  Randomized smoothing is the current state-of-the-art defense with provable
robustness against $\ell_2$ adversarial attacks. Many works have devised new
randomized smoothing schemes for other metrics, such as $\ell_1$ or
$\ell_\infty$; however, substantial effort was needed to derive such new
guarantees. This begs the question: can we find a general theory for randomized
smoothing?
  We propose a novel framework for devising and analyzing randomized smoothing
schemes, and validate its effectiveness in practice. Our theoretical
contributions are: (1) we show that for an appropriate notion of "optimal", the
optimal smoothing distributions for any "nice" norms have level sets given by
the norm's *Wulff Crystal*; (2) we propose two novel and complementary methods
for deriving provably robust radii for any smoothing distribution; and, (3) we
show fundamental limits to current randomized smoothing techniques via the
theory of *Banach space cotypes*. By combining (1) and (2), we significantly
improve the state-of-the-art certified accuracy in $\ell_1$ on standard
datasets. Meanwhile, we show using (3) that with only label statistics under
random input perturbations, randomized smoothing cannot achieve nontrivial
certified accuracy against perturbations of $\ell_p$-norm $\Omega(\min(1,
d^{\frac{1}{p} - \frac{1}{2}}))$, when the input dimension $d$ is large. We
provide code in github.com/tonyduan/rs4a.
