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.

翻译：本文提出异步触发梯度追踪（Asynchronous Triggered Gradient Tracking），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。作为基础模块，我们设计了连续时间梯度追踪（Continuous Gradient Tracking），即近期提出的（离散时间）分布式梯度追踪的连续时间对应物。通过采用李雅普诺夫方法，我们证明了与智能体估计到最优解达成共识的平衡点具有指数稳定性，且该稳定性对局部估计的任意初始化均成立。随后，我们提出了该算法的两种触发版本。第一种版本中，智能体持续积分其局部动力学，并以同步方式与邻居交换当前局部变量。在异步触发梯度追踪中，我们提出一种完全异步的机制：每个智能体基于本地可验证条件触发的条件，向邻居发送其当前局部变量。该触发协议保持了算法的线性收敛性，并避免了芝诺行为（即排除在有限时间区间内出现无限次触发事件的可能性）。利用连续梯度追踪的稳定性分析作为预备结果，我们证明了对两种触发算法及任意估计初始化，平衡点的指数稳定性仍然成立。最后，仿真实验在数据分析问题上验证了所提方法的有效性，并展示了其在智能体间通信性能方面的改进。