  For observational studies, we study the sensitivity of causal inference when
treatment assignments may depend on unobserved confounders. We develop a loss
minimization approach for estimating bounds on the conditional average
treatment effect (CATE) when unobserved confounders have a bounded effect on
the odds ratio of treatment selection. Our approach is scalable and allows
flexible use of model classes in estimation, including nonparametric and
black-box machine learning methods. Based on these bounds for the CATE, we
propose a sensitivity analysis for the average treatment effect (ATE). Our
semi-parametric estimator extends/bounds the augmented inverse propensity
weighted (AIPW) estimator for the ATE under bounded unobserved confounding. By
constructing a Neyman orthogonal score, our estimator of the bound for the ATE
is a regular root-$n$ estimator so long as the nuisance parameters are
estimated at the $o_p(n^{-1/4})$ rate. We complement our methodology with
optimality results showing that our proposed bounds are tight in certain cases.
We demonstrate our method on simulated and real data examples, and show
accurate coverage of our confidence intervals in practical finite sample
regimes with rich covariate information.
