Title: Analytical Moment Regularizer for Training Robust Networks. Abstract: Despite the impressive performance of deep neural networks (DNNs) on numerous learning  tasks, they still exhibit uncouth behaviours. One puzzling behaviour is the subtle sensitive reaction of DNNs to various noise attacks. Such a nuisance has strengthened the line of research around developing and training noise-robust networks. In this work, we propose a new training regularizer that aims to minimize the probabilistic expected training loss of a DNN subject to a generic Gaussian input. We provide an efficient and simple approach to approximate such a regularizer for arbitrarily deep networks. This is done by leveraging the analytic expression of the output mean of a shallow neural network, avoiding the need for memory and computation expensive data augmentation. We conduct extensive experiments on LeNet and AlexNet on various datasets including MNIST, CIFAR10, and CIFAR100 to demonstrate the effectiveness of our proposed regularizer. In particular, we show that networks that are trained with the proposed regularizer benefit from a boost in robustness against Gaussian noise to an equivalent amount of performing 3-21 folds of noisy data augmentation. Moreover, we empirically show on several architectures and datasets that improving robustness against Gaussian noise, by using the new regularizer, can improve the overall robustness against 6 other types of attacks by two orders of magnitude.