Title: Learning to Jointly Share and Prune Weights for Grounding Based Vision and Language Models. Abstract: Transformers have seen growing interest in processing different modalities,  including language and image data. As a result, we can process vision and language data using transformers that are architecturally similar. Leveraging this feature of transformers, we propose weight sharing across two transformer backbones and within the same transformer backbone and pruning across two backbones in a unified framework. More specifically, we investigate weight sharing and pruning for two components of the transformers: (1) Multi-Head Attention (MSA) and (2) Feed-Forward Network (FFN) layers. To jointly perform weight sharing and pruning, we propose to use a regularization term to align model weights and the desired structure during the multimodal pre-training step. The structure vectors of sharing and pruning are generated by using a hypernetwork, which can capture complex interactions between pruning and sharing across layers and modalities. We train the hypernetwork and model weights iteratively so that the learned structure evolves along with model weights. After minimizing the proposed objective in the pre-training step, we perform weight sharing and pruning and fine-tune the compressed model on downstream tasks. Finally, we perform experiments on vision and language tasks, including Referring Expression Comprehension (REC), Visual Question Answering (VQA), and Object Detection using the state-of-the-art grounding based models: MDETR and GLIP. Our experiments show that we can compress these models by $35-40\%$ by sharing and pruning MSA and FFN weights without almost any loss in accuracy.