Relative stability toward diffeomorphisms indicates performance in deep nets

TL;DR

This paper investigates the relationship between a neural network's stability to diffeomorphisms and its classification performance, revealing that relative stability correlates strongly with test error and training dynamics.

Contribution

It introduces a maximum-entropy framework for analyzing diffeomorphisms, demonstrating that relative stability to transformations predicts neural network performance better than absolute stability.

Findings

Stability to diffeomorphisms does not strongly correlate with benchmark accuracy.

Relative stability $R_f$ correlates with test error $\\epsilon_t$, decreasing during training.

Small $R_f$ is crucial for achieving low test error in deep networks.

Abstract

Understanding why deep nets can classify data in large dimensions remains a challenge. It has been proposed that they do so by becoming stable to diffeomorphisms, yet existing empirical measurements support that it is often not the case. We revisit this question by defining a maximum-entropy distribution on diffeomorphisms, that allows to study typical diffeomorphisms of a given norm. We confirm that stability toward diffeomorphisms does not strongly correlate to performance on benchmark data sets of images. By contrast, we find that the stability toward diffeomorphisms relative to that of generic transformations $R_f$ correlates remarkably with the test error $\epsilon_t$. It is of order unity at initialization but decreases by several decades during training for state-of-the-art architectures. For CIFAR10 and 15 known architectures, we find $\epsilon_t\approx 0.2\sqrt{R_f}$, suggesting that obtaining a small $R_f$ is important to achieve good performance. We study how $R_f$ depends on the size of the training set and compare it to a simple model of invariant learning.