Title: A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates,
Abstract: This paper considers the problem of decentralized optimization with a
composite objective containing smooth and non-smooth terms. To solve the
problem, a proximal-gradient scheme is studied. Specifically, the smooth and
nonsmooth terms are dealt with by gradient update and proximal update,
respectively. The studied algorithm is closely related to a previous
decentralized optimization algorithm, PG-EXTRA [37], but has a few advantages.
First of all, in our new scheme, agents use uncoordinated step-sizes and the
stable upper bounds on step-sizes are independent from network topology. The
step-sizes depend on local objective functions, and they can be as large as
that 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 our new scheme 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 also provide some
numerical experiments to demonstrate the efficacy of the introduced algorithms
and validate our theoretical discoveries.