Randomized Smoothing of All Shapes and Sizes
Greg Yang, Tony Duan, J. Edward Hu, Hadi Salman, Ilya Razenshteyn, Jerry Li
Abstract
Randomized smoothing is the current state-of-the-art defense with provable robustness against $\ell_2$ adversarial attacks. Many works have devised new randomized smoothing schemes for other metrics, such as $\ell_1$ or $\ell_\infty$; however, substantial effort was needed to derive such new guarantees. This begs the question: can we find a general theory for randomized smoothing? We propose a novel framework for devising and analyzing randomized smoothing schemes, and validate its effectiveness in practice. Our theoretical contributions are: (1) we show that for an appropriate notion of "optimal", the optimal smoothing distributions for any "nice" norms have level sets given by the norm’s *Wulff Crystal*; (2) we propose two novel and complementary methods for deriving provably robust radii for any smoothing distribution; and, (3) we show fundamental limits to current randomized smoothing techniques via the theory of *Banach space cotypes*. By combining (1) and (2), we significantly improve the state-of-the-art certified accuracy in $\ell_1$ on standard datasets. Meanwhile, we show using (3) that with only label statistics under random input perturbations, randomized smoothing cannot achieve nontrivial certified accuracy against perturbations of $\ell_p$-norm $\Omega(\min(1, d^{\frac{1}{p} - \frac{1}{2}}))$, when the input dimension $d$ is large. We provide code in github.com/tonyduan/rs4a.
BibTeX
@InProceedings{pmlr-v119-yang20c,
title = {Randomized Smoothing of All Shapes and Sizes},
author = {Yang, Greg and Duan, Tony and Hu, J. Edward and Salman, Hadi and Razenshteyn, Ilya and Li, Jerry},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
pages = {10693--10705},
year = {2020},
editor = {III, Hal Daumé and Singh, Aarti},
volume = {119},
series = {Proceedings of Machine Learning Research},
month = {13--18 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v119/yang20c/yang20c.pdf},
url = {https://proceedings.mlr.press/v119/yang20c.html},
abstract = {Randomized smoothing is the current state-of-the-art defense with provable robustness against $\ell_2$ adversarial attacks. Many works have devised new randomized smoothing schemes for other metrics, such as $\ell_1$ or $\ell_\infty$; however, substantial effort was needed to derive such new guarantees. This begs the question: can we find a general theory for randomized smoothing? We propose a novel framework for devising and analyzing randomized smoothing schemes, and validate its effectiveness in practice. Our theoretical contributions are: (1) we show that for an appropriate notion of "optimal", the optimal smoothing distributions for any "nice" norms have level sets given by the norm’s *Wulff Crystal*; (2) we propose two novel and complementary methods for deriving provably robust radii for any smoothing distribution; and, (3) we show fundamental limits to current randomized smoothing techniques via the theory of *Banach space cotypes*. By combining (1) and (2), we significantly improve the state-of-the-art certified accuracy in $\ell_1$ on standard datasets. Meanwhile, we show using (3) that with only label statistics under random input perturbations, randomized smoothing cannot achieve nontrivial certified accuracy against perturbations of $\ell_p$-norm $\Omega(\min(1, d^{\frac{1}{p} - \frac{1}{2}}))$, when the input dimension $d$ is large. We provide code in github.com/tonyduan/rs4a.}
}