  Recently, there has been significant progress in the development of
distributed first order methods. (At least) two different types of methods,
designed from very different perspectives, have been proposed that achieve both
exact and linear convergence when a constant step size is used -- a favorable
feature that was not achievable by most prior methods. In this paper, we unify,
generalize, and improve convergence speed of these exact distributed first
order methods. We first carry out a novel unifying analysis that sheds light on
how the different existing methods compare. The analysis reveals that a major
difference between the methods is on how a past dual gradient of an associated
augmented Lagrangian dual function is weighted. We then capitalize on the
insights from the analysis to derive a novel method -- with a tuned past
gradient weighting -- that improves upon the existing methods. We establish for
the proposed generalized method global R-linear convergence rate under strongly
convex costs with Lipschitz continuous gradients.
