Papers › Improved Estimation of Concentration Under ℓₚ-Norm Distance Metrics Using Half Spaces

Improved Estimation of Concentration Under ℓₚ-Norm Distance Metrics Using Half Spaces

24 Mar 2021ICLR 2021 1arXiv:2103.12913archive 2025-07-28

Jack Prescott, Xiao Zhang, David Evans

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 ℓₚ-norms (p ≥2). We leverage these theoretical insights to design a method that uses half-spaces to estimate the concentration of any empirical dataset under ℓₚ-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.

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