We address the problem of estimating statistics of hidden units in a neural network using a method of analytic moment propagation. These statistics are useful for approximate whitening of the inputs in front of saturating non-linearities such as a sigmoid function. This is important for initialization of training and for reducing the accumulated scale and bias dependencies (compensating covariate shift), which presumably eases the learning. In batch normalization, which is currently a very widely applied technique, sample estimates of statistics of hidden units over a batch are used. The proposed estimation uses an analytic propagation of mean and variance of the training set through the network. The result depends on the network structure and its current weights but not on the specific batch input. The estimates are suitable for initialization and normalization, efficient to compute and independent of the batch size. The experimental verification well supports these claims. However, the method does not share the generalization properties of BN, to which our experiments give some additional insight.
Batch normalization (BN) has become a de facto standard for training deep convolutional networks. However, BN accounts for a significant fraction of training run-time and is difficult to accelerate, since it is a memory-bandwidth bounded operation. Such a drawback of BN motivates us to explore recently proposed weight normalization algorithms (WN algorithms), i.e. weight normalization, normalization propagation and weight normalization with translated ReLU. These algorithms don't slow-down training iterations and were experimentally shown to outperform BN on relatively small networks and datasets. However, it is not clear if these algorithms could replace BN in practical, large-scale applications. We answer this question by providing a detailed comparison of BN and WN algorithms using ResNet-50 network trained on ImageNet. We found that although WN achieves better training accuracy, the final test accuracy is significantly lower ( @math ) than that of BN. This result demonstrates the surprising strength of the BN regularization effect which we were unable to compensate for using standard regularization techniques like dropout and weight decay. We also found that training of deep networks with WN algorithms is significantly less stable compared to BN, limiting their practical applications. We present weight normalization: a reparameterization of the weight vectors in a neural network that decouples the length of those weight vectors from their direction. By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time. We demonstrate the usefulness of our method on applications in supervised image recognition, generative modelling, and deep reinforcement learning.