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Curvature-based Robustness Certificates against Adversarial Examples

A robustness certificate against adversarial examples is the minimum distance of a given input to the decision boundary of the classifier (or its lower bound). For {\it any} perturbation of the input with a magnitude smaller than the certificate value, the classification output will provably remain unchanged. Computing exact robustness certificates for deep classifiers is difficult in general since it requires solving a non-convex optimization. In this paper, we provide computationally-efficient robustness certificates for deep classifiers with differentiable activation functions in two steps. First, we show that if the eigenvalues of the Hessian of the network (curvatures of the network) are bounded, we can compute a robustness certificate in the $l_2$ norm efficiently using convex optimization. Second, we derive a computationally-efficient differentiable upper bound on the curvature of a deep network. We also use the curvature bound as a regularization term during the training of the network to boost its certified robustness against adversarial examples. Putting these results together leads to our proposed {\bf C}urvature-based {\bf R}obustness {\bf C}ertificate (CRC) and {\bf C}urvature-based {\bf R}obust {\bf T}raining (CRT). Our numerical results show that CRC outperforms CROWN's certificate by an order of magnitude while CRT leads to higher certified accuracy compared to standard adversarial training and TRADES.

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