Geometrical Convergence Rate for Distributed Optimization with Time-Varying Directed Graphs and Uncoordinated Step-Sizes

This paper studies a class of distributed optimization algorithms by a set of agents, where each agent has only access to its own local convex objective function, and jointly minimizes the sum of the functions. The communications among agents are described by a sequence of time-varying directed graphs which are assumed to be uniformly strongly connected. A column stochastic mixing matrices is employed in the algorithm, which also exactly steers all the agents to asymptotically converge to a global and consensual optimal solution even under the assumption that the step-sizes are uncoordinated. Two fairly standard conditions for achieving the geometrical convergence rate are established under the assumption that the objective functions are strong convexity and have Lipschitz continuous gradient. The theoretical analysis shows that the distributed algorithm is capable of driving the whole network to geometrically converge to an optimal solution of the convex optimization problem as long as the uncoordinated step-sizes do not exceed some upper bounds. We also give an explicit analysis for the convergence rate of our algorithm through a different approach. Finally, simulation results illustrate the feasibility of the proposed algorithm and the theoretical analysis throughout this paper.