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The goal of unpaired image captioning (UIC) is to describe images without using image-caption pairs in the training phase. Although challenging, we except the task can be accomplished by leveraging a training set of images aligned with visual concepts. Most existing studies use off-the-shelf algorithms to obtain the visual concepts because the Bounding Box (BBox) labels or relationship-triplet labels used for the training are expensive to acquire. In order to resolve the problem in expensive annotations, we propose a novel approach to achieve cost-effective UIC. Specifically, we adopt image-level labels for the optimization of the UIC model in a weakly-supervised manner. For each image, we assume that only the image-level labels are available without specific locations and numbers. The image-level labels are utilized to train a weakly-supervised object recognition model to extract object information (e.g., instance) in an image, and the extracted instances are adopted to infer the relationships among different objects based on an enhanced graph neural network (GNN). The proposed approach achieves comparable or even better performance compared with previous methods without the expensive cost of annotations. Furthermore, we design an unrecognized object (UnO) loss combined with a visual concept reward to improve the alignment of the inferred object and relationship information with the images. It can effectively alleviate the issue encountered by existing UIC models about generating sentences with nonexistent objects. To the best of our knowledge, this is the first attempt to solve the problem of Weakly-Supervised visual concept recognition for UIC (WS-UIC) based only on image-level labels. Extensive experiments have been carried out to demonstrate that the proposed WS-UIC model achieves inspiring results on the COCO dataset while significantly reducing the cost of labeling.