Rajdeep Haldar

Rajdeep Haldar, Qifan Song

Adversarial attacks are usually expressed in terms of a gradient-based operation on the input data and model, this results in heavy computations every time an attack is generated. In this work, we solidify the idea of representing adversarial attacks as a trainable function, without further gradient computation. We first motivate that the theoretical best attacks, under proper conditions, can be represented as smooth piece-wise functions (piece-wise Hölder functions). Then we obtain an approximation result of such functions by a neural network. Subsequently, we emulate the ideal attack process by a neural network and reduce the adversarial training to a mathematical game between an attack network and a training model (a defense network). We also obtain convergence rates of adversarial loss in terms of the sample size $n$ for adversarial training in such a setting.

Rajdeep Haldar, Yue Xing, Qifan Song

The existence of adversarial attacks on machine learning models imperceptible to a human is still quite a mystery from a theoretical perspective. In this work, we introduce two notions of adversarial attacks: natural or on-manifold attacks, which are perceptible by a human/oracle, and unnatural or off-manifold attacks, which are not. We argue that the existence of the off-manifold attacks is a natural consequence of the dimension gap between the intrinsic and ambient dimensions of the data. For 2-layer ReLU networks, we prove that even though the dimension gap does not affect generalization performance on samples drawn from the observed data space, it makes the clean-trained model more vulnerable to adversarial perturbations in the off-manifold direction of the data space. Our main results provide an explicit relationship between the $\ell_2,\ell_{\infty}$ attack strength of the on/off-manifold attack and the dimension gap.