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Randomized smoothing, a method to certify a classifier's decision on an inputis invariant under adversarial noise, offers attractive advantages over othercertification methods. It operates in a black-box and so certification is notconstrained by the size of the classifier's architecture. Here, we extend thework of Li et al. \cite{li2018second}, studying how the choice of divergencebetween smoothing measures affects the final robustness guarantee, and how thechoice of smoothing measure itself can lead to guarantees in differing threatmodels. To this end, we develop a method to certify robustness against any$\ell_p$ ($p\in\mathbb{N}_{>0}$) minimized adversarial perturbation. We thendemonstrate a negative result, that randomized smoothing suffers from the curseof dimensionality; as $p$ increases, the effective radius around an input onecan certify vanishes.

Extensions and limitations of randomized smoothing for robustness guarantees