  We study offline reinforcement learning (RL), which aims to learn an optimal
policy based on a dataset collected a priori. Due to the lack of further
interactions with the environment, offline RL suffers from the insufficient
coverage of the dataset, which eludes most existing theoretical analysis. In
this paper, we propose a pessimistic variant of the value iteration algorithm
(PEVI), which incorporates an uncertainty quantifier as the penalty function.
Such a penalty function simply flips the sign of the bonus function for
promoting exploration in online RL, which makes it easily implementable and
compatible with general function approximators.
  Without assuming the sufficient coverage of the dataset, we establish a
data-dependent upper bound on the suboptimality of PEVI for general Markov
decision processes (MDPs). When specialized to linear MDPs, it matches the
information-theoretic lower bound up to multiplicative factors of the dimension
and horizon. In other words, pessimism is not only provably efficient but also
minimax optimal. In particular, given the dataset, the learned policy serves as
the ``best effort'' among all policies, as no other policies can do better. Our
theoretical analysis identifies the critical role of pessimism in eliminating a
notion of spurious correlation, which emerges from the ``irrelevant''
trajectories that are less covered by the dataset and not informative for the
optimal policy.
