Probabilistic Robustness Analysis for DNNs based on PAC Learning

This paper proposes a black box based approach for analysing deep neural networks (DNNs). We view a DNN as a function $\boldsymbol{f}$ from inputs to outputs, and consider the local robustness property for a given input. Based on scenario optimization technique in robust control design, we learn the score difference function $f_i-f_\ell$ with respect to the target label $\ell$ and attacking label $i$. We use a linear template over the input pixels, and learn the corresponding coefficients of the score difference function, based on a reduction to a linear programming (LP) problems. To make it scalable, we propose optimizations including components based learning and focused learning. The learned function offers a probably approximately correct (PAC) guarantee for the robustness property. Since the score difference function is an approximation of the local behaviour of the DNN, it can be used to generate potential adversarial examples, and the original network can be used to check whether they are spurious or not. Finally, we focus on the input pixels with large absolute coefficients, and use them to explain the attacking scenario. We have implemented our approach in a prototypical tool DeepPAC. Our experimental results show that our framework can handle very large neural networks like ResNet152 with $6.5$M neurons, and often generates adversarial examples which are very close to the decision boundary.