  In this paper, we consider minimizing a sum of local convex objective
functions in a distributed setting, where the cost of communication and/or
computation can be expensive. We extend and generalize the analysis for a class
of nested gradient-based distributed algorithms (NEAR-DGD; Berahas,
Bollapragada, Keskar and Wei, 2018) to account for multiple gradient steps at
every iteration. We show the effect of performing multiple gradient steps on
the rate of convergence and on the size of the neighborhood of convergence, and
prove R-Linear convergence to the exact solution with a fixed number of
gradient steps and increasing number of consensus steps. We test the
performance of the generalized method on quadratic functions and show the
effect of multiple consensus and gradient steps in terms of iterations, number
of gradient evaluations, number of communications and cost.
