  Given two or more Deep Neural Networks (DNNs) with the same or similar
architectures, and trained on the same dataset, but trained with different
solvers, parameters, hyper-parameters, regularization, etc., can we predict
which DNN will have the best test accuracy, and can we do so without peeking at
the test data? In this paper, we show how to use a new Theory of Heavy-Tailed
Self-Regularization (HT-SR) to answer this. HT-SR suggests, among other things,
that modern DNNs exhibit what we call Heavy-Tailed Mechanistic Universality
(HT-MU), meaning that the correlations in the layer weight matrices can be fit
to a power law (PL) with exponents that lie in common Universality classes from
Heavy-Tailed Random Matrix Theory (HT-RMT). From this, we develop a Universal
capacity control metric that is a weighted average of PL exponents. Rather than
considering small toy NNs, we examine over 50 different, large-scale
pre-trained DNNs, ranging over 15 different architectures, trained on
ImagetNet, each of which has been reported to have different test accuracies.
We show that this new capacity metric correlates very well with the reported
test accuracies of these DNNs, looking across each architecture
(VGG16/.../VGG19, ResNet10/.../ResNet152, etc.). We also show how to
approximate the metric by the more familiar Product Norm capacity measure, as
the average of the log Frobenius norm of the layer weight matrices. Our
approach requires no changes to the underlying DNN or its loss function, it
does not require us to train a model (although it could be used to monitor
training), and it does not even require access to the ImageNet data.
