  In this paper, we introduce a novel algorithm, $\mathsf{CO}_3$, for
communication-efficiency distributed Deep Neural Network (DNN) training.
$\mathsf{CO}_3$ is a joint training/communication protocol, which encompasses
three processing steps for the network gradients: (i) quantization through
floating-point conversion, (ii) lossless compression, and (iii) error
correction. These three components are crucial in the implementation of
distributed DNN training over rate-constrained links. The interplay of these
three steps in processing the DNN gradients is carefully balanced to yield a
robust and high-performance scheme. The performance of the proposed scheme is
investigated through numerical evaluations over CIFAR-10.
