Gaussian Universality of Perceptrons with Random Labels

TL;DR

This paper demonstrates that in high-dimensional generalized linear classification with random labels, the training loss exhibits universality across a wide class of input data distributions, not just Gaussian, especially as regularization vanishes.

Contribution

It establishes a universality class for high-dimensional perceptrons with random labels, extending Gaussian assumptions to broader data distributions, both theoretically and empirically.

Findings

Training loss is universal across data distributions with the same covariance.

Universality holds for mixtures of Gaussian clouds and real datasets.

Loss becomes independent of data covariance as regularization approaches zero.

Abstract

While classical in many theoretical settings - and in particular in statistical physics-inspired works - the assumption of Gaussian i.i.d. input data is often perceived as a strong limitation in the context of statistics and machine learning. In this study, we redeem this line of work in the case of generalized linear classification, a.k.a. the perceptron model, with random labels. We argue that there is a large universality class of high-dimensional input data for which we obtain the same minimum training loss as for Gaussian data with corresponding data covariance. In the limit of vanishing regularization, we further demonstrate that the training loss is independent of the data covariance. On the theoretical side, we prove this universality for an arbitrary mixture of homogeneous Gaussian clouds. Empirically, we show that the universality holds also for a broad range of real datasets.