Average Certified Radius is a Poor Metric for Randomized Smoothing

Abstract

Randomized smoothing (RS) is popular for providing certified robustness guarantees against adversarial attacks. The average certified radius (ACR) has emerged as a widely used metric for tracking progress in RS. However, in this work, for the first time we show that ACR is a poor metric for evaluating robustness guarantees provided by RS. We theoretically prove not only that a trivial classifier can have arbitrarily large ACR, but also that ACR is extremely sensitive to improvements on easy samples. In addition, the comparison using ACR has a strong dependence on the certification budget. Empirically, we confirm that existing training strategies, though improving ACR, reduce the model's robustness on hard samples consistently. To strengthen our findings, we propose strategies, including explicitly discarding hard samples, reweighing the dataset with approximate certified radius, and extreme optimization for easy samples, to replicate the progress in RS training and even achieve the state-of-the-art ACR on CIFAR-10, without training for robustness on the full data distribution. Overall, our results suggest that ACR has introduced a strong undesired bias to the field, and its application should be discontinued in RS. Finally, we suggest using the empirical distribution of $p_A$, the accuracy of the base model on noisy data, as an alternative metric for RS.