Large Norms of CNN Layers Do Not Hurt Adversarial Robustness
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Since the Lipschitz properties of convolutional neural network (CNN) are widely considered to be related to adversarial robustness, we theoretically characterize the ℓ_1 norm and ℓ_∞ norm of 2D multi-channel convolutional layers and provide efficient methods to compute the exact ℓ_1 norm and ℓ_∞ norm. Based on our theorem, we propose a novel regularization method termed norm decay, which can effectively reduce the norms of CNN layers. Experiments show that norm-regularization methods, including norm decay, weight decay, and singular value clipping, can improve generalization of CNNs. However, we are surprised to find that they can slightly hurt adversarial robustness. Furthermore, we compute the norms of layers in the CNNs trained with three different adversarial training frameworks and find that adversarially robust CNNs have comparable or even larger norms than their non-adversarially robust counterparts. Moreover, we prove that under a mild assumption, adversarially robust classifiers can be achieved with neural networks and an adversarially robust neural network can have arbitrarily large Lipschitz constant. For these reasons, enforcing small norms of CNN layers may be neither effective nor necessary in achieving adversarial robustness. Our code is available at https://github.com/youweiliang/norm_robustness.
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