Communication scheduling has been shown to be effective in accelerating distributed training, which enables all-reduce communications to be overlapped with backpropagation computations. This has been commonly adopted in popular distributed deep learning frameworks. However, there exist two fundamental problems: (1) excessive startup latency proportional to the number of workers for each all-reduce operation; (2) it only achieves sub-optimal training performance due to the dependency and synchronization requirement of the feed-forward computation in the next iteration. We propose a novel scheduling algorithm, DeAR, that decouples the all-reduce primitive into two continuous operations, which overlaps with both backpropagation and feed-forward computations without extra communications. We further design a practical tensor fusion algorithm to improve the training performance. Experimental results with five popular models show that DeAR achieves up to 83% and 15% training speedup over the state-of-the-art solutions on a 64-GPU cluster with 10Gb/s Ethernet and 100Gb/s InfiniBand interconnects, respectively.