Empirically Measuring Concentration: Fundamental Limits on Intrinsic Robustness
Abstract
Many recent works have shown that adversarial examples that fool classifiers can be found by minimally perturbing a normal input. Recent theoretical results, starting with Gilmer et al. (2018b), show that if the inputs are drawn from a concentrated metric probability space, then adversarial examples with small perturbation are inevitable. A concentrated space has the property that any subset with $\Omega(1)$ (e.g., 1/100) measure, according to the imposed distribution, has small distance to almost all (e.g., 99/100) of the points in the space. It is not clear, however, whether these theoretical results apply to actual distributions such as images. This paper presents a method for empirically measuring and bounding the concentration of a concrete dataset which is proven to converge to the actual concentration. We use it to empirically estimate the intrinsic robustness to $\ell_\infty$ and $\ell_2$ perturbations of several image classification benchmarks. Code for our experiments is available at https://github.com/xiaozhanguva/MeasureConcentration.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.12202
 Bibcode:
 2019arXiv190512202M
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Cryptography and Security;
 Computer Science  Information Theory;
 Statistics  Machine Learning
 EPrint:
 17 pages, 3 figures, 5 tables