Sup-Norm Error under Proportional Asymptotics: Phase Transitions under Linear Regression

This paper characterizes the sharp asymptotics of the $\ell_{\infty}$-norm estimation error for estimating the coefficient vector in linear regression under proportional asymptotics. While $\ell_2$-error is well-documented, we demonstrate that the $\ell_{\infty}$-error exhibits a distinct phase transition governed by the structure of the underlying signal. We explore this through the lens of a class of estimators that include ridge-regularized estimators, a method-of-moments estimator, and the ordinary least squares estimator for the under-parametrized regime. Our results provide a comparative study of these estimators and characterize regimes (as identified by the maximal signal component, number of large signals, total signal strength, and noise variance) where one estimator is preferable to the others. As a by-product, we obtain new double descent curves in $\ell_{\infty}$-error metric for the ridgeless regression, as well as ingredients for uniform confidence intervals for the signal coordinates. Our theoretical results are supported by extensive numerical simulations that confirm the predicted phase transitions and error limits.