Several practical applications of reinforcement learning involve an agent learning from past data without the possibility of further exploration. Often these applications require us to 1) identify a near optimal policy or to 2) estimate the value of a target policy. For both tasks we derive \emph{exponential} information-theoretic lower bounds in discounted infinite horizon MDPs with a linear function representation for the action value function even if 1) \emph{realizability} holds, 2) the batch algorithm observes the exact reward and transition \emph{functions}, and 3) the batch algorithm is given the \emph{best} a priori data distribution for the problem class. Furthermore, if the dataset does not come from policy rollouts then the lower bounds hold even if the action-value function of \emph{every} policy admits a linear representation. If the objective is to find a near-optimal policy, we discover that these hard instances are easily solved by an \emph{online} algorithm, showing that there exist RL problems where \emph{batch RL is exponentially harder than online RL} even under the most favorable batch data distribution. In other words, online exploration is critical to enable sample efficient RL with function approximation. A second corollary is the exponential separation between finite and infinite horizon batch problems under our assumptions. On a technical level, this work introduces a new `oracle + batch algorithm' framework to prove lower bounds that hold for every distribution, and automatically recovers traditional fixed distribution lower bounds as a special case. Finally this work helps formalize the issue known as \emph{deadly triad} and explains that the \emph{bootstrapping} problem \citep{sutton2018reinforcement} is potentially more severe than the \emph{extrapolation} issue for RL because unlike the latter, bootstrapping cannot be mitigated by adding more samples.
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