  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
$O(\frac{1}{t^{1.4-\epsilon}})$ convergence rate for all $\epsilon\in(0,1.4)$.
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 $\mu$-strongly convex and
$L$-smooth, we show that it achieves a linear convergence rate of $O([ 1 - C
(\frac{\mu}{L})^{5/7} ]^t)$, where $\frac{L}{\mu}$ is the condition number of
the objective, and $C>0$ is some constant that does not depend on
$\frac{L}{\mu}$.
