  Decentralized algorithms solve multi-agent problems over a connected network,
where the information can only be exchanged with the accessible neighbors.
Though there exist several decentralized optimization algorithms, there are
still gaps in convergence conditions and rates between decentralized and
centralized algorithms. In this paper, we fill some gaps by considering two
decentralized algorithms: EXTRA and NIDS. They both converge linearly with
strongly convex objective functions. We will answer two questions regarding
them. What are the optimal upper bounds for their stepsizes? Do decentralized
algorithms require more properties on the functions for linear convergence than
centralized ones? More specifically, we relax the required conditions for
linear convergence of both algorithms. For EXTRA, we show that the stepsize is
comparable to that of centralized algorithms. For NIDS, the upper bound of the
stepsize is shown to be exactly the same as the centralized ones. In addition,
we relax the requirement for the objective functions and the mixing matrices.
We provide the linear convergence results for both algorithms under the weakest
conditions.
