Doubly Robust Interval Estimation for Optimal Policy Evaluation in Online Learning

Evaluating the performance of an ongoing policy plays a vital role in many areas such as medicine and economics, to provide crucial instructions on the early-stop of the online experiment and timely feedback from the environment. Policy evaluation in online learning thus attracts increasing attention by inferring the mean outcome of the optimal policy (i.e., the value) in real-time. Yet, such a problem is particularly challenging due to the dependent data generated in the online environment, the unknown optimal policy, and the complex exploration and exploitation tradeoff in the adaptive experiment. In this article, we aim to overcome these difficulties in policy evaluation for online learning. We explicitly derive the probability of exploration that quantifies the probability of exploring nonoptimal actions under commonly used bandit algorithms. We use this probability to conduct valid inference on the online conditional mean estimator under each action and develop the doubly robust interval estimation (DREAM) method to infer the value under the estimated optimal policy in online learning. The proposed value estimator provides double protection for consistency and is asymptotically normal with a Wald-type confidence interval provided. Extensive simulation studies and real data applications are conducted to demonstrate the empirical validity of the proposed DREAM method. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.