The communication bottleneck has been a critical problem in large-scale distributed deep learning. In this work, we study distributed SGD with random block-wise sparsification as the gradient compressor, which is ring-allreduce compatible and highly computation-efficient but leads to inferior performance. To tackle this important issue, we improve the communication-efficient distributed SGD from a novel aspect, that is, the trade-off between the variance and second moment of the gradient. With this motivation, we propose a new detached error feedback (DEF) algorithm, which shows better convergence bound than error feedback for non-convex problems. We also propose DEF-A to accelerate the generalization of DEF at the early stages of the training, which shows better generalization bounds than DEF. Furthermore, we establish the connection between communication-efficient distributed SGD and SGD with iterate averaging (SGD-IA) for the first time. Extensive deep learning experiments show significant empirical improvement of the proposed methods under various settings. Our reproducible codes and scripts for all experiments in this work will be made publicly available.