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Communication is a crucial phase in the context of distributed training. Because parameter server (PS) frequently experiences network congestion, recent studies have found that training paradigms without a centralized server outperform the traditional server-based paradigms in terms of communication efficiency. However, with the increasing growth of model sizes, these server-free paradigms are also confronted with substantial communication overhead that seriously deteriorates the performance of distributed training. In this paper, we focus on communication efficiency of two serverless paradigms, i.e., Ring All-Reduce (RAR) and gossip, by proposing the Quantized Parallel Restarted Stochastic Gradient Descent (QPRSGD), an algorithm that allows multiple local SGD updates before a global synchronization, in synergy with the quantization to significantly reduce the communication overhead. We establish the bound of accumulative errors according to the synchronization mode and the network topology, which is essential to ensure the convergence property. Under both aggregation paradigms, the algorithm achieves the linear speedup property with respect to the number of local updates as well as the number of workers. Remarkably, the proposed algorithm achieves a convergence rate $O(1/\sqrt{NK^2M})$ under the gossip paradigm and outperforms all existing compression methods, where $N$ is the times of global synchronizations, and $K$ is the number of local updates, while $M$ is the number of nodes. An empirical study on various machine learning models demonstrates that the communication overhead is reduced by 90\%, and the convergence speed is boosted by up to 18.6 times, in a low bandwidth network, in comparison with Parallel SGD.