Keywords: Adversarial Examples, Concentration of Measure, Gaussian Isoperimetric Inequality
Abstract: Concentration of measure has been argued to be the fundamental cause of adversarial vulnerability. Mahloujifar et al. (2019) presented an empirical way to measure the concentration of a data distribution using samples, and employed it to find lower bounds on intrinsic robustness for several benchmark datasets. However, it remains unclear whether these lower bounds are tight enough to provide a useful approximation for the intrinsic robustness of a dataset. To gain a deeper understanding of the concentration of measure phenomenon, we first extend the Gaussian Isoperimetric Inequality to non-spherical Gaussian measures and arbitrary $\ell_p$-norms ($p \geq 2$). We leverage these theoretical insights to design a method that uses half-spaces to estimate the concentration of any empirical dataset under $\ell_p$-norm distance metrics. Our proposed algorithm is more efficient than Mahloujifar et al. (2019)'s, and experiments on synthetic datasets and image benchmarks demonstrate that it is able to find much tighter intrinsic robustness bounds. These tighter estimates provide further evidence that rules out intrinsic dataset concentration as a possible explanation for the adversarial vulnerability of state-of-the-art classifiers.
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics
One-sentence Summary: We show that concentration of measure does not prohibit the existence of adversarially robust classifiers using a novel method of empirical concentration estimation.
Supplementary Material: zip
Code: [![github](/images/github_icon.svg) jackbprescott/EMC_HalfSpaces](https://github.com/jackbprescott/EMC_HalfSpaces)