Off-Policy Evaluation and Learning for External Validity under a Covariate Shift

We consider evaluating and training a new policy for the evaluation data by using the historical data obtained from a different policy. The goal of off-policy evaluation (OPE) is to estimate the expected reward of a new policy over the evaluation data, and that of off-policy learning (OPL) is to find a new policy that maximizes the expected reward over the evaluation data. Although the standard OPE and OPL methods assume the same distribution of covariate between the historical and evaluation data, a covariate shift often exists in real-world applications, i.e., the distribution of the covariate of the historical data is different from that of the evaluation data. In this paper, we derive the efficiency bound of an OPE estimator under a covariate shift. Then, we propose doubly robust and efficient estimators for OPE and OPL under a covariate shift by using a nonparametric estimator of the density ratio between the historical and evaluation data distributions. We also discuss other possible estimators and compare their theoretical properties. Finally, we conduct experiments to confirm the effectiveness of the proposed estimators.