We propose a method for analyzing the distributed random coordinate descent algorithm for solving separable resource allocation problems in the context of an open multiagent system, where agents can be replaced during the process. In particular, we characterize the evolution of the distance to the minimizer in expectation by following a time-varying optimization approach which builds on two components. First, we establish the linear convergence of the algorithm in closed systems, in terms of the estimate towards the minimizer, for general graphs and appropriate step-size. Second, we estimate the change of the optimal solution after a replacement, in order to evaluate its effect on the distance between the current estimate and the minimizer. From these two elements, we derive stability conditions in open systems and establish the linear convergence of the algorithm towards a steady-state expected error. Our results enable to characterize the trade-off between speed of convergence and robustness to agent replacements, under the assumptions that local functions are smooth, strongly convex, and have their minimizers located in a given ball. The approach proposed in this paper can moreover be extended to other algorithms guaranteeing linear convergence in closed system.

翻译：本文提出了一种方法，用于分析开放多智能体系统背景下求解可分离资源分配问题的分布式随机坐标下降算法。在该系统中，智能体可在运行过程中被替换。具体地，我们通过构建基于两个组成部分的时变优化方法，刻画了期望意义上与最小化点之间距离的演化过程。首先，针对一般图结构及适当步长，建立了封闭系统中算法关于估计值向最小化点收敛的线性收敛性。其次，我们估计了智能体替换后最优解的变化量，以评估该替换对当前估计值与最小化点之间距离的影响。基于这两点，我们推导了开放系统的稳定性条件，并建立了算法向稳态期望误差线性收敛的结论。在局部函数光滑、强凸且其最小化点位于给定球内的假设下，研究结果刻画了收敛速度与对智能体替换鲁棒性之间的权衡关系。此外，本文提出的方法可推广至其他保证封闭系统线性收敛的算法。