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Many existing face anti-spoofing (FAS) methods focus on modeling the decision boundaries for some predefined spoof types. However, the diversity of the spoof samples including the unknown ones hinders the effective decision boundary modeling and leads to weak generalization capability. In this paper, we reformulate FAS in an anomaly detection perspective and propose a residual-learning framework to learn the discriminative live-spoof differences which are defined as the spoof cues. The proposed framework consists of a spoof cue generator and an auxiliary classifier. The generator minimizes the spoof cues of live samples while imposes no explicit constraint on those of spoof samples to generalize well to unseen attacks. In this way, anomaly detection is implicitly used to guide spoof cue generation, leading to discriminative feature learning. The auxiliary classifier serves as a spoof cue amplifier and makes the spoof cues more discriminative. We conduct extensive experiments and the experimental results show the proposed method consistently outperforms the state-of-the-art methods. The code will be publicly available at https://github.com/vis-var/lgsc-for-fas.