It is well-known that  classifiers are vulnerable to adversarial perturbations. To defend against adversarial perturbations, various certified robustness results have been derived. However, existing certified robustnesses are limited to top-1 predictions. In many real-world applications, top-$k$ predictions are more relevant. In this work, we aim to derive certified robustness for top-$k$ predictions. In particular, our certified robustness is based on randomized smoothing, which turns any classifier to a new classifier via adding noise to an input example. We adopt randomized smoothing because it is scalable to large-scale neural networks and applicable to any classifier. We derive a tight robustness in $\ell_2$ norm for top-$k$ predictions  when using randomized smoothing with Gaussian noise. We find that generalizing the certified robustness  from top-1 to top-$k$ predictions faces significant technical challenges. We also empirically evaluate our method on CIFAR10 and ImageNet. For example, our method can obtain an ImageNet classifier with a certified top-5 accuracy of 62.8\% when the $\ell_2$-norms of the adversarial perturbations are less than 0.5 (=127/255). Our code is publicly available at: \url{https://github.com/jjy1994/Certify_Topk}.
We study the certified robustness for top-k predictions via randomized smoothing under Gaussian noise and derive a tight robustness bound in L_2 norm. This paper extends work on deducing a certified radius using randomized smoothing, and shows the radius at which a smoothed classifier under Gaussian perturbations is certified for the top k predictions. This paper builds upon the random smoothing technique for top-1 prediction, and aims to provide certification on top-k predictions.