Papers › Random Smoothing Might be Unable to Certify ℓ_∞ Robustness for High-Dimensional Images
Random Smoothing Might be Unable to Certify ℓ_∞ Robustness for High-Dimensional Images
Avrim Blum, Travis Dick, Naren Manoj, Hongyang Zhang
We show a hardness result for random smoothing to achieve certified adversarial robustness against attacks in the ℓₚ ball of radius ϵ when p>2. Although random smoothing has been well understood for the ℓ₂ case using the Gaussian distribution, much remains unknown concerning the existence of a noise distribution that works for the case of p>2. This has been posed as an open problem by Cohen et al. (2019) and includes many significant paradigms such as the ℓ_∞ threat model. In this work, we show that any noise distribution 𝒟 over ℝᵈ that provides ℓₚ robustness for all base classifiers with p>2 must satisfy 𝔼ηᵢ²=Ω(d^(1-2/p)ϵ²(1-δ)/δ²) for 99% of the features (pixels) of vector η∼𝒟, where ϵ is the robust radius and δ is the score gap between the highest-scored class and the runner-up. Therefore, for high-dimensional images with pixel values bounded in [0,255], the required noise will eventually dominate the useful information in the images, leading to trivial smoothed classifiers.
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