Distributed Smooth and Strongly Convex Optimization with Inexact Dual Methods

In this paper, we consider a class of decentralized convex optimization problems in which a network of agents aims to minimize a global objective function that is a sum of private (local) smooth and strongly convex objectives. More specifically, we study decentralized inexact dual ascent method, in which the agents only approximately solve their private minimization and then update their dual variables using inexact dual ascent. We study the effect of inexact inner minimization on the convergence rate. In particular, we show that the overall convergence rate will not be affected by inexact minimization if the errors are decreased at an appropriate rate. We illustrate our findings in a distributed binary classification problem.