Due to the explosion in the size of the training datasets, distributed\nlearning has received growing interest in recent years. One of the major\nbottlenecks is the large communication cost between the central server and the\nlocal workers. While error feedback compression has been proven to be\nsuccessful in reducing communication costs with stochastic gradient descent\n(SGD), there are much fewer attempts in building communication-efficient\nadaptive gradient methods with provable guarantees, which are widely used in\ntraining large-scale machine learning models. In this paper, we propose a new\ncommunication-compressed AMSGrad for distributed nonconvex optimization\nproblem, which is provably efficient. Our proposed distributed learning\nframework features an effective gradient compression strategy and a worker-side\nmodel update design. We prove that the proposed communication-efficient\ndistributed adaptive gradient method converges to the first-order stationary\npoint with the same iteration complexity as uncompressed vanilla AMSGrad in the\nstochastic nonconvex optimization setting. Experiments on various benchmarks\nback up our theory.\n