Minimax rates for heterogeneous causal effect estimation

TL;DR

This paper establishes the minimax rate for estimating heterogeneous causal effects in a nonparametric setting and introduces a new local polynomial estimator that achieves this optimal rate, advancing theoretical understanding in causal inference.

Contribution

It derives the minimax rate for CATE estimation in a Holder-smooth model and proposes a new estimator that attains this rate under specific conditions.

Findings

Derived the minimax rate for CATE estimation.

Proposed a local polynomial estimator that is minimax optimal.

Identified an unusual interpolation between regression and functional estimation rates.

Abstract

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 Holder-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.