Classification and Adversarial examples in an Overparameterized Linear Model: A Signal Processing Perspective

TL;DR

This paper investigates the adversarial vulnerability of overparameterized linear models with Fourier features, revealing that spatial localization causes sensitivity near training points, despite good classification performance.

Contribution

It introduces a signal processing perspective to understand adversarial susceptibility in overparameterized models using Fourier features, highlighting the role of spatial localization.

Findings

Models are susceptible to adversarial perturbations near training points.

Classification can be easier than in models with independent features.

Susceptibility occurs even without model mis-specification or label noise.

Abstract

State-of-the-art deep learning classifiers are heavily overparameterized with respect to the amount of training examples and observed to generalize well on "clean" data, but be highly susceptible to infinitesmal adversarial perturbations. In this paper, we identify an overparameterized linear ensemble, that uses the "lifted" Fourier feature map, that demonstrates both of these behaviors. The input is one-dimensional, and the adversary is only allowed to perturb these inputs and not the non-linear features directly. We find that the learned model is susceptible to adversaries in an intermediate regime where classification generalizes but regression does not. Notably, the susceptibility arises despite the absence of model mis-specification or label noise, which are commonly cited reasons for adversarial-susceptibility. These results are extended theoretically to a random-Fourier-sum setup that exhibits double-descent behavior. In both feature-setups, the adversarial vulnerability arises because of a phenomenon we term spatial localization: the predictions of the learned model are markedly more sensitive in the vicinity of training points than elsewhere. This sensitivity is a consequence of feature lifting and is reminiscent of Gibb's and Runge's phenomena from signal processing and functional analysis. Despite the adversarial susceptibility, we find that classification with these features can be easier than the more commonly studied "independent feature" models.