We propose and investigate probabilistic guarantees for the adversarial robustness of classification algorithms. While traditional formal verification approaches for robustness are intractable and sampling-based approaches do not provide formal guarantees, our approach is able to efficiently certify a probabilistic relaxation of robustness. The key idea is to sample an $\epsilon$-net and invoke a local robustness oracle on the sample. Remarkably, the size of the sample needed to achieve probably approximately global robustness guarantees is independent of the input dimensionality, the number of classes, and the learning algorithm itself. Our approach can, therefore, be applied even to large neural networks that are beyond the scope of traditional formal verification. Experiments empirically confirm that it characterizes robustness better than state-of-the-art sampling-based approaches and scales better than formal methods.

Online social media provide a platform for people to discuss current events and exchange opinions with their peers. While interactions are predominantly positive, in recent years, there has been a lot of research to understand the conflicts in social networks and how they are based on different views and opinions. In this paper, we ask whether the conflicts in a network reveal a small and interpretable set of prevalent opinion ranges that explain the users' interactions. More precisely, we consider signed graphs, where the edge signs indicate positive and negative interactions of node pairs, and our goal is to infer opinion intervals that are consistent with the edge signs. We introduce an optimization problem that models this question, and we give strong hardness results and a polynomial-time approximation scheme by utilizing connections to interval graphs and the Correlation Clustering problem. We further provide scalable heuristics and show that in experiments they yield more expressive solutions than Correlation Clustering baselines. We also present a case study on a novel real-world dataset from the German parliament, showing that our algorithms can recover the political leaning of German parties based on co-voting behavior.