Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions

Abstract

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.

12 pages + 34 pages of Appendix, 10 figures

References in corpus (10)

• Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms
• Prevalence of Neural Collapse during the terminal phase of deep learning training
• A framework to characterize performance of LASSO algorithms
• Learning curves of generic features maps for realistic datasets with a teacher-student model
• The Gaussian equivalence of generative models for learning with shallow neural networks
• The Lasso with general Gaussian designs with applications to hypothesis testing
• Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula)
• Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed
• Kernel Alignment Risk Estimator: Risk Prediction from Training Data
• Risk Bounds for Over-parameterized Maximum Margin Classification on Sub-Gaussian Mixtures