Analysis of Bootstrap and Subsampling in High-dimensional Regularized
  Regression
Lucas Clarté Adrien Vandenbroucque Guillaume Dalle Bruno Loureiro Florent Krzakala Lenka Zdeborová
We investigate popular resampling methods for estimating the uncertainty of
statistical models, such as subsampling, bootstrap and the jackknife, and their
performance in high-dimensional supervised regression tasks. We provide a tight
asymptotic description of the biases and variances estimated by these methods
in the context of generalized linear models, such as ridge and logistic
regression, taking the limit where the number of samples $n$ and dimension $d$
of the covariates grow at a comparable fixed rate $\alpha\!=\! n/d$. Our
findings are three-fold: i) resampling methods are fraught with problems in
high dimensions and exhibit the double-descent-like behavior typical of these
situations; ii) only when $\alpha$ is large enough do they provide consistent
and reliable error estimations (we give convergence rates); iii) in the
over-parametrized regime $\alpha\!<\!1$ relevant to modern machine learning
practice, their predictions are not consistent, even with optimal
regularization.