  Let $(x_{i}, y_{i})_{i=1,\dots,n}$ denote independent samples from a general
mixture distribution $\sum_{c\in\mathcal{C}}\rho_{c}P_{c}^{x}$, and consider
the hypothesis class of generalized linear models $\hat{y} =
F(\Theta^{\top}x)$. In this work, we investigate the asymptotic joint
statistics of the family of generalized linear estimators $(\Theta_{1}, \dots,
\Theta_{M})$ obtained either from (a) minimizing an empirical risk
$\hat{R}_{n}(\Theta;X,y)$ or (b) sampling from the associated Gibbs measure
$\exp(-\beta n \hat{R}_{n}(\Theta;X,y))$. Our main contribution is to
characterize under which conditions the asymptotic joint statistics of this
family depends (on a weak sense) only on the means and covariances of the class
conditional features distribution $P_{c}^{x}$. In particular, this allow us to
prove the universality of different quantities of interest, such as the
training and generalization errors, redeeming a recent line of work in
high-dimensional statistics working under the Gaussian mixture hypothesis.
Finally, we discuss the applications of our results to different machine
learning tasks of interest, such as ensembling and uncertainty
