text: 480,136,1227303297880657921,2733642475,Soheil Feizi,"Check out our new work on ""Curse of Dimensionality on Randomized Smoothing"". This is a joint work with @tomgoldsteincs and @umdcs students Aounon Kumar and Alex Levine. Paper: <LINK> We have three main results (see the sub-tweets): <LINK> (1) We show that the robustness radius of almost any i.i.d. smoothing distribution to defend against L_p attacks decreases as 1/(d^(1/2-1/p)). Note that unlike the standard case of L_2, for p&gt;2, this decreases with the input dimension d. <LINK> (2) Using an isotropic Gaussian smoothing as in Cohen @deepcohen , Kolter @zicokolte et al. is almost as good as any other i.i.d. smoothing distributions for p&gt;2 within a constant factor of the robustness radius. The gap becomes tighter by using generalized Gaussian dists <LINK> (3) We show that for smoothing over an L_{\infty} ball, this dependency to the input dimension d becomes worst as 1/d^(1-1/p) <LINK>",https://arxiv.org/abs/2002.03239,"Randomized smoothing, using just a simple isotropic Gaussian distribution, has been shown to produce good robustness guarantees against $\ell_2$-norm bounded adversaries. In this work, we show that extending the smoothing technique to defend against other attack models can be challenging, especially in the high-dimensional regime. In particular, for a vast class of i.i.d.~smoothing distributions, we prove that the largest $\ell_p$-radius that can be certified decreases as $O(1/d^{\frac{1}{2} - \frac{1}{p}})$ with dimension $d$ for $p > 2$. Notably, for $p \geq 2$, this dependence on $d$ is no better than that of the $\ell_p$-radius that can be certified using isotropic Gaussian smoothing, essentially putting a matching lower bound on the robustness radius. When restricted to {\it generalized} Gaussian smoothing, these two bounds can be shown to be within a constant factor of each other in an asymptotic sense, establishing that Gaussian smoothing provides the best possible results, up to a constant factor, when $p \geq 2$. We present experimental results on CIFAR to validate our theory. For other smoothing distributions, 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