  We compute precise asymptotic expressions for the learning curves of least
squares random feature (RF) models with either a separable strongly convex
regularization or the $\ell_1$ regularization. We propose a novel multi-level
application of the convex Gaussian min max theorem (CGMT) to overcome the
traditional difficulty of finding computable expressions for random features
models with correlated data. Our result takes the form of a computable
4-dimensional scalar optimization. In contrast to previous results, our
approach does not require solving an often intractable proximal operator, which
scales with the number of model parameters. Furthermore, we extend the
universality results for the training and generalization errors for RF models
to $\ell_1$ regularization. In particular, we demonstrate that under mild
conditions, random feature models with elastic net or $\ell_1$ regularization
are asymptotically equivalent to a surrogate Gaussian model with the same first
and second moments. We numerically demonstrate the predictive capacity of our
results, and show experimentally that the predicted test error is accurate even
in the non-asymptotic regime.
