Disentangling trainability and generalization in deep learning

A BSTRACT A fundamental goal in deep learning is the characterization of trainability and generalization of neural networks as a function of their architecture and hyper-parameters. 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 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 a necessary condition for generalization across a range of architectures including Fully Connected Networks (FCNs) and Con-volutional Neural Networks (CNNs). In particular, we find that there are large regions of hyperparameter space where networks can only memorize the training set in the sense they reach perfect training accuracy but completely fail to generalize outside the training set, 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. Historically, the rampant success of deep learning models has lacked a sturdy theoretical foundation; architectures, hyperparameters, and learning algorithms are often selected by brute force search (Bergstra & Bengio, 2012) and heuristics (Glorot & Bengio, 2010). Recently, significant theoretical progress has been made on several fronts that have shown promise in making neural network design more systematic. In particular, in the infinite width (or channel) limit, the distribution of functions induced by neural networks with random weights and biases has been precisely characterized before, during, and after training. The study of infinite networks dates back to seminal work by Neal (1994) who showed that the distribution of functions given by single hidden-layer networks with random weights and biases in the infinite-width limit are Gaussian Processes (GPs). Recently, there has been renewed interest in studying random, infinite, networks starting with concurrent work on "conjugate kernels" (Daniely et al., 2016; Daniely, 2017) and "mean-field theory" (Poole et al., 2016; Schoenholz et al., 2017).