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We consider a high-dimensional mixture of two Gaussians in the noisy regimewhere even an oracle knowing the centers of the clusters misclassifies a smallbut finite fraction of the points. We provide a rigorous analysis of thegeneralization error of regularized convex classifiers, including ridge, hingeand logistic regression, in the high-dimensional limit where the number $n$ ofsamples and their dimension $d$ go to infinity while their ratio is fixed to$\alpha= n/d$. We discuss surprising effects of the regularization that in somecases allows to reach the Bayes-optimal performances. We also illustrate theinterpolation peak at low regularization, and analyze the role of therespective sizes of the two clusters.

The role of regularization in classification of high-dimensional noisy Gaussian mixture