Analysis of Bootstrap and Subsampling in High-dimensional Regularized Regression

Lucas Clarté, Adrien Vandenbroucque, Guillaume Dalle, Bruno Loureiro, Florent Krzakala, Lenka Zdeborova

Keywords: high-dimension, bootstrap, convex problems, statistical physics, uncertainty, approximate message passing, high-dimensional statistics

TL;DR: We evaluate the effectiveness of bootstrap and subsampling for estimating bias and variances in high-dimensional regularized regression, revealing their limitations in this context

Abstract: 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 rate: $\alpha=n/d$ fixed. 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.

List Of Authors: Clart\'e, Lucas and Vandenbroucque, Adrien and Dalle, Guillaume and Loureiro, Bruno and Krzakala, Florent and Zdeborov\'a, Lenka

Code Url: https://github.com/SPOC-group/BootstrapAsymptotics

Submission Number: 361