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Deep learning's success has led to larger and larger models to handle more and more complex tasks; trained models can contain millions of parameters. These large models are compute- and memory-intensive, which makes it a challenge to deploy them with minimized latency, throughput, and storage requirements. Some model compression methods have been successfully applied to image classification and detection or language models, but there has been very little work compressing generative adversarial networks (GANs) performing complex tasks. In this paper, we show that a standard model compression technique, weight pruning, cannot be applied to GANs using existing methods. We then develop a self-supervised compression technique which uses the trained discriminator to supervise the training of a compressed generator. We show that this framework has a compelling performance to high degrees of sparsity, can be easily applied to new tasks and models, and enables meaningful comparisons between different pruning granularities.