Semiparametric Efficiency in Policy Learning with General Treatments

Recent literature on policy learning has primarily focused on regret bounds of the learned policy. We provide a new perspective by developing a unified semiparametric efficiency framework for policy learning, allowing for general treatments that are discrete, continuous, or mixed. We provide a characterization of the failure of pathwise differentiability for parameters arising from deterministic policies. We then establish efficiency bounds for pathwise differentiable parameters in randomized policies, both when the propensity score is known and when it must be estimated. Building on the convolution theorem, we introduce a notion of efficiency for the asymptotic distribution of welfare regret, showing that inefficient policy estimators not only inflate the variance of the asymptotic regret but also shift its mean upward. We derive the asymptotic theory of several common policy estimators, with a key contribution being a policy-learning analogue of the Hirano-Imbens-Ridder (HIR) phenomenon: the inverse propensity weighting estimator with an estimated propensity is efficient, whereas the same estimator using the true propensity is not. We illustrate the theoretical results with an empirically calibrated simulation study based on data from a job training program and an empirical application to a commitment savings program.