AUCPro: AUC-Oriented Provable Robustness Learning

The current studies of provable robustness for deep neural networks (DNNs) usually assume that the class distribution is overall balanced. However, in real-world applications especially for safety-sensitive systems, the class distribution often exhibits a long-tailed property. It is well-known that the Area Under the ROC Curve (AUC) is a more proper metric for long-tailed learning problems. Motivated by this fact, an AUC-oriented provable robustness learning framework (named AUCPro) is first proposed in this paper. The key is to construct a proxy model smoothed by the isotropic Gaussian noise and then consider optimizing the proxy model from the AUC-oriented learning point of view. Theoretically, we provide a certified safety region for AUCPro within which the model would be free from the $\ell _{2}$ adversarial attacks. Most importantly, we propose a novel standard to theoretically study the robustness generalization toward unseen data for provable robustness learning approaches. To the best of our knowledge, such a problem remains barely considered in the machine learning community. To be specific, under a general principle for performance-robustness trade-off, we prove that the generalization ability of the resulting model could be equivalently expressed as the expected adversarial risk of AUC under $\ell _{2}$ perturbation. On top of this, we present two practical settings to explore the excess risk formed by the difference between the empirical risk of AUCPro and the derived generalization performance. Finally, comprehensive experiments speak to the efficacy of our proposed algorithm.