Twice Regularized Markov Decision Processes: The Equivalence between Robustness and Regularization
This work addresses the problem of handling uncertainty in MDPs for reinforcement learning practitioners by providing a more efficient and scalable approach, though it is incremental in extending regularization methods to robustness.
The paper tackles the computational complexity and scalability issues in robust Markov decision processes (MDPs) by showing that robust MDPs can be reduced to regularized MDPs, establishing an equivalence that enables planning and learning schemes with convergence guarantees and similar time complexity. It introduces twice regularized MDPs (R² MDPs) with both value and policy regularization, demonstrating efficacy in tabular and continuous environments.
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.