  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.
