Feature Whitening via Gradient Transformation for Improved Convergence

Abstract

Feature whitening is a known technique for speeding up training of DNN. Under certain assumptions, whitening the activations reduces the Fisher information matrix to a simple identity matrix, in which case stochastic gradient descent is equivalent to the faster natural gradient descent. Due to the additional complexity resulting from transforming the layer inputs and their corresponding gradients in the forward and backward propagation, and from repeatedly computing the Eigenvalue decomposition (EVD), this method is not commonly used to date. In this work, we address the complexity drawbacks of feature whitening. Our contribution is twofold. First, we derive an equivalent method, which replaces the sample transformations by a transformation to the weight gradients, applied to every batch of B samples. The complexity is reduced by a factor of S=(2B), where S denotes the feature dimension of the layer output. As the batch size increases with distributed training, the benefit of using the proposed method becomes more compelling. Second, motivated by the theoretical relation between the condition number of the sample covariance matrix and the convergence speed, we derive an alternative sub-optimal algorithm which recursively reduces the condition number of the latter matrix. Compared to EVD, complexity is reduced by a factor of the input feature dimension M. We exemplify the proposed algorithms with ResNet-based networks for image classification demonstrated on the CIFAR and Imagenet datasets. Parallelizing the proposed algorithms is straightforward and we implement a distributed version thereof. Improved convergence, in terms of speed and attained accuracy, can be observed in our experiments.