  Communication compression techniques are of growing interests for solving the
decentralized optimization problem under limited communication, where the
global objective is to minimize the average of local cost functions over a
multi-agent network using only local computation and peer-to-peer
communication. In this paper, we first propose a novel compressed gradient
tracking algorithm (C-GT) that combines gradient tracking technique with
communication compression. In particular, C-GT is compatible with a general
class of compression operators that unifies both unbiased and biased
compressors. We show that C-GT inherits the advantages of gradient
tracking-based algorithms and achieves linear convergence rate for strongly
convex and smooth objective functions. In the second part of this paper, we
propose an error feedback based compressed gradient tracking algorithm
(EF-C-GT) to further improve the algorithm efficiency for biased compression
operators. Numerical examples complement the theoretical findings and
demonstrate the efficiency and flexibility of the proposed algorithms.
