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Our algorithm combines the marginalized importance sampling framework with the actor-critic paradigm, where the critic returns evaluations of the actor (policy) that are pessimistic relative to the offline data and have a small average (importance-weighted) Bellman error. Compared to existing methods, our algorithm simultaneously offers a number of advantages: (1) It achieves the optimal statistical rate of $1/\\sqrt{N}$ -- where $N$ is the size of offline dataset -- in converging to the best policy covered in the offline dataset, even when combined with general function approximators. (2) It relies on a weaker average notion of policy coverage (compared to the $\\ell_\\infty$ single-policy concentrability) that exploits the structure of policy visitations. (3) It outperforms the data-collection behavior policy over a wide range of specific hyperparameters. 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