Disentangling Trainability and Generalization in Deep Learning

A fundamental goal in deep learning is the characterization of trainability and generalization of neural networks as a function of their architecture and hyperparameters. In this paper, we discuss these challenging issues in the context of wide neural networks at large depths where we will see that the situation simplifies considerably. To do this, we leverage recent advances that have separately shown: (1) that in the wide network limit, random networks before training are Gaussian Processes governed by a kernel known as the Neural Network Gaussian Process (NNGP) kernel, (2) that at large depths the spectrum of the NNGP kernel simplifies considerably and becomes ``weakly data-dependent'', and (3) that gradient descent training of wide neural networks is described by a kernel called the Neural Tangent Kernel (NTK) that is related to the NNGP. Here we show that by combining the in the large depth limit the spectrum of the NTK simplifies in much the same way as that of the NNGP kernel. By analyzing this spectrum, we arrive at a precise characterization of trainability and generalization across a range of architectures including Fully Connected Networks (FCNs) and Convolutional Neural Networks (CNNs). We find that there are large regions of hyperparameter space where networks will train but will fail to generalize, in contrast with several recent results. By comparing CNNs with- and without-global average pooling, we show that CNNs without average pooling have very nearly identical learning dynamics to FCNs while CNNs with pooling contain a correction that alters its generalization performance. We perform a thorough empirical investigation of these theoretical results and finding excellent agreement on real datasets.