Certified Adversarial Robustness Within Multiple Perturbation Bounds

Randomized smoothing (RS) is a well known certified defense against adversarial attacks, which creates a smoothed classifier by predicting the most likely class under random noise perturbations of inputs during inference. While initial work focused on robustness to $\ell_2$ norm perturbations using noise sampled from a Gaussian distribution, subsequent works have shown that different noise distributions can result in robustness to other $\ell_p$ norm bounds as well. In general, a specific noise distribution is optimal for defending against a given $\ell_p$ norm based attack. In this work, we aim to improve the certified adversarial robustness against multiple perturbation bounds simultaneously. Towards this, we firstly present a novel \textit{certification scheme}, that effectively combines the certificates obtained using different noise distributions to obtain optimal results against multiple perturbation bounds. We further propose a novel \textit{training noise distribution} along with a \textit{regularized training scheme} to improve the certification within both $\ell_1$ and $\ell_2$ perturbation norms simultaneously. Contrary to prior works, we compare the certified robustness of different training algorithms across the same natural (clean) accuracy, rather than across fixed noise levels used for training and certification. We also empirically invalidate the argument that training and certifying the classifier with the same amount of noise gives the best results. The proposed approach achieves improvements on the ACR (Average Certified Radius) metric across both $\ell_1$ and $\ell_2$ perturbation bounds.