Improved Estimation of Concentration Under $\ell_p$-Norm Distance Metrics Using Half Spaces

TL;DR

This paper introduces a more efficient method using half-spaces to estimate data concentration under $\, ext{l}_p$-norms, providing tighter robustness bounds and challenging the role of dataset concentration in adversarial vulnerability.

Contribution

It extends the Gaussian Isoperimetric Inequality to non-spherical measures and develops a half-space based algorithm for better concentration estimation.

Findings

The new method is more efficient than previous approaches.

It produces tighter bounds on intrinsic robustness.

Results suggest dataset concentration alone does not explain adversarial vulnerability.

Abstract

Concentration of measure has been argued to be the fundamental cause of adversarial vulnerability. Mahloujifar et al. 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.'s, and our 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.