Why Adversarial Training of ReLU Networks Is Difficult?

Xu Cheng, Hao Zhang, Yue Xin, Wen Shen, Quanshi Zhang

Keywords: Adversarial attack, adversarial training

TL;DR: This paper theoretically analyzes the dynamics of adversarial perturbations, and further theoretically explains the difficulty of adversarial training.

Abstract: This paper mathematically derives an analytic solution of the adversarial perturbation on a ReLU network, and theoretically explains the difficulty of adversarial training. Specifically, we formulate the dynamics of the adversarial perturbation generated by the multi-step attack, which shows that the adversarial perturbation tends to strengthen eigenvectors corresponding to a few top-ranked eigenvalues of the Hessian matrix of the loss w.r.t. the input. We also prove that adversarial training tends to strengthen the influence of unconfident input samples with large gradient norms in an exponential manner. Besides, we find that adversarial training strengthens the influence of the Hessian matrix of the loss w.r.t. network parameters, which makes the adversarial training more likely to oscillate along directions of a few samples, and boosts the difficulty of adversarial training. Crucially, our proofs provide a unified explanation for previous findings in understanding adversarial training.

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