Abstract: Deep neural networks (DNNs) easily cause security issues due to the lack of adversarial robustness. An emerging research topic for this problem is to design adversarially robust architectures via neural architecture search (NAS), i.e., robust NAS. However, robust NAS needs to train numerous DNNs for robustness estimation, making the search process prohibitively expensive. In this paper, we propose a zero-cost proxy to evaluate the adversarial robustness without training. Specifically, the proposed zero-cost proxy formulates the upper bound of adversarial loss, which can directly reflect the adversarial robustness. The formulation involves only the initialized weights of DNNs, thus the training process is no longer needed. Moreover, we theoretically justify the validity of the proposed proxy based on the theory of neural tangent kernel and input loss landscape. Experimental results show that the proposed zero-cost proxy can bring more than $20\times$ speedup compared with the state-of-the-art robust NAS methods, while the searched architecture has superior robustness and transferability under white-box and black-box attacks. Furthermore, compared with the state-of-the-art zero-cost proxies, the calculation of the proposed method has the strongest correlation with adversarial robustness. Our source code is available at https://github.com/fyqsama/Robust_ZCP.