Wide flat minima and optimal generalization in classifying high-dimensional Gaussian mixtures

TL;DR

This paper investigates the relationship between wide flat minima and optimal generalization in high-dimensional Gaussian mixture classifiers, showing that Bayes-optimal solutions reside in wide flat regions of the loss landscape.

Contribution

It analytically demonstrates the connection between flat minima and optimal generalization, including in unbalanced Gaussian mixtures, and explores the error landscape near Bayes-optimal solutions.

Findings

Bayes-optimal solutions are in wide flat regions of the loss landscape.

Wide flat minima correlate with better generalization, especially in unbalanced mixtures.

Performance improvements are possible by targeting flat minima in the loss landscape.

Abstract

We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations that achieve the Bayes-optimal generalization error, even in the case of unbalanced clusters. We explore analytically the error-counting loss landscape in the vicinity of a Bayes-optimal solution, and show that the closer we get to such configurations, the higher the local entropy, implying that the Bayes-optimal solution lays inside a wide flat region. We also consider the algorithmically relevant case of targeting wide flat minima of the (differentiable) mean squared error loss. Our analytical and numerical results show not only that in the balanced case the dependence on the norm of the weights is mild, but also, in the unbalanced case, that the performances can be improved.