The past few years have witnessed the flourishing of large-scale deep neural network models with ever-growing parameter numbers. Training such large-scale models typically requires massive memory and computing resources that exceed those of a single GPU, necessitating distributed training. As GPU performance has rapidly evolved in recent years, computation time has shrunk, thereby increasing the proportion of communication in the overall training time. Therefore, optimizing communication for distributed training has become an urgent issue. In this article, we briefly introduce the general architecture of distributed deep neural network training and analyze relationships among Parallelization Strategy, Collective Communication Library, and Network from the perspective of communication optimization, which forms a three-layer paradigm. We then review current representative research advances with this three-layer paradigm. We find that layers in the current three-layer paradigm are relatively independent, but there is a rich design space for cross-layer collaborative optimization in distributed training scenarios. Therefore, we further advocate a communication-efficient five-layer paradigm underlining opportunities for collaboration designs and look forward to the perspectives of "Vertical", "Horizontal", "Intra-Inter" and "Host-Net" collaboration designs. We hope this article can shed some light on future research on communication optimization for distributed training.