Distributed Heavy-Ball Over Time-Varying Digraphs With Barzilai-Borwein Step Sizes

In this paper, we consider the problem of distributed optimization over time-varying directed graphs, where each agent maintains a private objective function and the goal of all agents is to cooperatively minimize the sum of their objects. By combining Heavy-Ball method with Barzilai-Borwein step sizes, a novel discrete-time accelerated distributed algorithm termed as HBBB is proposed. Compared with existing distributed algorithms over digraphs, HBBB exploits the merit of both momentum and adaptive step sizes for acceleration but requiring very few additional computational costs. It is proved that the algorithm converges to the exact optimal solution at a geometric rate as long as a scale factor for step sizes and the momentum coefficient do not exceed certain bounds. Finally, numerical experiments are performed to show the effectiveness of HBBB as well as its performance, which is comparable to and sometimes even better than existing distributed optimization algorithms over general directed graphs.