Data-adaptive inference of the optimal treatment rule and its mean reward. The masked bandit

This article studies the data-adaptive inference of an optimal treatment rule. A treatment rule is an individualized treatment strategy in which treatment assignment for a patient is based on her measured baseline covariates. Eventually, a reward is measured on the patient. We also infer the mean reward under the optimal treatment rule. We do so in the so called non-exceptional case, i.e., assuming that there is no stratum of the baseline covariates where treatment is neither beneficial nor harmful, and under a companion margin assumption. Our pivotal estimator, whose definition hinges on the targeted minimum loss estimation (TMLE) principle, actually infers the mean reward under the current estimate of the optimal treatment rule. This data-adaptive statistical parameter is worthy of interest on its own. Our main result is a central limit theorem which enables the construction of confidence intervals on both mean rewards under the current estimate of the optimal treatment rule and under the optimal treatment rule itself. The asymptotic variance of the estimator takes the form of the variance of an efficient influence curve at a limiting distribution, allowing to discuss the efficiency of inference. As a by product, we also derive confidence intervals on two cumulated pseudoregrets, a key notion in the study of bandits problems. Seen as two additional dataadaptive statistical parameters, they compare the sum of the rewards actually received during the course of the experiment with, either the sum of the means of the rewards, or the counterfactual rewards we would have obtained if we had used from the start the current estimate of the optimal treatment rule to assign treatment. A simulation study illustrates the procedure. One of the cornerstones of the theoretical study is a new maximal inequality for martingales with respect to the uniform entropy integral.