标题： 关于具有有限时间一致性的去中心化随机梯度跟踪的收敛性 显示英文标题

标题： On the Convergence of Decentralized Stochastic Gradient-Tracking with Finite-Time Consensus

摘要： 去中心化优化和学习算法依赖于局部优化步骤以及图上的组合步骤。近期的研究表明，使用一个能够实现有限时间一致性的时变矩阵序列可以提高基于梯度跟踪的去中心化优化算法的通信复杂度和迭代复杂度。在实际应用中，由于网络拓扑结构的知识不完善、序列长度的限制或者数值不稳定等问题，可能无法获得满足精确有限时间一致性性质的矩阵序列。在这项工作中，我们量化了近似有限时间一致性序列对基于梯度跟踪的去中心化优化算法收敛性的影响，阐明了序列的准确性和长度与其他典型问题参数（如平滑性和梯度噪声）之间的相互作用关系。 显示英文摘要

摘要： Algorithms for decentralized optimization and learning rely on local optimization steps coupled with combination steps over a graph. Recent works have demonstrated that using a time-varying sequence of matrices that achieve finite-time consensus can improve the communication and iteration complexity of decentralized optimization algorithms based on gradient tracking. In practice, a sequence of matrices satisfying the exact finite-time consensus property may not be available due to imperfect knowledge of the network topology, a limit on the length of the sequence, or numerical instabilities. In this work, we quantify the impact of approximate finite-time consensus sequences on the convergence of a gradient-tracking based decentralized optimization algorithm, clarifying the interplay between accuracy and length of the sequence as well as typical problem parameters such as smoothness and gradient noise.