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This paper proposes a novel proximal-gradient algorithm for a decentralizedoptimization problem with a composite objective containing smooth andnon-smooth terms. Specifically, the smooth and nonsmooth terms are dealt withby gradient and proximal updates, respectively. The proposed algorithm isclosely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, buthas a few advantages. First of all, agents use uncoordinated step-sizes, andthe stable upper bounds on step-sizes are independent of network topologies.The step-sizes depend on local objective functions, and they can be as large asthose of the gradient descent. Secondly, for the special case withoutnon-smooth terms, linear convergence can be achieved under the strong convexityassumption. The dependence of the convergence rate on the objective functionsand the network are separated, and the convergence rate of the new algorithm isas good as one of the two convergence rates that match the typical rates forthe general gradient descent and the consensus averaging. We provide numericalexperiments to demonstrate the efficacy of the introduced algorithm andvalidate our theoretical discoveries.

A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates