Uniform Consistency of Cross-Validation Estimators for High-Dimensional Ridge Regression

Pratik Patil, Yuting Wei, Alessandro Rinaldo, Ryan Tibshirani
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:3178-3186, 2021.

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.

Cite this Paper

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. } }
Endnote
%0 Conference Paper %T Uniform Consistency of Cross-Validation Estimators for High-Dimensional Ridge Regression %A Pratik Patil %A Yuting Wei %A Alessandro Rinaldo %A Ryan Tibshirani %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-patil21a %I PMLR %P 3178--3186 %U https://proceedings.mlr.press/v130/patil21a.html %V 130 %X 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.
APA
Patil, P., Wei, Y., Rinaldo, A. & Tibshirani, R.. (2021). Uniform Consistency of Cross-Validation Estimators for High-Dimensional Ridge Regression . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:3178-3186 Available from https://proceedings.mlr.press/v130/patil21a.html.

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