Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions
This work provides foundational theoretical insights for multi-class classification in high-dimensional settings, extending previous results but being incremental in nature.
The paper tackles the problem of learning Gaussian mixtures with generalized linear models via empirical risk minimization, proving exact asymptotics in high dimensions and applying the theory to tasks like sparse classification and max-margin multi-class classification, showing phase transitions and efficiency comparisons.
Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. In this manuscript, we characterise the learning of a mixture of $K$ Gaussians with generic means and covariances via empirical risk minimisation (ERM) with any convex loss and regularisation. In particular, we prove exact asymptotics characterising the ERM estimator in high-dimensions, extending several previous results about Gaussian mixture classification in the literature. We exemplify our result in two tasks of interest in statistical learning: a) classification for a mixture with sparse means, where we study the efficiency of $\ell_1$ penalty with respect to $\ell_2$; b) max-margin multi-class classification, where we characterise the phase transition on the existence of the multi-class logistic maximum likelihood estimator for $K>2$. Finally, we discuss how our theory can be applied beyond the scope of synthetic data, showing that in different cases Gaussian mixtures capture closely the learning curve of classification tasks in real data sets.