Title: Global Robustness Evaluation of Deep Neural Networks with Provable Guarantees for the $L_0$ Norm,
Abstract: Deployment of deep neural networks (DNNs) in safety- or security-critical
systems requires provable guarantees on their correct behaviour. A common
requirement is robustness to adversarial perturbations in a neighbourhood
around 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 around
the input within which there are no adversarial examples. Then we define global
robustness 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 approximate
approach to iteratively compute lower and upper bounds on the network's
robustness. The approach is \emph{anytime}, i.e., it returns intermediate
bounds and robustness estimates that are gradually, but strictly, improved as
the computation proceeds; \emph{tensor-based}, i.e., the computation is
conducted over a set of inputs simultaneously, instead of one by one, to enable
efficient GPU computation; and has \emph{provable guarantees}, i.e., both the
bounds and the robustness estimates can converge to their optimal values.
Finally, we demonstrate the utility of the proposed approach in practice to
compute tight bounds by applying and adapting the anytime algorithm to a set of
challenging problems, including global robustness evaluation, competitive $L_0$
attacks, test case generation for DNNs, and local robustness evaluation on
large-scale ImageNet DNNs. We release the code of all case studies via GitHub.