On Measuring Excess Capacity in Neural Networks

TL;DR

This paper investigates the excess capacity of deep neural networks, especially residual architectures, using Rademacher complexity bounds, revealing significant capacity beyond what is necessary for empirical error and suggesting a form of weight norm-based compressibility.

Contribution

The authors extend Rademacher complexity bounds to modern architectures, providing new insights into the excess capacity and its relation to weight norms and compressibility.

Findings

Substantial excess capacity exists per task.

Capacity remains similar across different tasks.

Weight norm-based measures relate to model compressibility.

Abstract

We study the excess capacity of deep networks in the context of supervised classification. That is, given a capacity measure of the underlying hypothesis class - in our case, empirical Rademacher complexity - to what extent can we (a priori) constrain this class while retaining an empirical error on a par with the unconstrained regime? To assess excess capacity in modern architectures (such as residual networks), we extend and unify prior Rademacher complexity bounds to accommodate function composition and addition, as well as the structure of convolutions. The capacity-driving terms in our bounds are the Lipschitz constants of the layers and an (2, 1) group norm distance to the initializations of the convolution weights. Experiments on benchmark datasets of varying task difficulty indicate that (1) there is a substantial amount of excess capacity per task, and (2) capacity can be kept at a surprisingly similar level across tasks. Overall, this suggests a notion of compressibility with respect to weight norms, complementary to classic compression via weight pruning. Source code is available at https://github.com/rkwitt/excess_capacity.