Randomized smoothing is currently the state-of-the-art method that provides certified robustness for deep neural networks. However, due to its excessively conservative nature, this method of incomplete verification often cannot achieve an adequate certified radius on real-world datasets. One way to obtain a larger certified radius is to use an input-specific algorithm instead of using a fixed Gaussian filter for all data points. Several methods based on this idea have been proposed, but they either suffer from high computational costs or gain marginal improvement in certified radius. In this work, we show that by exploiting the quasiconvex problem structure, we can find the optimal certified radii for most data points with slight computational overhead. This observation leads to an efficient and effective input-specific randomized smoothing algorithm. We conduct extensive experiments and empirical analysis on CIFAR-10 and ImageNet. The results show that the proposed method significantly enhances the certified radii with low computational overhead.

Adversarially robust classification seeks a classifier that is insensitive to adversarial perturbations of test patterns. This problem is often formulated via a minimax objective, where the target loss is the worst-case value of the 0-1 loss subject to a bound on the size of perturbation. Recent work has proposed convex surrogates for the adversarial 0-1 loss, in an effort to make optimization more tractable. In this work, we consider the question of which surrogate losses are calibrated with respect to the adversarial 0-1 loss, meaning that minimization of the former implies minimization of the latter. We show that no convex surrogate loss is calibrated with respect to the adversarial 0-1 loss when restricted to the class of linear models. We further introduce a class of nonconvex losses and offer necessary and sufficient conditions for losses in this class to be calibrated.