Linear Convergent Decentralized Optimization with Compression

Communication compression has been extensively adopted to speed up large-scale distributed optimization. However, most existing decentralized algorithms with compression are unsatisfactory in terms of convergence rate and stability. In this paper, we delineate two key obstacles in the algorithm design -- data heterogeneity and compression error. Our attempt to explicitly overcome these obstacles leads to a novel decentralized algorithm named LEAD. This algorithm is the first \underline{L}in\underline{EA}r convergent \underline{D}ecentralized algorithm with communication compression. Our theory describes the coupled dynamics of the inaccurate model propagation and optimization process. We also provide the first consensus error bound without assuming bounded gradients. Empirical experiments validate our theoretical analysis and show that the proposed algorithm achieves state-of-the-art computation and communication efficiency.