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We show a hardness result for random smoothing to achieve certifiedadversarial robustness against attacks in the $\ell_p$ ball of radius$\epsilon$ when $p>2$. Although random smoothing has been well understood forthe $\ell_2$ case using the Gaussian distribution, much remains unknownconcerning 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) andincludes many significant paradigms such as the $\ell_\infty$ threat model. Inthis work, we show that any noise distribution $\mathcal{D}$ over$\mathbb{R}^d$ that provides $\ell_p$ robustness for all base classifiers with$p>2$ must satisfy$\mathbb{E}\eta_i^2=\Omega(d^{1-2/p}\epsilon^2(1-\delta)/\delta^2)$ for 99% ofthe features (pixels) of vector $\eta\sim\mathcal{D}$, where $\epsilon$ is therobust radius and $\delta$ is the score gap between the highest-scored classand the runner-up. Therefore, for high-dimensional images with pixel valuesbounded in $[0,255]$, the required noise will eventually dominate the usefulinformation in the images, leading to trivial smoothed classifiers.

Random Smoothing Might be Unable to Certify $\ell_\infty$ Robustness for High-Dimensional Images