Classification and Adversarial examples in an Overparameterized Linear Model: A Signal Processing Perspective
This work addresses the problem of adversarial vulnerability in machine learning models for researchers, showing it can occur even without model mis-specification or label noise, though it is incremental as it focuses on a specific linear model setup.
The paper investigates why overparameterized models generalize well on clean data but are vulnerable to adversarial perturbations, identifying that adversarial susceptibility arises in an intermediate regime where classification generalizes but regression does not, due to spatial localization effects from feature lifting.
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.