Title: Orthogonal Machine Learning: Power and Limitations,
Abstract: Double machine learning provides $\sqrt{n}$-consistent estimates of
parameters of interest even when high-dimensional or nonparametric nuisance
parameters are estimated at an $n^{-1/4}$ rate. The key is to employ
Neyman-orthogonal moment equations which are first-order insensitive to
perturbations in the nuisance parameters. We show that the $n^{-1/4}$
requirement can be improved to $n^{-1/(2k+2)}$ by employing a $k$-th order
notion of orthogonality that grants robustness to more complex or
higher-dimensional nuisance parameters. In the partially linear regression
setting popular in causal inference, we show that we can construct second-order
orthogonal moments if and only if the treatment residual is not normally
distributed. Our proof relies on Stein's lemma and may be of independent
interest. We conclude by demonstrating the robustness benefits of an explicit
doubly-orthogonal estimation procedure for treatment effect.