Analysis of Bootstrap and Subsampling in High-dimensional Regularized Regression

• URL: http://arxiv.org/abs/2402.13622v1
• Date: Wed, 21 Feb 2024 08:50:33 GMT
• Title: Analysis of Bootstrap and Subsampling in High-dimensional Regularized Regression
• Authors: Lucas Clart\'e, Adrien Vandenbroucque, Guillaume Dalle, Bruno Loureiro, Florent Krzakala, Lenka Zdeborov\'a
• Abstract summary: We investigate popular resampling methods for estimating the uncertainty of statistical models. We provide a tight description of the biases and variances estimated by these methods in the context of generalized linear models.
• Score: 20.348736537045916
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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