Mitigating the Curse of Dimensionality for Certified Robustness via Dual
  Randomized Smoothing
Song Xia Yu Yi Xudong Jiang Henghui Ding
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.