  Neural networks (NNs) are known to be vulnerable against adversarial
perturbations, and thus there is a line of work aiming to provide robustness
certification for NNs, such as randomized smoothing, which samples smoothing
noises from a certain distribution to certify the robustness for a smoothed
classifier. However, as shown by previous work, the certified robust radius in
randomized smoothing suffers from scaling to large datasets ("curse of
dimensionality"). To overcome this hurdle, we propose a Double Sampling
Randomized Smoothing (DSRS) framework, which exploits the sampled probability
from an additional smoothing distribution to tighten the robustness
certification of the previous smoothed classifier. Theoretically, under mild
assumptions, we prove that DSRS can certify $\Theta(\sqrt d)$ robust radius
under $\ell_2$ norm where $d$ is the input dimension, implying that DSRS may be
able to break the curse of dimensionality of randomized smoothing. We
instantiate DSRS for a generalized family of Gaussian smoothing and propose an
efficient and sound computing method based on customized dual optimization
considering sampling error. Extensive experiments on MNIST, CIFAR-10, and
ImageNet verify our theory and show that DSRS certifies larger robust radii
than existing baselines consistently under different settings. Code is
available at https://github.com/llylly/DSRS.
