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We introduce a challenging training scheme of conditional GANs, called open-set semi-supervised image generation, where the training dataset consists of two parts: (i) labeled data and (ii) unlabeled data with samples belonging to one of the labeled data classes, namely, a closed-set, and samples not belonging to any of the labeled data classes, namely, an open-set. Unlike the existing semi-supervised image generation task, where unlabeled data only contain closed-set samples, our task is more general and lowers the data collection cost in practice by allowing open-set samples to appear. Thanks to entropy regularization, the classifier that is trained on labeled data is able to quantify sample-wise importance to the training of cGAN as confidence, allowing us to use all samples in unlabeled data. We design OSSGAN, which provides decision clues to the discriminator on the basis of whether an unlabeled image belongs to one or none of the classes of interest, smoothly integrating labeled and unlabeled data during training. The results of experiments on Tiny ImageNet and ImageNet show notable improvements over supervised BigGAN and semi-supervised methods. Our code is available at https://github.com/raven38/OSSGAN.