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Over the past decade, characterizing the exact asymptotic risk of regularizedestimators in high-dimensional regression has emerged as a popular line ofwork. This literature considers the proportional asymptotics framework, wherethe number of features and samples both diverge, at a rate proportional to eachother. Substantial work in this area relies on Gaussianity assumptions on theobserved covariates. Further, these studies often assume the design entries tobe independent and identically distributed. Parallel research investigates theuniversality of these findings, revealing that results based on thei.i.d.~Gaussian assumption extend to a broad class of designs, such asi.i.d.~sub-Gaussians. However, universality results examining dependentcovariates so far focused on correlation-based dependence or a highlystructured form of dependence, as permitted by right rotationally invariantdesigns. In this paper, we break this barrier and study a dependence structurethat in general falls outside the purview of these established classes. We seekto pin down the extent to which results based on i.i.d.~Gaussian assumptionspersist. We identify a class of designs characterized by a block dependencestructure that ensures the universality of i.i.d.~Gaussian-based results. Weestablish that the optimal values of the regularized empirical risk and therisk associated with convex regularized estimators, such as the Lasso andridge, converge to the same limit under block dependent designs as they do fori.i.d.~Gaussian entry designs. Our dependence structure differs significantlyfrom correlation-based dependence, and enables, for the first time,asymptotically exact risk characterization in prevalent nonparametricregression problems in high dimensions. Finally, we illustrate throughexperiments that this universality becomes evident quite early, even forrelatively moderate sample sizes.

Universality in block dependent linear models with applications to nonparametric regression