Gaussian Universality of Linear Classifiers with Random Labels in High-Dimension

While classical in many theoretical settings, the assumption of Gaussian i.i.d. inputs is often perceived as a strong limitation in the analysis of high-dimensional learning. In this study, we redeem this line of work in the case of generalized linear classification with random labels. Our main contribution is a rigorous proof that data coming from a range of generative models in high-dimensions have the same minimum training loss as Gaussian data with corresponding data covariance. In particular, our theorem covers data created by an arbitrary mixture of homogeneous Gaussian clouds, as well as multi-modal generative neural networks. In the limit of vanishing regularization, we further demonstrate that the training loss is independent of the data covariance. Finally, we show that this universality property is observed in practice with real datasets and random labels.