Empirically Measuring Concentration:
Fundamental Limits on Intrinsic Robustness

Saeed Mahloujifar, Xiao Zhang, Mohammad Mahmoody, and David Evans
33rd Conference on Neural Information Processing Systems (NeurIPS)
Vancouver, Canada
December 2019


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. (2018), 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 Ω(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 linf and l2 perturbations of several image classification benchmarks.