Estimating Optimal Treatment Rules with an Instrumental Variable: A Partial Identification Learning Approach

Individualized treatment rules (ITRs) are considered a promising recipe to deliver better policy interventions. One key ingredient in optimal ITR estimation problems is to estimate the average treatment effect conditional on a subject's covariate information, which is often challenging in observational studies due to the universal concern of unmeasured confounding. Instrumental variables (IVs) are widely-used tools to infer the treatment effect when there is unmeasured confounding between the treatment and outcome. In this work, we propose a general framework of approaching the optimal ITR estimation problem when a valid IV is allowed to only partially identify the treatment effect. We introduce a novel notion of optimality called "IV-optimality". A treatment rule is said to be IV-optimal if it minimizes the maximum risk with respect to the putative IV and the set of IV identification assumptions. We derive a bound on the risk of an IV-optimal rule that illuminates when an IV-optimal rule has favorable generalization performance. We propose a classification-based statistical learning method that estimates such an IV-optimal rule, design computationally-efficient algorithms, and prove theoretical guarantees. We contrast our proposed method to the popular outcome weighted learning (OWL) approach via extensive simulations, and apply our method to study which mothers would benefit from traveling to deliver their premature babies at hospitals with high level neonatal intensive care units.