Mitigating the Curse of Dimensionality for Certified Robustness via Dual Randomized Smoothing
arxiv(2024)
摘要
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 ℓ_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/√(d). This paper explores the feasibility of providing ℓ_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 ℓ_2 certified robustness radius for the original
input and reveal that DRS attains a superior upper bound on the ℓ_2
robustness radius, which decreases proportionally at a rate of (1/√(m) +
1/√(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 ℓ_2 certified robustness baselines of RS on the
CIFAR-10 and ImageNet datasets. Code is available at
https://github.com/xiasong0501/DRS.
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