Tight Second-Order Certificates for Randomized Smoothing

Alexander Levine, Aounon Kumar, Tom Goldstein, Soheil Feizi

Keywords: certificates, adversarial, robustness, defenses, smoothing, curvature

Abstract: Randomized smoothing is a popular way of providing robustness guarantees against adversarial attacks: randomly-smoothed functions have a universal Lipschitz-like bound, allowing for robustness certificates to be easily computed. In this work, we show that there also exists a universal curvature-like bound for Gaussian random smoothing: given the exact value and gradient of a smoothed function, we compute a lower bound on the distance of a point to its closest adversarial example, called the Second-order Smoothing (SoS) robustness certificate. In addition to proving the correctness of this novel certificate, we show that SoS certificates are realizable and therefore tight. Interestingly, we show that the maximum achievable benefits, in terms of certified robustness, from using the additional information of the gradient norm are relatively small: because our bounds are tight, this is a fundamental negative result. The gain of SoS certificates further diminishes if we consider the estimation error of the gradient norms, for which we have developed an estimator. We therefore additionally develop a variant of Gaussian smoothing, called Gaussian dipole smoothing, which provides similar bounds to randomized smoothing with gradient information, but with much-improved sample efficiency. This allows us to achieve (marginally) improved robustness certificates on high-dimensional datasets such as CIFAR-10 and ImageNet.

One-sentence Summary: We provide a tight robustness certificate for Gaussian smoothed classifiers using the gradient of the smoothed classifier in addition to its value.

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2010.10549/code)

Reviewed Version (pdf): https://openreview.net/references/pdf?id=WBA-GsC4w9