  The bootstrap is a popular method of constructing confidence intervals due to
its ease of use and broad applicability. Theoretical properties of bootstrap
procedures have been established in a variety of settings. However, there is
limited theoretical research on the use of the bootstrap in the context of
estimation of a differentiable functional in a nonparametric or semiparametric
model when nuisance functions are estimated using machine learning. In this
article, we provide general conditions for consistency of the bootstrap in such
scenarios. Our results cover a range of estimator constructions, nuisance
estimation methods, bootstrap sampling distributions, and bootstrap confidence
interval types. We provide refined results for the empirical bootstrap and
smoothed bootstraps, and for one-step estimators, plug-in estimators, empirical
mean plug-in estimators, and estimating equations-based estimators. We
illustrate the use of our general results by demonstrating the asymptotic
validity of bootstrap confidence intervals for the average density value and
G-computed conditional mean parameters, and compare their performance in finite
samples using numerical studies. Throughout, we emphasize whether and how the
bootstrap can produce asymptotically valid confidence intervals when standard
methods fail to do so.
