Expected Tight Bounds for Robust Training

Training Deep Neural Networks that are robust to norm bounded adversarial attacks remains an elusive problem. While exact and inexact verification-based methods are generally too expensive to train large networks, it was demonstrated that bounded input intervals can be inexpensively propagated from a layer to another through deep networks. This interval bound propagation approach (IBP) not only has improved both robustness and certified accuracy but was the first to be employed on large/deep networks. However, due to the very loose nature of the IBP bounds, the required training procedure is complex and involved. In this paper, we closely examine the bounds of a block of layers composed in the form of Affine-ReLU-Affine. To this end, we propose expected tight bounds (true bounds in expectation), referred to as ETB, which are provably tighter than IBP bounds in expectation. We then extend this result to deeper networks through blockwise propagation and show that we can achieve orders of magnitudes tighter bounds compared to IBP. Furthermore, using a simple standard training procedure, we can achieve impressive robustness-accuracy trade-off on both MNIST and CIFAR10.