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Recent studies show deep neural networks (DNNs) are extremely vulnerable tothe elaborately designed adversarial examples. Adversarial learning with thoseadversarial examples has been proved as one of the most effective methods todefend against such an attack. At present, most existing adversarial examplesgeneration methods are based on first-order gradients, which can hardly furtherimprove models' robustness, especially when facing second-order adversarialattacks. Compared with first-order gradients, second-order gradients provide amore accurate approximation of the loss landscape with respect to naturalexamples. Inspired by this, our work crafts second-order adversarial examplesand uses them to train DNNs. Nevertheless, second-order optimization involvestime-consuming calculation for Hessian-inverse. We propose an approximationmethod through transforming the problem into an optimization in the Krylovsubspace, which remarkably reduce the computational complexity to speed up thetraining procedure. Extensive experiments conducted on the MINIST and CIFAR-10datasets show that our adversarial learning with second-order adversarialexamples outperforms other fisrt-order methods, which can improve the modelrobustness against a wide range of attacks.

Hessian-Free Second-Order Adversarial Examples for Adversarial Learning