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Evaluating the performance of an ongoing policy plays a vital role in manyareas such as medicine and economics, to provide crucial instruction on theearly-stop of the online experiment and timely feedback from the environment.Policy evaluation in online learning thus attracts increasing attention byinferring the mean outcome of the optimal policy (i.e., the value) inreal-time. Yet, such a problem is particularly challenging due to the dependentdata generated in the online environment, the unknown optimal policy, and thecomplex exploration and exploitation trade-off in the adaptive experiment. Inthis paper, we aim to overcome these difficulties in policy evaluation foronline learning. We explicitly derive the probability of exploration thatquantifies the probability of exploring the non-optimal actions under commonlyused bandit algorithms. We use this probability to conduct valid inference onthe online conditional mean estimator under each action and develop the doublyrobust interval estimation (DREAM) method to infer the value under theestimated optimal policy in online learning. The proposed value estimatorprovides double protection on the consistency and is asymptotically normal witha Wald-type confidence interval provided. Extensive simulations and real dataapplications are conducted to demonstrate the empirical validity of theproposed DREAM method.

Doubly Robust Interval Estimation for Optimal Policy Evaluation in Online Learning