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This paper probes intrinsic factors behind typical failure cases (e.g. spatial inconsistency and boundary confusion) produced by the existing state-of-the-art method in face parsing. To tackle these problems, we propose a novel Decoupled Multi-task Learning with Cyclical Self-Regulation (DML-CSR) for face parsing. Specifically, DML-CSR designs a multi-task model which comprises face parsing, binary edge, and category edge detection. These tasks only share low-level encoder weights without high-level interactions between each other, enabling to decouple auxiliary modules from the whole network at the inference stage. To address spatial inconsistency, we develop a dynamic dual graph convolutional network to capture global contextual information without using any extra pooling operation. To handle boundary confusion in both single and multiple face scenarios, we exploit binary and category edge detection to jointly obtain generic geometric structure and fine-grained semantic clues of human faces. Besides, to prevent noisy labels from degrading model generalization during training, cyclical self-regulation is proposed to self-ensemble several model instances to get a new model and the resulting model then is used to self-distill subsequent models, through alternating iterations. Experiments show that our method achieves the new state-of-the-art performance on the Helen, CelebAMask-HQ, and Lapa datasets. The source code is available at https://github.com/deepinsight/insightface/tree/master/parsing/dml_csr.