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Deployment of deep neural networks (DNNs) in safety- or security-criticalsystems requires provable guarantees on their correct behaviour. A commonrequirement is robustness to adversarial perturbations in a neighbourhoodaround an input. In this paper we focus on the $L_0$ norm and aim to compute,for a trained DNN and an input, the maximal radius of a safe norm ball aroundthe input within which there are no adversarial examples. Then we define globalrobustness as an expectation of the maximal safe radius over a test data set.We first show that the problem is NP-hard, and then propose an approximateapproach to iteratively compute lower and upper bounds on the network'srobustness. The approach is \emph{anytime}, i.e., it returns intermediatebounds and robustness estimates that are gradually, but strictly, improved asthe computation proceeds; \emph{tensor-based}, i.e., the computation isconducted over a set of inputs simultaneously, instead of one by one, to enableefficient GPU computation; and has \emph{provable guarantees}, i.e., both thebounds and the robustness estimates can converge to their optimal values.Finally, we demonstrate the utility of the proposed approach in practice tocompute tight bounds by applying and adapting the anytime algorithm to a set ofchallenging problems, including global robustness evaluation, competitive $L_0$attacks, test case generation for DNNs, and local robustness evaluation onlarge-scale ImageNet DNNs. We release the code of all case studies via GitHub.

Global Robustness Evaluation of Deep Neural Networks with Provable Guarantees for the $L_0$ Norm