When is Agnostic Reinforcement Learning Statistically Tractable?

We study the problem of agnostic PAC reinforcement learning (RL): given a policy class $\Pi$, how many rounds of interaction with an unknown MDP (with a potentially large state and action space) are required to learn an $\epsilon$-suboptimal policy with respect to \(\Pi\)? Towards that end, we introduce a new complexity measure, called the \emph{spanning capacity}, that depends solely on the set \(\Pi\) and is independent of the MDP dynamics. With a generative model, we show that the spanning capacity characterizes PAC learnability for every policy class $\Pi$. However, for online RL, the situation is more subtle. We show there exists a policy class $\Pi$ with a bounded spanning capacity that requires a superpolynomial number of samples to learn. This reveals a surprising separation for agnostic learnability between generative access and online access models (as well as between deterministic/stochastic MDPs under online access). On the positive side, we identify an additional \emph{sunflower} structure which in conjunction with bounded spanning capacity enables statistically efficient online RL via a new algorithm called POPLER, which takes inspiration from classical importance sampling methods as well as recent developments for reachable-state identification and policy evaluation in reward-free exploration.