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This work aims at reconstructing image from compressed-measured data in the presence of symmetric α-stable (SαS) noise. We first employ the ℓ1-norm as the estimator to depress the influence of SαS noise, and then the ADMM framework is employed to address the resulting optimization problem. Moreover, a smoothing strategy is adopted to address the ℓ1-norm resulting in a nonsmooth optimization problem. To exploit more prior knowledge and image features, a robust composite regularization model is proposed for training by deep neural network (DNN). In the training phase, the DNN can be utilized to train the samples for the optimal parameters, the optimal shrinkage function and the optimal transform domain. Experiments show that our proposed algorithm can obtain higher reconstruction Peak Signal to Noise Ratio (PSNRs) than some existing state-of-the-art robust CS methods.