This paper proposes Asynchronous Triggered Gradient Tracking, i.e., a distributed optimization algorithm to solve consensus optimization over networks with asynchronous communication. As a building block, we devise the continuous-time counterpart of the recently proposed (discrete-time) distributed gradient tracking called Continuous Gradient Tracking. By using a Lyapunov approach, we prove exponential stability of the equilibrium corresponding to agents’ estimates being consensual to the optimal solution, with arbitrary initialization of the local estimates. Then, we propose two triggered versions of the algorithm. In the first one, the agents continuously integrate their local dynamics and exchange with neighbors their current local variables in a synchronous way. In Asynchronous Triggered Gradient Tracking, we propose a totally asynchronous scheme in which each agent sends to neighbors its current local variables based on a triggering condition that depends on a locally verifiable condition. The triggering protocol preserves the linear convergence of the algorithm and avoids the Zeno behavior, i.e., an infinite number of triggering events over a finite interval of time is excluded. By using the stability analysis of Continuous Gradient Tracking as a preparatory result, we show exponential stability of the equilibrium point holds for both triggered algorithms and any estimate initialization. Finally, the simulations validate the effectiveness of the proposed methods on a data analytics problem, showing also improved performance in terms of inter-agent communication.

本文提出异步触发梯度跟踪，即一种分布式优化算法，用于解决异步通信网络上的共识优化问题。作为构建块，我们设计了最近提出的（离散时间）分布式梯度跟踪的连续时间对应物，称为连续梯度跟踪。通过使用 Lyapunov 方法，我们证明了与代理人的估计一致的均衡的指数稳定性是对最优解的共识，具有局部估计的任意初始化。然后，我们提出了该算法的两个触发版本。在第一个中，代理不断地整合他们的本地动态，并以同步的方式与邻居交换他们当前的本地变量。在Asynchronous Triggered Gradient Tracking，我们提出了一种完全异步的方案，其中每个代理根据取决于本地可验证条件的触发条件向邻居发送其当前局部变量。触发协议保持算法的线性收敛性并避免芝诺行为，即在有限时间间隔内排除无限数量的触发事件。通过使用连续梯度跟踪的稳定性分析作为预备结果，我们展示了平衡点的指数稳定性适用于触发算法和任何估计初始化。最后，仿真验证了所提出的方法在数据分析问题上的有效性，还显示了在代理间通信方面的改进性能。