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 ∞ norm less than = 0.1. This represents the largest verified network that we are aware of, and we discuss future challenges in scaling the approach to much larger domains.
In contrast, robust training attempts to minimize the "worst-case loss" REF for each example -that is, the maximum loss over all bounded perturbations of that example.
Provable defenses against adversarial examples via the convex outer adversarial polytope
ArXiv
ArXiv
Mathematics Computer Science