Statistical Inference After Adaptive Sampling for Longitudinal Data

Online reinforcement learning and other adaptive sampling algorithms are increasingly used in digital intervention experiments to optimize treatment delivery for users over time. In this work, we focus on longitudinal user data collected by a large class of adaptive sampling algorithms that are designed to optimize treatment decisions online using accruing data from multiple users. Combining or "pooling" data across users allows adaptive sampling algorithms to potentially learn faster. However, by pooling, these algorithms induce dependence between the collected user data trajectories; we show that this can cause standard variance estimators for i.i.d. data to underestimate the true variance of common estimators on this data type. We develop novel methods to perform a variety of statistical analyses on such adaptively collected data via Z-estimation. Specifically, we introduce the adaptive sandwich variance estimator, a corrected sandwich estimator that leads to consistent variance estimates under adaptive sampling. Additionally, to prove our results we develop significant theory for empirical processes on non-i.i.d., adaptively collected, longitudinal data. This work is motivated by our efforts in designing experiments in which online reinforcement learning algorithms pool data across users to learn to optimize treatment decisions, yet reliable statistical inference is essential for conducting a variety of statistical analyses after the experiment is over.