPs) aim to handle changing or partially
known system dynamics. To solve them, one typically resorts to robust
optimization methods. However, this significantly increases computational
complexity and limits scalability in both learning and planning. On the other
hand, regularized MDPs show more stability in policy learning without impairing
time complexity. Yet, they generally do not encompass uncertainty in the model
dynamics. In this work, we aim to learn robust MDPs using regularization. We
first show that regularized MDPs are a particular instance of robust MDPs with
uncertain reward. We thus establish that policy iteration on reward-robust MDPs
can have the same time complexity as on regularized MDPs. We further extend
this relationship to MDPs with uncertain transitions: this leads to a
regularization term with an additional dependence on the value function. We
then generalize regularized MDPs to twice regularized MDPs ($\text{R}^2$ MDPs),
i.e., MDPs with $\textit{both}$ value and policy regularization. The
corresponding Bellman operators enable us to derive planning and learning
schemes with convergence and generalization guarantees, thus reducing
robustness to regularization. We numerically show this two-fold advantage on
tabular and physical domains, highlighting the fact that $\text{R}^2$ preserves
its efficacy in continuous environments.
