  In this paper, we determine the optimal convergence rates for strongly convex
and smooth distributed optimization in two settings: centralized and
decentralized communications over a network. For centralized (i.e.
master/slave) algorithms, we show that distributing Nesterov's accelerated
gradient descent is optimal and achieves a precision $\varepsilon > 0$ in time
$O(\sqrt{\kappa_g}(1+\Delta\tau)\ln(1/\varepsilon))$, where $\kappa_g$ is the
condition number of the (global) function to optimize, $\Delta$ is the diameter
of the network, and $\tau$ (resp. $1$) is the time needed to communicate values
between two neighbors (resp. perform local computations). For decentralized
algorithms based on gossip, we provide the first optimal algorithm, called the
multi-step dual accelerated (MSDA) method, that achieves a precision
$\varepsilon > 0$ in time
$O(\sqrt{\kappa_l}(1+\frac{\tau}{\sqrt{\gamma}})\ln(1/\varepsilon))$, where
$\kappa_l$ is the condition number of the local functions and $\gamma$ is the
(normalized) eigengap of the gossip matrix used for communication between
nodes. We then verify the efficiency of MSDA against state-of-the-art methods
for two problems: least-squares regression and classification by logistic
regression.
