Mitigating the Curse of Dimensionality for Certified Robustness via Dual Randomized Smoothing

Randomized Smoothing (RS) has been proven a promising method for endowing an arbitrary image classifier with certified robustness. However, the substantial uncertainty inherent in the high-dimensional isotropic Gaussian noise imposes the curse of dimensionality on RS. Specifically, the upper bound of {\ell_2} certified robustness radius provided by RS exhibits a diminishing trend with the expansion of the input dimension d, proportionally decreasing at a rate of 1/\sqrt{d}. This paper explores the feasibility of providing {\ell_2} certified robustness for high-dimensional input through the utilization of dual smoothing in the lower-dimensional space. The proposed Dual Randomized Smoothing (DRS) down-samples the input image into two sub-images and smooths the two sub-images in lower dimensions. Theoretically, we prove that DRS guarantees a tight {\ell_2} certified robustness radius for the original input and reveal that DRS attains a superior upper bound on the {\ell_2} robustness radius, which decreases proportionally at a rate of (1/\sqrt m + 1/\sqrt n ) with m+n=d. Extensive experiments demonstrate the generalizability and effectiveness of DRS, which exhibits a notable capability to integrate with established methodologies, yielding substantial improvements in both accuracy and {\ell_2} certified robustness baselines of RS on the CIFAR-10 and ImageNet datasets. Code is available at https://github.com/xiasong0501/DRS.