This work concerns the development of deep networks that are certifiably robust to adversarial attacks. Joint robust classification-detection was recently introduced as a certified defense mechanism, where adversarial examples are either correctly classified or assigned to the "abstain" class. In this work, we show that such a provable framework can benefit by extension to networks with multiple explicit abstain classes, where the adversarial examples are adaptively assigned to those. We show that naively adding multiple abstain classes can lead to "model degeneracy", then we propose a regularization approach and a training method to counter this degeneracy by promoting full use of the multiple abstain classes. Our experiments demonstrate that the proposed approach consistently achieves favorable standard vs. robust verified accuracy tradeoffs, outperforming state-of-the-art algorithms for various choices of number of abstain classes.

We explore adversarial robustness in the setting in which it is acceptable for a classifier to abstain---that is, output no class---on adversarial examples. Adversarial examples are small perturbations of normal inputs to a classifier that cause the classifier to give incorrect output; they present security and safety challenges for machine learning systems. In many safety-critical applications, it is less costly for a classifier to abstain on adversarial examples than to give incorrect output for them. We first introduce a novel objective function for adversarial robustness with an abstain option which characterizes an explicit tradeoff between robustness and accuracy. We then present a simple baseline in which an adversarially-trained classifier abstains on all inputs within a certain distance of the decision boundary, which we theoretically and experimentally evaluate. Finally, we propose Combined Abstention Robustness Learning (CARL), a method for jointly learning a classifier and the region of the input space on which it should abstain. We explore different variations of the PGD and DeepFool adversarial attacks on CARL in the abstain setting. Evaluating against these attacks, we demonstrate that training with CARL results in a more accurate, robust, and efficient classifier than the baseline.