We propose a method to learn deep ReLU-based classifiers that are provably robust against norm-bounded adversarial perturbations (on the training data; for previously unseen examples, the approach will be guaranteed to detect all adversarial examples, though it may flag some non-adversarial examples as well). The basic idea of the approach is to consider a convex outer approximation of the set of activations reachable through a norm-bounded perturbation, and we develop a robust optimization procedure that minimizes the worst case loss over this outer region (via a linear program). Crucially, we show that the dual problem to this linear program can be represented itself as a deep network similar to the backpropagation network, leading to very efficient optimization approaches that produce guaranteed bounds on the robust loss. The end result is that by executing a few more forward and backward passes through a slightly modified version of the original network (though possibly with much larger batch sizes), we can learn a classifier that is provably robust to any norm-bounded adversarial attack. We illustrate the approach on a toy 2D robust classification task, and on a simple convolutional architecture applied to MNIST, where we produce a classifier that provably has less than 8.4 test error for any adversarial attack with bounded @math norm less than @math . This represents the largest verified network that we are aware of, and we discuss future challenges in scaling the approach to much larger domains.
We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networks when their input is subject to an adversarial perturbation. The most important feature of Parseval networks is to maintain weight matrices of linear and convolutional layers to be (approximately) Parseval tight frames, which are extensions of orthogonal matrices to non-square matrices. We describe how these constraints can be maintained efficiently during SGD. We show that Parseval networks match the state-of-the-art in terms of accuracy on CIFAR-10 100 and Street View House Numbers (SVHN) while being more robust than their vanilla counterpart against adversarial examples. Incidentally, Parseval networks also tend to train faster and make a better usage of the full capacity of the networks.