Distributed Stochastic Subgradient Projection Algorithms Based on Weight-Balancing over Time-Varying Directed Graphs

We consider a distributed constrained optimization problem over graphs, where cost function of each agent is private. Moreover, we assume that the graphs are time-varying and directed. In order to address such problem, a fully decentralized stochastic subgradient projection algorithm is proposed over time-varying directed graphs. However, since the graphs are directed, the weight matrix may not be a doubly stochastic matrix. Therefore, we overcome this difficulty by using weight-balancing technique. By choosing appropriate step-sizes, we show that iterations of all agents asymptotically converge to some optimal solutions. Further, by our analysis, convergence rate of our proposed algorithm is O(ln Γ/Γ) under local strong convexity, where Γ is the number of iterations. In addition, under local convexity, we prove that our proposed algorithm can converge with rate O(ln Γ/Γ). In addition, we verify the theoretical results through simulations.