  We study distributed optimization problems when $N$ nodes minimize the sum of
their individual costs subject to a common vector variable. The costs are
convex, have Lipschitz continuous gradient (with constant $L$), and bounded
gradient. We propose two fast distributed gradient algorithms based on the
centralized Nesterov gradient algorithm and establish their convergence rates
in terms of the per-node communications $\mathcal{K}$ and the per-node gradient
evaluations $k$. Our first method, Distributed Nesterov Gradient, achieves
rates $O\left({\log \mathcal{K}}/{\mathcal{K}}\right)$ and $O\left({\log
k}/{k}\right)$. Our second method, Distributed Nesterov gradient with Consensus
iterations, assumes at all nodes knowledge of $L$ and $\mu(W)$ -- the second
largest singular value of the $N \times N$ doubly stochastic weight matrix $W$.
It achieves rates $O\left({1}/{\mathcal{K}^{2-\xi}}\right)$ and
$O\left({1}/{k^2}\right)$ ($\xi>0$ arbitrarily small). Further, we give with
both methods explicit dependence of the convergence constants on $N$ and $W$.
Simulation examples illustrate our findings.
