Active Adaptive Experimental Design for Treatment Effect Estimation with Covariate Choices

TL;DR

This work studies efficient estimation of the average treatment effect (ATE) under an adaptive experimental design that jointly optimizes covariate density and treatment-assignment probabilities. By deriving the semiparametric efficiency bound and the corresponding efficient $p^*(x)$ and $w^*(a|x)$, it designs an Active Adaptive Sampling (AAS)-AIPWIW experiment that sequentially estimates these probabilities and uses an AIPW-based estimator with importance weighting. The main result shows that the proposed design achieves an asymptotic variance equal to the minimized bound, improving over traditional propensity-only adaptations. Simulation studies across heterogeneous variances and covariate distributions demonstrate substantial MSE reductions and robust performance, including oracle-style results in several settings, highlighting practical impact for efficient causal inference in fields such as medicine and economics.

Abstract

This study designs an adaptive experiment for efficiently estimating average treatment effects (ATEs). In each round of our adaptive experiment, an experimenter sequentially samples an experimental unit, assigns a treatment, and observes the corresponding outcome immediately. At the end of the experiment, the experimenter estimates an ATE using the gathered samples. The objective 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 optimizing the covariate density as well as the propensity score. First, we derive the efficient covariate density and propensity score that minimize the semiparametric efficiency bound and find that optimizing both covariate density and propensity score minimizes the semiparametric efficiency bound more effectively than optimizing only the 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.

Figures (11)

• Figure 1: Active adaptive experiment.
• Figure 2: Efficient covariate density. The distribution with the blue line represents the covariate density of interest, while that with the dotted black line represents the covariate density in an experiment.
• Figure 3: Heterogeneous variances.
• Figure 6: Results of simulation studies with covariates following with a Gaussian distribution $\mathcal{N}(1, 25)$.
• Figure 8: MSEs at $T = 500, 750, 1,000, 1,250, 1,500, 1,750, 2,000, 2,250, 2,500, 2,750, 3,000$ in simulation studies with Gaussian covariates.

Theorems & Definitions (21)

• Remark 2.3: Stable unit treatment value assumption (SUTVA)
• Theorem 3.1
• Theorem 4.1
• Theorem 5.3
• Corollary 5.4
• Lemma 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['thm:tabular']}
• Proposition 1.3: From Theorem 25.20 in Vaart1998.
• Lemma 1.4