  We study the problem of learning conditional average treatment effects (CATE)
from observational data with unobserved confounders. The CATE function maps
baseline covariates to individual causal effect predictions and is key for
personalized assessments. Recent work has focused on how to learn CATE under
unconfoundedness, i.e., when there are no unobserved confounders. Since CATE
may not be identified when unconfoundedness is violated, we develop a
functional interval estimator that predicts bounds on the individual causal
effects under realistic violations of unconfoundedness. Our estimator takes the
form of a weighted kernel estimator with weights that vary adversarially. We
prove that our estimator is sharp in that it converges exactly to the tightest
bounds possible on CATE when there may be unobserved confounders. Further, we
study personalized decision rules derived from our estimator and prove that
they achieve optimal minimax regret asymptotically. We assess our approach in a
simulation study as well as demonstrate its application in the case of hormone
replacement therapy by comparing conclusions from a real observational study
and clinical trial.
