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.
翻译:在培训实例数量方面,最先进的深层次学习分类方法被严重地过分地夸大了培训实例的数量,并被观察对“清洁”数据进行概括化,但很容易受到无限的对抗性扰动。在本文中,我们确定了一个过于分解的线性共性,使用“提升”的Fourier地貌图,以显示这两种行为。输入是一维的,而对手只能对这些投入进行扰动,而不是直接的非线性特征。我们发现,在分类一般但回归不明显的中间系统中,学习的模型很容易被对手所利用。值得注意的是,尽管没有模型的错误区分或标签的噪音,但这种易感性还是出现,而这正是常被引证的对抗性共同感知性。这些结果在理论上被扩展为随机的Fourier地貌组合,显示的是两种行为。在这两个特征设置中,对抗性脆弱性的产生是因为一种现象,我们称之为空间本地化:所学模型的预测在培训点附近比其他地方更敏感。这种敏感度是,尽管没有模型的模型缺乏模型的模型,但这种易感,但这种易感应变的特性是常规性分析的结果。