DEXTRA: A Fast Algorithm for Optimization Over Directed Graphs

Abstract

This paper develops a fast distributed algorithm, termed DEXTRA, to solve the optimization problem when n agents reach agreement and collaboratively minimize the sum of their local objective functions over the network, where the communication between the agents is described by a directed graph. Existing algorithms solve the problem restricted to directed graphs with convergence √ rates of O(ln k/ √k) for general convex objective functions and O(ln k/k) when the objective functions are strongly convex, where k is the number of iterations. We show that, with the appropriate step-size, DEXTRA converges at a linear rate O(τ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ) for 0 <; τ <; 1, given that the objective functions are restricted strongly convex. The implementation of DEXTRA requires each agent to know its local out-degree. Simulation examples further illustrate our findings.