Gradient-Consensus: Linearly Convergent Distributed Optimization Algorithm over Directed Graphs

In this article, we focus on a multi-agent optimization problem of minimizing a sum, $f = \sum_{i=1}^n f_i$, of convex objective functions $f_i$'s, where, $f_i$ is only available locally to the agent $i$, over a network of $n$ agents. The agents communicate only to their neighbors connected through a directed edge governed by a directed graph. In this article, we propose a "optimize then agree" framework to decouple the gradient-descent step and the consensus step in the distributed optimization algorithms. Using this framework we develop a novel distributed algorithm to solve the above multi-agent convex optimization problem. In this method, each agent maintains an estimate of the optimal solution. During each iteration of the proposed algorithm, the agents utilize locally available gradient information along with a finite-time approximate consensus protocol to move towards the optimal solution (hence the name "gradient-consensus" method). We establish that the proposed algorithm converges with a global linear rate if the aggregate function $f$ is strongly convex and smooth. We also show that the proposed method has a linear rate of convergence (in terms of number of iterations), compared to prior state-of-the-art, until reaching an $O(\eta)$ neighborhood of the optimal objective function value, for a given $\eta > 0$, under the relaxed assumption of $f_i$'s being convex and smooth. To the best of our knowledge, the proposed method achieves a convergence rate better than the best known rate estimates existing in the literature under these assumptions. Further, we numerically evaluate the proposed algorithm by solving two distributed optimization problems. The results show that the proposed \textit{GradConsensus} algorithm has a reduced computational footprint compared to the existing distributed optimization schemes for a similar accuracy solution.