  This paper proposes a novel proximal-gradient algorithm for a decentralized
optimization problem with a composite objective containing smooth and
non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with
by gradient and proximal updates, respectively. The proposed algorithm is
closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but
has a few advantages. First of all, agents use uncoordinated step-sizes, and
the stable upper bounds on step-sizes are independent of network topologies.
The step-sizes depend on local objective functions, and they can be as large as
those of the gradient descent. Secondly, for the special case without
non-smooth terms, linear convergence can be achieved under the strong convexity
assumption. The dependence of the convergence rate on the objective functions
and the network are separated, and the convergence rate of the new algorithm is
as good as one of the two convergence rates that match the typical rates for
the general gradient descent and the consensus averaging. We provide numerical
experiments to demonstrate the efficacy of the introduced algorithm and
validate our theoretical discoveries.
