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Francesca Mignacco, Florent Krzakala, Yue Lu, Pierfrancesco Urbani, Lenka Zdeborova
We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number $n$ of samples and their dimension $d$ goes to infinity while their ratio is fixed to $\alpha=n/d$. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances, we illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.