Active Adaptive Experimental Design for Treatment Effect Estimation with Covariate Choices

This study designs an adaptive experiment for efficiently estimating average treatment effect (ATEs). We consider an adaptive experiment where an experimenter sequentially samples an experimental unit from a covariate density decided by the experimenter and assigns a treatment. After assigning a treatment, the experimenter observes the corresponding outcome immediately. At the end of the experiment, the experimenter estimates an ATE using gathered samples. The objective of the experimenter is to estimate the ATE with a smaller asymptotic variance. Existing studies have designed experiments that adaptively optimize the propensity score (treatment-assignment probability). As a generalization of such an approach, we propose a framework under which an experimenter optimizes the covariate density, as well as the propensity score, and find that optimizing both covariate density and propensity score reduces the asymptotic variance more than optimizing only the propensity score. Based on this idea, in each round of our experiment, the experimenter optimizes the covariate density and propensity score based on past observations. To design an adaptive experiment, we first derive the efficient covariate density and propensity score that minimizes the semiparametric efficiency bound, a lower bound for the asymptotic variance given a fixed covariate density and a fixed propensity score. Next, we design an adaptive experiment using the efficient covariate density and propensity score sequentially estimated during the experiment. Lastly, we propose an ATE estimator whose asymptotic variance aligns with the minimized semiparametric efficiency bound.