Communication bottleneck has been identified as a significant issue in distributed optimization of large-scale learning models. Recently, several approaches to mitigate this problem have been proposed, including different forms of gradient compression or computing local models and mixing them iteratively. In this paper we propose Qsparse-local-SGD algorithm, which combines aggressive sparsification with quantization and local computation along with error compensation, by keeping track of the difference between the true and compressed gradients. We propose both synchronous and asynchronous implementations of Qsparse-local-SGD. We analyze convergence for Qsparse-local-SGD in the distributed case, for smooth non-convex and convex objective functions. We demonstrate that Qsparse-local-SGD converges at the same rate as vanilla distributed SGD for many important classes of sparsifiers and quantizers. We use Qsparse-local-SGD to train ResNet-50 on ImageNet, and show that it results in significant savings over the state-of-the-art, in the number of bits transmitted to reach target accuracy.