The Gradient Tracking Is a Distributed Integral Action

We revisit the recent gradient tracking algorithm for distributed consensus optimization from a control theoretic viewpoint. We show that the algorithm can be constructed by solving a servomechanism control problem stemming from the distributed implementation of a centralized gradient method. Moreover, we show that, if expressed in proper coordinates, the gradient tracking embeds an integral action fed by a signal related to the consensus error. Finally, we provide an alternative convergence analysis based on Lyapunov arguments that also proves exponential asymptotic stability of the optimal equilibrium.