We investigate the effects of initialization and architecture on the start of training in deep ReLU nets. We identify two common failure modes for early training in which the mean and variance of activations are poorly behaved. For each failure mode, we give a rigorous proof of when it occurs at initialization and how to avoid it. The first failure mode, exploding vanishing mean activation length, can be avoided by initializing weights from a symmetric distribution with variance 2 fan-in. The second failure mode, exponentially large variance of activation length, can be avoided by keeping constant the sum of the reciprocals of layer widths. We demonstrate empirically the effectiveness of our theoretical results in predicting when networks are able to start training. In particular, we note that many popular initializations fail our criteria, whereas correct initialization and architecture allows much deeper networks to be trained.
We give a rigorous analysis of the statistical behavior of gradients in randomly initialized feed-forward networks with ReLU activations. Our results show that a fully connected depth @math ReLU net with hidden layer widths @math will have exploding and vanishing gradients if and only if @math is large. The point of view of this article is that whether a given neural net will have exploding vanishing gradients is a function mainly of the architecture of the net, and hence can be tested at initialization. Our results imply that a fully connected network that produces manageable gradients at initialization must have many hidden layers that are about as wide as the network is deep. This work is related to the mean field theory approach to random neural nets. From this point of view, we give a rigorous computation of the @math corrections to the propagation of gradients at the so-called edge of chaos. We examine the role of memorization in deep learning, drawing connections to capacity, generalization, and adversarial robustness. While deep networks are capable of memorizing noise data, our results suggest that they tend to prioritize learning simple patterns first. In our experiments, we expose qualitative differences in gradient-based optimization of deep neural networks (DNNs) on noise vs. real data. We also demonstrate that for appropriately tuned explicit regularization (e.g., dropout) we can degrade DNN training performance on noise datasets without compromising generalization on real data. Our analysis suggests that the notions of effective capacity which are dataset independent are unlikely to explain the generalization performance of deep networks when trained with gradient based methods because training data itself plays an important role in determining the degree of memorization.