Abstract: Vision transformers (ViTs) have recently received explosive popularity, but their enormous model sizes and training costs remain daunting. Conventional post-training pruning often incurs higher training budgets. In contrast, this paper aims to trim down both the training memory overhead and the inference complexity, without sacrificing the achievable accuracy. We carry out the first-of-its-kind comprehensive exploration, on taking a unified approach of integrating sparsity in ViTs "from end to end''. Specifically, instead of training full ViTs, we dynamically extract and train sparse subnetworks, while sticking to a fixed small parameter budget. Our approach jointly optimizes model parameters and explores connectivity throughout training, ending up with one sparse network as the final output. The approach is seamlessly extended from unstructured to structured sparsity, the latter by considering to guide the prune-and-grow of self-attention heads inside ViTs. We further co-explore data and architecture sparsity for additional efficiency gains by plugging in a novel learnable token selector to adaptively determine the currently most vital patches. Extensive results on ImageNet with diverse ViT backbones validate the effectiveness of our proposals which obtain significantly reduced computational cost and almost unimpaired generalization. Perhaps most surprisingly, we find that the proposed sparse (co-)training can sometimes \textit{improve the ViT accuracy} rather than compromising it, making sparsity a tantalizing "free lunch''. For example, our sparsified DeiT-Small at ($5\%$, $50\%$) sparsity for (data, architecture), improves $\mathbf{0.28\%}$ top-1 accuracy, and meanwhile enjoys $\mathbf{49.32\%}$ FLOPs and $\mathbf{4.40\%}$ running time savings. Our codes are available at https://github.com/VITA-Group/SViTE.