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Randomized smoothing, using just a simple isotropic Gaussian distribution,has been shown to produce good robustness guarantees against $\ell_2$-normbounded adversaries. In this work, we show that extending the smoothingtechnique to defend against other attack models can be challenging, especiallyin the high-dimensional regime. In particular, for a vast class ofi.i.d.~smoothing distributions, we prove that the largest $\ell_p$-radius thatcan be certified decreases as $O(1/d^{\frac{1}{2} - \frac{1}{p}})$ withdimension $d$ for $p > 2$. Notably, for $p \geq 2$, this dependence on $d$ isno better than that of the $\ell_p$-radius that can be certified usingisotropic Gaussian smoothing, essentially putting a matching lower bound on therobustness radius. When restricted to {\it generalized} Gaussian smoothing,these two bounds can be shown to be within a constant factor of each other inan asymptotic sense, establishing that Gaussian smoothing provides the bestpossible results, up to a constant factor, when $p \geq 2$. We presentexperimental results on CIFAR to validate our theory. For other smoothingdistributions, such as, a uniform distribution within an $\ell_1$ or an$\ell_\infty$-norm ball, we show upper bounds of the form $O(1 / d)$ and $O(1 /d^{1 - \frac{1}{p}})$ respectively, which have an even worse dependence on $d$.

Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness