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

Adversarial Training and Robustness for Multiple Perturbations