Higher-Order Certification for Randomized Smoothing

Randomized smoothing is a recently proposed defense against adversarial attacks that has achieved SOTA provable robustness against ℓ2 perturbations. A number of publications have extended the guarantees to other metrics, such as ℓ1 or ℓ_∞, by using different smoothing measures. Although the current framework has been shown to yield near-optimal ℓp radii, the total safety region certified by the current framework can be arbitrarily small compared to the optimal. In this work, we propose a framework to improve the certified safety region for these smoothed classifiers without changing the underlying smoothing scheme. The theoretical contributions are as follows: 1) We generalize the certification for randomized smoothing by reformulating certified radius calculation as a nested optimization problem over a class of functions. 2) We provide a method to calculate the certified safety region using 0th-order and 1st-order information for Gaussian-smoothed classifiers. We also provide a framework that generalizes the calculation for certification using higher-order information. 3) We design efficient, high-confidence estimators for the relevant statistics of the first-order information. Combining the theoretical contribution 2) and 3) allows us to certify safety region that are significantly larger than the ones provided by the current methods. On CIFAR10 and Imagenet datasets, the new regions certified by our approach achieve significant improvements on general ℓ1 certified radii and on the ℓ2 certified radii for color-space attacks (ℓ2 restricted to 1 channel) while also achieving smaller improvements on the general ℓ2 certified radii. Our framework can also provide a way to circumvent the current impossibility results on achieving higher magnitude of certified radii without requiring the use of data-dependent smoothing techniques.