Instance-dependent $\ell_\infty$-bounds for policy evaluation in tabular reinforcement learning
This work addresses the policy evaluation problem in reinforcement learning, providing instance-dependent bounds that are incremental improvements over existing methods.
The paper tackles the problem of estimating the value function in Markov reward processes for policy evaluation in reinforcement learning, establishing non-asymptotic and data-dependent bounds in the ℓ∞-norm and showing that plug-in and robust variants are minimax-optimal up to constants.
Markov reward processes (MRPs) are used to model stochastic phenomena arising in operations research, control engineering, robotics, and artificial intelligence, as well as communication and transportation networks. In many of these cases, such as in the policy evaluation problem encountered in reinforcement learning, the goal is to estimate the long-term value function of such a process without access to the underlying population transition and reward functions. Working with samples generated under the synchronous model, we study the problem of estimating the value function of an infinite-horizon, discounted MRP on finitely many states in the $\ell_\infty$-norm. We analyze both the standard plug-in approach to this problem and a more robust variant, and establish non-asymptotic bounds that depend on the (unknown) problem instance, as well as data-dependent bounds that can be evaluated based on the observations of state-transitions and rewards. We show that these approaches are minimax-optimal up to constant factors over natural sub-classes of MRPs. Our analysis makes use of a leave-one-out decoupling argument tailored to the policy evaluation problem, one which may be of independent interest.