Relative stability toward diffeomorphisms indicates performance in deep nets
This addresses the problem of understanding generalization in deep learning for researchers, but it is incremental as it refines existing theories without a major breakthrough.
The study found that relative stability to diffeomorphisms, not absolute stability, correlates strongly with test error in deep neural networks, with a relationship approximated as ε_t ≈ 0.2√R_f for CIFAR10 across 15 architectures, showing a decrease by several decades during training.
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 $ε_t\approx 0.2\sqrt{R_f}$, 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.