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Color constancy aims to restore the constant colors of a scene under different illuminants. However, due to the existence of camera spectral sensitivity, the network trained on a certain sensor, cannot work well on others. Also, since the training datasets are collected in certain environments, the diversity of illuminants is limited for complex real world prediction. In this paper, we tackle these problems via two aspects. First, we propose cross-sensor self-supervised training to train the network. In detail, we consider both the general sRGB images and the white-balanced RAW images from current available datasets as the white-balanced agents. Then, we train the network by randomly sampling the artificial illuminants in a sensor-independent manner for scene relighting and supervision. Second, we analyze a previous cascaded framework and present a more compact and accurate model by sharing the backbone parameters with learning attention specifically. Experiments show that our cross-sensor model and single-sensor model outperform other state-of-the-art methods by a large margin on cross and single sensor evaluations, respectively, with only 16% parameters of the previous best model.