Towards Certifying L-infinity Robustness using Neural Networks with L-inf-dist Neurons

It is well-known that standard neural networks, even with a high classification accuracy, are vulnerable to small \(\ell_\infty\)-norm bounded adversarial perturbations. Although many attempts have been made, most previous works either can only provide empirical verification of the defense to a particular attack method, or can only develop a certified guarantee of the model robustness in limited scenarios. In this paper, we seek for a new approach to develop a theoretically principled neural network that inherently resists \(\ell_\infty\) perturbations. In particular, we design a novel neuron that uses \(\ell_\infty\)-distance as its basic operation (which we call \(\ell_\infty\)-dist neuron), and show that any neural network constructed with \(\ell_\infty\)-dist neurons (called \(\ell_{\infty}\)-dist net) is naturally a 1-Lipschitz function with respect to \(\ell_\infty\)-norm. This directly provides a rigorous guarantee of the certified robustness based on the margin of prediction outputs. We also prove that such networks have enough expressive power to approximate any 1-Lipschitz function with robust generalization guarantee. Our experimental results show that the proposed network is promising. Using \(\ell_{\infty}\)-dist nets as the basic building blocks, we consistently achieve state-of-the-art performance on commonly used datasets: 93.09% certified accuracy on MNIST (\(\epsilon=0.3\)), 79.23% on Fashion MNIST (\(\epsilon=0.1\)) and 35.10% on CIFAR-10 (\(\epsilon=8/255\)).