Gradient-type distributed optimization methods have blossomed into one of the most important tools for solving a minimization learning task over a networked agent system. However, only one gradient update per iteration is difficult to achieve a substantive acceleration of convergence. In this paper, we propose an accelerated framework named as MUSIC allowing each agent to perform multiple local updates and a single combination in each iteration. More importantly, we equip inexact and exact distributed optimization methods into this framework, thereby developing two new algorithms that exhibit accelerated linear convergence and high communication efficiency. Our rigorous convergence analysis reveals the sources of steady-state errors arising from inexact policies and offers effective solutions. Numerical results based on synthetic and real datasets demonstrate both our theoretical motivations and analysis, as well as performance advantages.
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