High-dimensional ridge regression with random features for non-identically distributed data with a variance profile

Abstract

Random feature ridge regression is often analyzed in the high-dimensional regime under the homogeneous sampling model $x_i=\Sigma^{1/2}x_i'$, where the vectors $x_i'$ have iid entries and the same covariance matrix $\Sigma$ is shared by all samples. In this paper, we move beyond this setting and study non-identically distributed data through a variance-profile model in which the training and test covariates have row-dependent diagonal covariance matrices $\Sigma_i=\diag(\gamma_{i1}^2,\ldots,\gamma_{ip}^2)$ and $\widetilde{\Sigma}_i=\diag(\tilde\gamma_{i1}^2,\ldots,\tilde\gamma_{ip}^2)$. Our main contribution is the derivation of asymptotic equivalents for the training and test risks of ridge regression with random features when $n$, $p$, and $m$ grow proportionally. The first set of equivalents is obtained by combining the linear-plus-chaos approximation with traffic-probability arguments, whereas the second set is deterministic and follows from operator-valued free probability through an amalgamation-over-the-diagonal argument. These equivalents are sharp in numerical experiments. They also reveal how heterogeneous variance profiles, including mixture-type profiles inspired by MNIST, can modify generalization and exhibit double-descent behavior when the ridge parameter is small.