Recent advances in deep learning have given us some very promising results on the generalization ability of deep neural networks, however literature still lacks a comprehensive theory explaining why heavily over-parametrized models are able to generalize well while fitting the training data. In this paper we propose a PAC type bound on the generalization error of feedforward ReLU networks via estimating the Rademacher complexity of the set of networks available from an initial parameter vector via gradient descent. The key idea is to bound the sensitivity of the network's gradient to perturbation of the input data along the optimization trajectory. The obtained bound does not explicitly depend on the depth of the network. Our results are experimentally verified on the MNIST and CIFAR-10 datasets.

Recent articles indicate that deep neural networks are efficient models for various learning problems. However they are often highly sensitive to various changes that cannot be detected by an independent observer. As our understanding of deep neural networks with traditional generalization bounds still remains incomplete, there are several measures which capture the behaviour of the model in case of small changes at a specific state. In this paper we consider adversarial stability in the tangent space and suggest tangent sensitivity in order to characterize stability. We focus on a particular kind of stability with respect to changes in parameters that are induced by individual examples without known labels. We derive several easily computable bounds and empirical measures for feed-forward fully connected ReLU (Rectified Linear Unit) networks and connect tangent sensitivity to the distribution of the activation regions in the input space realized by the network. Our experiments suggest that even simple bounds and measures are associated with the empirical generalization gap.