🤖 AI Summary
In high-dimensional regularized regression, standard $k$-fold cross-validation (CV) and generalized cross-validation (GCV) fail under sample-dependent or heavy-tailed covariates, leading to inconsistent regularization parameter tuning and generalization risk estimation. To address this, we propose a robust cross-validation framework within the proportional asymptotic regime. Our method is the first to incorporate right-rotationally invariant covariate distributions into GCV theory, enabling consistent estimation of signal-to-noise ratio and noise variance under structured dependence and heavy-tailed settings. It integrates tools from random matrix theory, compressed sensing, and GCV correction techniques. Experiments on diverse synthetic and semi-synthetic datasets demonstrate that our approach significantly improves tuning accuracy and generalization risk estimation compared to standard GCV and $k$-fold CV, while exhibiting superior stability and reliability across challenging dependency and tail regimes.
📝 Abstract
Two key tasks in high-dimensional regularized regression are tuning the regularization strength for accurate predictions and estimating the out-of-sample risk. It is known that the standard approach -- $k$-fold cross-validation -- is inconsistent in modern high-dimensional settings. While leave-one-out and generalized cross-validation remain consistent in some high-dimensional cases, they become inconsistent when samples are dependent or contain heavy-tailed covariates. As a first step towards modeling structured sample dependence and heavy tails, we use right-rotationally invariant covariate distributions -- a crucial concept from compressed sensing. In the proportional asymptotics regime where the number of features and samples grow comparably, which is known to better reflect the empirical behavior in moderately sized datasets, we introduce a new framework, ROTI-GCV, for reliably performing cross-validation under these challenging conditions. Along the way, we propose new estimators for the signal-to-noise ratio and noise variance. We conduct experiments that demonstrate the accuracy of our approach in a variety of synthetic and semi-synthetic settings.