The Curse of Concentration in Robust Learning: Evasion and Poisoning Attacks from Concentration of Measure

TL;DR

This paper connects the vulnerability of classifiers to adversarial attacks with the mathematical phenomenon of measure concentration, showing that high-dimensional, concentrated spaces inherently lead to adversarial examples and poisoning vulnerabilities.

Contribution

It provides a theoretical framework linking measure concentration to adversarial and poisoning attacks, extending existing results and introducing new attack bounds for various distributions.

Findings

Adversarial vulnerability is inherent in concentrated metric spaces.

New attack bounds of O(√n) for Levy families and product distributions.

Poisoning attacks can significantly increase failure probability with few data modifications.

Abstract

Many modern machine learning classifiers are shown to be vulnerable to adversarial perturbations of the instances. Despite a massive amount of work focusing on making classifiers robust, the task seems quite challenging. In this work, through a theoretical study, we investigate the adversarial risk and robustness of classifiers and draw a connection to the well-known phenomenon of concentration of measure in metric measure spaces. We show that if the metric probability space of the test instance is concentrated, any classifier with some initial constant error is inherently vulnerable to adversarial perturbations. One class of concentrated metric probability spaces are the so-called Levy families that include many natural distributions. In this special case, our attacks only need to perturb the test instance by at most $O(\sqrt n)$ to make it misclassified, where $n$ is the data dimension. Using our general result about Levy instance spaces, we first recover as special case some of the previously proved results about the existence of adversarial examples. However, many more Levy families are known (e.g., product distribution under the Hamming distance) for which we immediately obtain new attacks that find adversarial examples of distance $O(\sqrt n)$. Finally, we show that concentration of measure for product spaces implies the existence of forms of "poisoning" attacks in which the adversary tampers with the training data with the goal of degrading the classifier. In particular, we show that for any learning algorithm that uses $m$ training examples, there is an adversary who can increase the probability of any "bad property" (e.g., failing on a particular test instance) that initially happens with non-negligible probability to $\approx 1$ by substituting only $\tilde{O}(\sqrt m)$ of the examples with other (still correctly labeled) examples.