Gradient-Push Algorithm for Distributed Optimization With Event-Triggered Communications

Decentralized optimization problems consist of multiple agents connected by a network. The agents have each local cost function, and the goal is to minimize the sum of the functions cooperatively. It requires the agents to communicate with each other, and reducing the cost for communication is desired for a communication-limited environment. Recently, the decentralized gradient descent algorithms involving event-triggered communication have been proposed when the network of the agents is an undirected graph. On the other hand, the network of agents is often directed graph for realistic scenarios whose communication resources are limited. In this work, we first propose a gradient-push algorithm involving event-triggered communication on a directed network. Each agent sends its current states to its neighbors only when the differences between the latest sent states and the current states are larger than thresholds. The convergence of the algorithm is established under suitable decays and summability conditions on a stepsize and triggering thresholds. Numerical experiments are presented to support the effectiveness and the convergence results of the algorithm. More precisely, the numerical results reveals that the proposed algorithm may reduce the communication cost significantly compared to the gradient-push algorithm not involving the event-triggered communication.