  There has been considerable effort to better understand the generalization
capabilities of deep neural networks both as a means to unlock a theoretical
understanding of their success as well as providing directions for further
improvements. In this paper, we investigate margin-based multiclass
generalization bounds for neural networks which rely on a recent complexity
measure, the geometric complexity, developed for neural networks. We derive a
new upper bound on the generalization error which scales with the
margin-normalized geometric complexity of the network and which holds for a
broad family of data distributions and model classes. Our generalization bound
is empirically investigated for a ResNet-18 model trained with SGD on the
CIFAR-10 and CIFAR-100 datasets with both original and random labels.
