  A longstanding goal in the theory of deep learning is to characterize the
conditions under which a given neural network architecture will be trainable,
and if so, how well it might generalize to unseen data. In this work, we
provide such a characterization in the limit of very wide and very deep
networks, for which the analysis simplifies considerably. For wide networks,
the trajectory under gradient descent is governed by the Neural Tangent Kernel
(NTK), and for deep networks the NTK itself maintains only weak data
dependence. By analyzing the spectrum of the NTK, we formulate necessary
conditions for trainability and generalization across a range of architectures,
including Fully Connected Networks (FCNs) and Convolutional Neural Networks
(CNNs). We identify large regions of hyperparameter space for which networks
can memorize the training set but completely fail to generalize. We find that
CNNs without global average pooling behave almost identically to FCNs, but that
CNNs with pooling have markedly different and often better generalization
performance. These theoretical results are corroborated experimentally on
CIFAR10 for a variety of network architectures and we include a colab notebook
that reproduces the essential results of the paper.
