Mitigating the Curse of Dimensionality for Certified Robustness via Dual Randomized Smoothing
<|reference_start|>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.<|reference_end|>
@article{xia2024mitigating,
    title={Mitigating the Curse of Dimensionality for Certified Robustness via Dual
  Randomized Smoothing},
    author={Song Xia, Yi Yu, Xudong Jiang, Henghui Ding},
    journal={arXiv preprint arXiv:2404.09586},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.09586},
    primaryClass={cs.CV cs.LG}
}