$L_p$-norm Distortion-Efficient Adversarial Attack

Adversarial examples have shown a powerful ability to make a well-trained model misclassified. Current mainstream adversarial attack methods only consider one of the distortions among $L_0$-norm, $L_2$-norm, and $L_\infty$-norm. $L_0$-norm based methods cause large modification on a single pixel, resulting in naked-eye visible detection, while $L_2$-norm and $L_\infty$-norm based methods suffer from weak robustness against adversarial defense since they always diffuse tiny perturbations to all pixels. A more realistic adversarial perturbation should be sparse and imperceptible. In this paper, we propose a novel $L_p$-norm distortion-efficient adversarial attack, which not only owns the least $L_2$-norm loss but also significantly reduces the $L_0$-norm distortion. To this aim, we design a new optimization scheme, which first optimizes an initial adversarial perturbation under $L_2$-norm constraint, and then constructs a dimension unimportance matrix for the initial perturbation. Such a dimension unimportance matrix can indicate the adversarial unimportance of each dimension of the initial perturbation. Furthermore, we introduce a new concept of adversarial threshold for the dimension unimportance matrix. The dimensions of the initial perturbation whose unimportance is higher than the threshold will be all set to zero, greatly decreasing the $L_0$-norm distortion. Experimental results on three benchmark datasets show that under the same query budget, the adversarial examples generated by our method have lower $L_0$-norm and $L_2$-norm distortion than the state-of-the-art. Especially for the MNIST dataset, our attack reduces 8.1$\%$ $L_2$-norm distortion meanwhile remaining 47$\%$ pixels unattacked. This demonstrates the superiority of the proposed method over its competitors in terms of adversarial robustness and visual imperceptibility.