Title: Improving Adversarial Robustness via Frequency Regularization. Abstract: Deep neural networks (DNNs) are incredibly vulnerable to crafted, human-imperceptible adversarial perturbations. While adversarial training (AT) has proven to be an effective defense approach, the properties of AT for robustness improvement remain an open issue. In this paper, we investigate AT from a spectral perspective, providing new insights into the design of effective defenses. Our analyses show that AT induces the deep model to focus more on the low-frequency region, which retains the shape-biased representations, to gain robustness. Further, we find that the spectrum of a white-box attack is primarily distributed in regions the model focuses on, and the perturbation attacks the spectral bands where the model is vulnerable. To train a model tolerant to frequency-varying perturbation, we propose a frequency regularization (FR) such that the spectral output inferred by an attacked input stays as close as possible to its natural input counterpart. Experiments demonstrate that FR and its weight averaging (WA) extension could significantly improve the robust accuracy by 1.14% ~ 4.57%, across multiple datasets (SVHN, CIFAR-10, CIFAR-100, and Tiny ImageNet), and various attacks (PGD, C&W, and Autoattack), without any extra data.