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We characterise the learning of a mixture of two clouds of data points withgeneric centroids via empirical risk minimisation in the high dimensionalregime, under the assumptions of generic convex loss and convex regularisation.Each cloud of data points is obtained via a double-stochastic process, wherethe sample is obtained from a Gaussian distribution whose variance is itself arandom parameter sampled from a scalar distribution $\varrho$. As a result, ouranalysis covers a large family of data distributions, including the case ofpower-law-tailed distributions with no covariance, and allows us to test recent"Gaussian universality" claims. We study the generalisation performance of theobtained estimator, we analyse the role of regularisation, and we analyticallycharacterise the separability transition.

Classification of Heavy-tailed Features in High Dimensions: a Superstatistical Approach