Relative stability toward diffeomorphisms indicates performance in deep nets*This article is an updated version of: Petrini L, Favero A, Geiger M and Wyart M 2021 Relative stability toward diffeomorphisms indicates performance in deep nets Advances in Neural Information Processing Systems vol 34 ed M Ranzato, A Beygelzimer, Y Dauphin, P S Liang and J Wortman Vaughan (New York: Curran Associates) pp 8727–39. (1st November 2022) - Petrini, Leonardo; Favero, Alessandro; Geiger, Mario; Wyart, Matthieu
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 ϵ 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 ϵ t ≈ 0.2 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.