  Network-distributed optimization has attracted significant attention in
recent years due to its ever-increasing applications. However, the classic
decentralized gradient descent (DGD) algorithm is communication-inefficient for
large-scale and high-dimensional network-distributed optimization problems. To
address this challenge, many compressed DGD-based algorithms have been
proposed. However, most of the existing works have high complexity and assume
compressors with bounded noise power. To overcome these limitations, in this
paper, we propose a new differential-coded compressed DGD (DC-DGD) algorithm.
The key features of DC-DGD include: i) DC-DGD works with general
SNR-constrained compressors, relaxing the bounded noise power assumption; ii)
The differential-coded design entails the same convergence rate as the original
DGD algorithm; and iii) DC-DGD has the same low-complexity structure as the
original DGD due to a {\em self-noise-reduction effect}. Moreover, the above
features inspire us to develop a hybrid compression scheme that offers a
systematic mechanism to minimize the communication cost. Finally, we conduct
extensive experiments to verify the efficacy of the proposed DC-DGD and hybrid
compressor.
