We study the \emph{offline reinforcement learning} (offline RL) problem, where the goal is to learn a rewardmaximizing policy in an unknown \emph{Markov Decision Process} (MDP) using the data coming from a policy $\mu$. In particular, we consider the sample complexity problems of offline RL for the finite horizon MDPs. Prior works derive the informationtheoretical lower bounds based on different datacoverage assumptions and their upper bounds are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the \emph{Adaptive Pessimistic Value Iteration} (APVI) algorithm and derive the suboptimality upper bound that nearly matches
$
O\left(\sum_{h=1}^H\sum_{s_h,a_h}d^{\pi^\star}_h(s_h,a_h)\sqrt{\frac{\mathrm{Var}_{P_{s_h,a_h}}{(V^\star_{h+1}+r_h)}}{d^\mu_h(s_h,a_h)}}\sqrt{\frac{1}{n}}\right).
$
We also prove an informationtheoretical lower bound to show this quantity is required under the weak assumption that $d^\mu_h(s_h,a_h)>0$ if $d^{\pi^\star}_h(s_h,a_h)>0$. Here $\pi^\star$ is a optimal policy, $\mu$ is the behavior policy and $d(s_h,a_h)$ is the marginal stateaction probability. We call this adaptive bound the \emph{intrinsic offline reinforcement learning bound} since it directly implies all the existing optimal results: minimax rate under uniform datacoverage assumption, horizonfree setting, single policy concentrability, and the tight problemdependent results. Later, we extend the result to the \emph{assumptionfree} regime (where we make no assumption on $
\mu$) and obtain the assumptionfree intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL.
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FiniteSample Regret Bound for Distributionally Robust Offline Tabular Reinforcement Learning
While reinforcement learning has witnessed tremendous success recently in a wide range of domains, robustness–or the lack thereof–remains an important issue that remains inadequately addressed. In this paper, we provide a distributionally robust formulation of offline learning policy in tabular RL that aims to learn a policy from historical data (collected by some other behavior policy) that is robust to the future environment arising as a perturbation of the training environment. We first develop a novel policy evaluation scheme that accurately estimates the robust value (i.e. how robust it is in a perturbed environment) of any given policy and establish its finitesample estimation error. Building on this, we then develop a novel and minimaxoptimal distributionally robust learning algorithm that achieves $O_P\left(1/\sqrt{n}\right)$ regret, meaning that with high probability, the policy learned from using $n$ training data points will be $O\left(1/\sqrt{n}\right)$ close to the optimal distributionally robust policy. Finally, our simulation results demonstrate the superiority of our distributionally robust approach compared to nonrobust RL algorithms.
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 Award ID(s):
 1915967
 NSFPAR ID:
 10344976
 Editor(s):
 Banerjee, Arindam and
 Date Published:
 Journal Name:
 Proceedings of The 24th International Conference on Artificial Intelligence and Statistics
 Volume:
 130
 Issue:
 2021
 Page Range / eLocation ID:
 33313339
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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