Defenses against adversarial examples, such as adversarial training, are typically tailored to a single perturbation type (e.g., small $\ell_\infty$-noise). For other perturbations, these defenses offer no guarantees and, at times, even increase the model's vulnerability. Our aim is to understand the reasons underlying this robustness trade-off, and to train models that are simultaneously robust to multiple perturbation types. We prove that a trade-off in robustness to different types of $\ell_p$-bounded and spatial perturbations must exist in a natural and simple statistical setting. We corroborate our formal analysis by demonstrating similar robustness trade-offs on MNIST and CIFAR10. Building upon new multi-perturbation adversarial training schemes, and a novel efficient attack for finding $\ell_1$-bounded adversarial examples, we show that no model trained against multiple attacks achieves robustness competitive with that of models trained on each attack individually. In particular, we uncover a pernicious gradient-masking phenomenon on MNIST, which causes adversarial training with first-order $\ell_\infty, \ell_1$ and $\ell_2$ adversaries to achieve merely $50\%$ accuracy. Our results question the viability and computational scalability of extending adversarial robustness, and adversarial training, to multiple perturbation types.

针对对抗样本的防御，如对抗训练，通常针对单个干扰类型（例如小的l∞-噪声），对于其它类型的干扰，这些防御没有保障，甚至会增加模型的脆弱性。我们的目标是了解这种鲁棒性取舍背后的原因，并训练同时对多种扰动类型具有鲁棒性的模型。

多扰动下的对抗训练和鲁棒性