  We study distributed (strongly convex) optimization problems over a network
of agents, with no centralized nodes. The loss functions of the agents are
assumed to be \textit{similar}, due to statistical data similarity or
otherwise. In order to reduce the number of communications to reach a solution
accuracy, we proposed a {\it preconditioned, accelerated} distributed method.
An $\varepsilon$-solution is achieved in
$\tilde{\mathcal{O}}\big(\sqrt{\frac{\beta/\mu}{1-\rho}}\log1/\varepsilon\big)$
number of communications steps, where $\beta/\mu$ is the relative condition
number between the global and local loss functions, and $\rho$ characterizes
the connectivity of the network. This rate matches (up to poly-log factors)
lower complexity communication bounds of distributed gossip-algorithms applied
to the class of problems of interest. Numerical results show significant
communication savings with respect to existing accelerated distributed schemes,
especially when solving ill-conditioned problems.
