  Consider supervised learning from i.i.d. samples $\{{\boldsymbol
x}_i,y_i\}_{i\le n}$ where ${\boldsymbol x}_i \in\mathbb{R}^p$ are feature
vectors and ${y} \in \mathbb{R}$ are labels. We study empirical risk
minimization over a class of functions that are parameterized by $\mathsf{k} =
O(1)$ vectors ${\boldsymbol \theta}_1, . . . , {\boldsymbol \theta}_{\mathsf k}
\in \mathbb{R}^p$ , and prove universality results both for the training and
test error. Namely, under the proportional asymptotics $n,p\to\infty$, with
$n/p = \Theta(1)$, we prove that the training error depends on the random
features distribution only through its covariance structure. Further, we prove
that the minimum test error over near-empirical risk minimizers enjoys similar
universality properties. In particular, the asymptotics of these quantities can
be computed $-$to leading order$-$ under a simpler model in which the feature
vectors ${\boldsymbol x}_i$ are replaced by Gaussian vectors ${\boldsymbol
g}_i$ with the same covariance. Earlier universality results were limited to
strongly convex learning procedures, or to feature vectors ${\boldsymbol x}_i$
with independent entries. Our results do not make any of these assumptions. Our
assumptions are general enough to include feature vectors ${\boldsymbol x}_i$
that are produced by randomized featurization maps. In particular we explicitly
check the assumptions for certain random features models (computing the output
of a one-layer neural network with random weights) and neural tangent models
(first-order Taylor approximation of two-layer networks).
