$\ell_1$ Adversarial Robustness Certificates: a Randomized Smoothing Approach

Jiaye Teng, Guang-He Lee, Yang Yuan

TL;DR: We derive the first tight $\ell_1$ robustness certificate under isotropic Laplace distributions.

Abstract: Robustness is an important property to guarantee the security of machine learning models. It has recently been demonstrated that strong robustness certificates can be obtained on ensemble classifiers generated by input randomization. However, tight robustness certificates are only known for symmetric norms including $\ell_0$ and $\ell_2$, while for asymmetric norms like $\ell_1$, the existing techniques do not apply. By converting the likelihood ratio into a one-dimensional mixed random variable, we derive the first tight $\ell_1$ robustness certificate under isotropic Laplace distributions. Empirically, the deep networks smoothed by Laplace distributions yield the state-of-the-art certified robustness in $\ell_1$ norm on CIFAR-10 and ImageNet.