Accelerated Distributed Nesterov Gradient Descent

This paper considers the distributed optimization problem over a network, where the objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. We develop an Accelerated Distributed Nesterov Gradient Descent (Acc-DNGD) method. When the objective function is convex and $L$-smooth, we show that it achieves a $O(\frac{1}{t^{1.4-ε}})$ convergence rate for all $ε\in(0,1.4)$. We also show the convergence rate can be improved to $O(\frac{1}{t^2})$ if the objective function is a composition of a linear map and a strongly-convex and smooth function. When the objective function is $μ$-strongly convex and $L$-smooth, we show that it achieves a linear convergence rate of $O([ 1 - C (\fracμ{L})^{5/7} ]^t)$, where $\frac{L}μ$ is the condition number of the objective, and $C>0$ is some constant that does not depend on $\frac{L}μ$.

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