Go to ICLR 2025 Conference homepageAre Probabilistic Robust Accuracy Bounded
Adversarial samples pose a security threat to many critical systems built on neural networks. It has recently been proven that achieving deterministic robustness (i.e., complete elimination of adversarial samples) always comes at an unbearable cost to accuracy. As a result, probabilistic robustness (where the probability of retaining the same label within a vicinity is at least $1 - \kappa$) has been proposed as a promising compromise. However, existing training methods for probabilistic robustness still experience non-trivial accuracy loss. It remains an open question what the upper limit on accuracy is when optimizing for probabilistic robustness, and whether there is a specific relationship between $\kappa$ and this potential bound. This work studies these problems from a Bayes error perspective. We find that while Bayes uncertainty does affect probabilistic robustness, its impact is smaller than that on deterministic robustness. This reduced Bayes uncertainty allows a higher upper bound on probabilistic robust accuracy than that on deterministic robust accuracy. Further, we show that voting within the vicinity always improves probabilistic robust accuracy and the upper bound of probabilistic robust accuracy monotonically increases as $\kappa$ grows. Our empirical findings also align with our results. This study thus presents a theoretical argument supporting probabilistic robustness as the appropriate target for achieving neural network robustness.