Bridging the Gap between Empirical Welfare Maximization and Conditional Average Treatment Effect Estimation in Policy Learning
Masahiro Kato
TL;DR
The paper addresses policy learning by unifying two mainstream approaches: empirical welfare maximization (EWM) and the plug-in method. It shows a precise equivalence between EWM over a policy class and least-squares regression on the conditional average treatment effect, via the reparameterization g = 2π − 1, and extends this equivalence to regularized, [0,1]-valued policies. This leads to a practical, continuous surrogate objective (a least-squares problem) that preserves welfare-maximization goals and can be optimized with standard tools, while acknowledging that exact combinatorial EWM remains NP-hard for many classes. The framework links IPW/AIPW-based welfare estimation with CATE regression guarantees, enabling unified analysis and easier implementation, and it generalizes to multiple treatments and constrained settings.
Abstract
The goal of policy learning is to train a policy function that recommends a treatment given covariates to maximize population welfare. There are two major approaches in policy learning: the empirical welfare maximization (EWM) approach and the plug-in approach. The EWM approach is analogous to a classification problem, where one first builds an estimator of the population welfare, which is a functional of policy functions, and then trains a policy by maximizing the estimated welfare. In contrast, the plug-in approach is based on regression, where one first estimates the conditional average treatment effect (CATE) and then recommends the treatment with the highest estimated outcome. This study bridges the gap between the two approaches by showing that both are based on essentially the same optimization problem. In particular, we prove an exact equivalence between EWM and least squares over a reparameterization of the policy class. As a consequence, the two approaches are interchangeable in several respects and share the same theoretical guarantees under common conditions. Leveraging this equivalence, we propose a regularization method for policy learning. The reduction to least squares yields a smooth surrogate that is typically easier to optimize in practice. At the same time, for many natural policy classes the inherent combinatorial hardness of exact EWM generally remains, so the reduction should be viewed as an optimization aid rather than a universal bypass of NP-hardness.