Spectral Batch Normalization: Normalization in the Frequency Domain

Regularization is a set of techniques that are used to improve the generalization ability of deep neural networks. In this paper, we introduce spectral batch normalization (SBN), a novel effective method to improve generalization by normalizing feature maps in the frequency (spectral) domain. The activations of residual networks without batch normalization (BN) tend to explode exponentially in the depth of the network at initialization. This leads to extremely large feature map norms even though the parameters are relatively small. These explosive dynamics can be very detrimental to learning. BN makes weight decay regularization on the scaling factors gamma, beta approximately equivalent to an additive penalty on the norm of the feature maps, which prevents extremely large feature map norms to a certain degree. It was previously shown that preventing explosive growth at the final layer at initialization and during training in ResNets can recover a large part of Batch Normalization's generalization boost. However, we show experimentally that, despite the approximate additive penalty of BN, feature maps in deep neural networks (DNNs) tend to explode at the beginning of the training and that feature maps of DNNs contain large values during the whole training. This phenomenon also occurs in a weakened form in non-residual networks. Intuitively, it is not preferred to have large values in feature maps since they have too much influence on the prediction in contrast to other parts of the feature map. SBN addresses large feature maps by normalizing them in the frequency domain. In our experiments, we empirically show that SBN prevents exploding feature maps at initialization and large feature map values during the training. Moreover, the normalization of feature maps in the frequency domain leads to more uniform distributed frequency components. This discourages the DNNs to rely on single frequency components of feature maps. These, together with other effects (e.g. noise injection, scaling and shifting of the feature map) of SBN, have a regularizing effect on the training of residual and non-residual networks. We show experimentally that using SBN in addition to standard regularization methods improves the performance of DNNs by a relevant margin, e.g. ResNet50 on CIFAR-100 by 2.31%, on ImageNet by 0.71% (from 76.80% to 77.51%) and VGG19 on CIFAR-100 by 0.66%.