Minimax rates for heterogeneous causal effect estimation

• Published in: The Annals of statistics Vol. 52; no. 2; p. 793
• Main Authors: Kennedy, Edward H, Balakrishnan, Sivaraman, Robins, James M, Wasserman, Larry
• Format: Journal Article
• Language: English
• Published: United States 01.04.2024
• Subjects: nonparametric regression; higher order influence functions; causal inference; functional estimation; optimal rates of convergence
• ISSN: 0090-5364
• DOI: 10.1214/24-aos2369

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference. Many methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a Hölder-smooth nonparametric model, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. Our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid.