  We study distributed composite optimization over networks: agents minimize
the sum of a smooth (strongly) convex function, the agents' sum-utility, plus a
non-smooth (extended-valued) convex one. We propose a general algorithmic
framework for such a class of problems and provide a unified convergence
analysis leveraging the theory of operator splitting. Our results unify several
approaches proposed in the literature of distributed optimization for special
instances of our formulation. Distinguishing features of our scheme are: (i)
when the agents' functions are strongly convex, the algorithm converges at a
linear rate, whose dependencies on the agents' functions and the network
topology are decoupled, matching the typical rates of centralized optimization;
(ii) the step-size does not depend on the network parameters but only on the
optimization ones; and (iii) the algorithm can adjust the ratio between the
number of communications and computations to achieve the same rate of the
centralized proximal gradient scheme (in terms of computations). This is the
first time that a distributed algorithm applicable to composite optimization
enjoys such properties.
