Relative Flatness and Generalization

TL;DR

This paper explores the theoretical link between flatness of loss landscapes and model generalization, proposing a new relative flatness measure that addresses reparameterization issues and correlates strongly with generalization.

Contribution

It introduces a novel relative flatness measure based on data representativeness and feature robustness, providing a rigorous connection to generalization and resolving reparameterization concerns.

Findings

The new flatness measure correlates strongly with generalization.

It simplifies to ridge regression in linear models.

Addresses reparameterization invariance in flatness measures.

Abstract

Flatness of the loss curve is conjectured to be connected to the generalization ability of machine learning models, in particular neural networks. While it has been empirically observed that flatness measures consistently correlate strongly with generalization, it is still an open theoretical problem why and under which circumstances flatness is connected to generalization, in particular in light of reparameterizations that change certain flatness measures but leave generalization unchanged. We investigate the connection between flatness and generalization by relating it to the interpolation from representative data, deriving notions of representativeness, and feature robustness. The notions allow us to rigorously connect flatness and generalization and to identify conditions under which the connection holds. Moreover, they give rise to a novel, but natural relative flatness measure that correlates strongly with generalization, simplifies to ridge regression for ordinary least squares, and solves the reparameterization issue.