Uniform Consistency of Cross-Validation Estimators for High-Dimensional Ridge Regression
Pratik Patil, Yuting Wei, Alessandro Rinaldo, Ryan Tibshirani
Abstract
We examine generalized and leave-one-out cross-validation for ridge regression in a proportional asymptotic framework where the dimension of the feature space grows proportionally with the number of observations. Given i.i.d. samples from a linear model with an arbitrary feature covariance and a signal vector that is bounded in $\ell_2$ norm, we show that generalized cross-validation for ridge regression converges almost surely to the expected out-of-sample prediction error, uniformly over a range of ridge regularization parameters that includes zero (and even negative values). We prove the analogous result for leave-one-out cross-validation. As a consequence, we show that ridge tuning via minimization of generalized or leave-one-out cross-validation asymptotically almost surely delivers the optimal level of regularization for predictive accuracy, whether it be positive, negative, or zero.
BibTeX
@InProceedings{pmlr-v130-patil21a,
title = { Uniform Consistency of Cross-Validation Estimators for High-Dimensional Ridge Regression },
author = {Patil, Pratik and Wei, Yuting and Rinaldo, Alessandro and Tibshirani, Ryan},
booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics},
pages = {3178--3186},
year = {2021},
editor = {Banerjee, Arindam and Fukumizu, Kenji},
volume = {130},
series = {Proceedings of Machine Learning Research},
month = {13--15 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v130/patil21a/patil21a.pdf},
url = {https://proceedings.mlr.press/v130/patil21a.html},
abstract = { We examine generalized and leave-one-out cross-validation for ridge regression in a proportional asymptotic framework where the dimension of the feature space grows proportionally with the number of observations. Given i.i.d. samples from a linear model with an arbitrary feature covariance and a signal vector that is bounded in $\ell_2$ norm, we show that generalized cross-validation for ridge regression converges almost surely to the expected out-of-sample prediction error, uniformly over a range of ridge regularization parameters that includes zero (and even negative values). We prove the analogous result for leave-one-out cross-validation. As a consequence, we show that ridge tuning via minimization of generalized or leave-one-out cross-validation asymptotically almost surely delivers the optimal level of regularization for predictive accuracy, whether it be positive, negative, or zero. }
}