We consider a discrete-time model of continuous-time distributed optimization over dynamic directed-graphs (digraphs) with applications to distributed learning. Our optimization algorithm works over general strongly connected dynamic networks under switching topologies, e.g., in mobile multi-agent systems and volatile networks due to link failures. Compared to many existing lines of work, there is no need for bi-stochastic weight designs on the links. The existing literature mostly needs the link weights to be stochastic using specific weight-design algorithms needed both at the initialization and at all times when the topology of the network changes. This paper eliminates the need for such algorithms and paves the way for distributed optimization over time-varying digraphs. We derive the bound on the gradient-tracking step-size and discrete time-step for convergence and prove dynamic stability using arguments from consensus algorithms, matrix perturbation theory, and Lyapunov theory. This work, particularly, is an improvement over existing stochastic-weight undirected networks in case of link removal or packet drops. This is because the existing literature may need to rerun time-consuming and computationally complex algorithms for stochastic design, while the proposed strategy works as long as the underlying network is weight-symmetric and balanced. The proposed optimization framework finds applications to distributed classification and learning.

翻译：我们考虑了一种基于动态有向图的连续时间分布式优化的离散时间模型，并将其应用于分布式学习。我们的优化算法适用于切换拓扑下的一般强连通动态网络，例如移动多智能体系统及因链路故障而不稳定的网络。与现有诸多工作相比，该方法无需在链路上设计双随机权重。现有文献通常需要在网络初始化以及拓扑结构发生变化的任何时刻，使用特定的权重设计算法来确保链路权重具有随机性。本文消除了对此类算法的需求，为时变有向图上的分布式优化开辟了新路径。我们推导了梯度跟踪步长和离散时间步长的收敛边界，并利用一致性算法、矩阵摄动理论和Lyapunov理论论证了动态稳定性。尤其值得指出的是，在链路移除或数据包丢失场景下，本工作是对现有基于随机权重无向网络方法的改进。这是因为现有文献可能需要重新运行耗时且计算复杂的随机设计算法，而所提出的策略仅在底层网络满足权重对称且平衡的条件下即可生效。本文提出的优化框架可应用于分布式分类与学习。