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Training Deep Neural Networks that are robust to norm bounded adversarialattacks remains an elusive problem. While exact and inexact verification-basedmethods are generally too expensive to train large networks, it wasdemonstrated that bounded input intervals can be inexpensively propagated froma layer to another through deep networks. This interval bound propagationapproach (IBP) not only has improved both robustness and certified accuracy butwas the first to be employed on large/deep networks. However, due to the veryloose nature of the IBP bounds, the required training procedure is complex andinvolved. In this paper, we closely examine the bounds of a block of layerscomposed in the form of Affine-ReLU-Affine. To this end, we propose expectedtight bounds (true bounds in expectation), referred to as ETB, which areprovably tighter than IBP bounds in expectation. We then extend this result todeeper networks through blockwise propagation and show that we can achieveorders of magnitudes tighter bounds compared to IBP. Furthermore, using asimple standard training procedure, we can achieve impressiverobustness-accuracy trade-off on both MNIST and CIFAR10.

Expected Tight Bounds for Robust Training