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 would also like to point out a previous appearance in Hanin @cite_34 of the sum of reciprocals of layer widths, which we here show determines the variance of the sizes of the activations (see Theorem ) in randomly initialized fully connected @math nets. The article @cite_34 studied the more delicate question of the variance for gradients computed by random @math nets. Finally, we point the reader to the discussion around Figure 6 in @cite_3 , which also finds that time to convergence is better for wider networks.
@cite_34 @cite_3
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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. Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find @math on the training objective of DNNs in @math . We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter means the network width is sufficiently large: @math in @math , the number of layers and in @math , the number of samples. Our key technique is to derive that, in a sufficiently large neighborhood of the random initialization, the optimization landscape is almost-convex and semi-smooth even with ReLU activations. This implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. As concrete examples, starting from randomly initialized weights, we prove that SGD can attain 100 training accuracy in classification tasks, or minimize regression loss in linear convergence speed, with running time polynomial in @math . Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet). Layer-sequential unit-variance (LSUV) initialization - a simple method for weight initialization for deep net learning - is proposed. The method consists of the two steps. First, pre-initialize weights of each convolution or inner-product layer with orthonormal matrices. Second, proceed from the first to the final layer, normalizing the variance of the output of each layer to be equal to one. Experiment with different activation functions (maxout, ReLU-family, tanh) show that the proposed initialization leads to learning of very deep nets that (i) produces networks with test accuracy better or equal to standard methods and (ii) is at least as fast as the complex schemes proposed specifically for very deep nets such as FitNets ( (2015)) and Highway ( (2015)). Performance is evaluated on GoogLeNet, CaffeNet, FitNets and Residual nets and the state-of-the-art, or very close to it, is achieved on the MNIST, CIFAR-10 100 and ImageNet datasets. In this paper, we explore theoretical properties of training a two-layered ReLU network @math with centered @math -dimensional spherical Gaussian input @math ( @math =ReLU). We train our network with gradient descent on @math to mimic the output of a teacher network with the same architecture and fixed parameters @math . We show that its population gradient has an analytical formula, leading to interesting theoretical analysis of critical points and convergence behaviors. First, we prove that critical points outside the hyperplane spanned by the teacher parameters ("out-of-plane") are not isolated and form manifolds, and characterize in-plane critical-point-free regions for two ReLU case. On the other hand, convergence to @math for one ReLU node is guaranteed with at least @math probability, if weights are initialized randomly with standard deviation upper-bounded by @math , consistent with empirical practice. For network with many ReLU nodes, we prove that an infinitesimal perturbation of weight initialization results in convergence towards @math (or its permutation), a phenomenon known as spontaneous symmetric-breaking (SSB) in physics. We assume no independence of ReLU activations. Simulation verifies our findings.