The aim of decentralized gradient descent (DGD) is to minimize a sum of $n$ functions held by interconnected agents. We study the stability of DGD in open contexts where agents can join or leave the system, resulting each time in the addition or the removal of their function from the global objective. Assuming all functions are smooth, strongly convex, and their minimizers all lie in a given ball, we characterize the sensitivity of the global minimizer of the sum of these functions to the removal or addition of a new function and provide bounds in $ O\left(\min \left(\kappa^{0.5}, \kappa/n^{0.5},\kappa^{1.5}/n\right)\right)$ where $\kappa$ is the condition number. We also show that the states of all agents can be eventually bounded independently of the sequence of arrivals and departures. The magnitude of the bound scales with the importance of the interconnection, which also determines the accuracy of the final solution in the absence of arrival and departure, exposing thus a potential trade-off between accuracy and sensitivity. Our analysis relies on the formulation of DGD as gradient descent on an auxiliary function. The tightness of our results is analyzed using the PESTO Toolbox.

中文翻译：

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开放式多智能体系统中分散梯度下降的稳定性

分散梯度下降 (DGD) 的目标是最小化互连代理持有的 $n$ 函数的总和。我们研究了 DGD 在开放环境中的稳定性，在这种环境中，代理可以加入或离开系统，每次都会导致他们的功能从全局目标中添加或删除。 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 假设所有函数都是光滑的、强凸的，并且它们的极小值都位于给定的球中，我们描述了这些函数总和的全局极小值对移除或添加新函数的敏感性，并提供了 $O\left 的边界(\min \left(\kappa^{0.5}, \kappa/n^{0.5},\kappa^{1.5}/n\right)\right)$ 其中 $\kappa$ 是条件数。我们还表明，所有代理的状态最终都可以独立于到达和离开的顺序而有界。界限的大小与互连的重要性成比例，这也决定了在没有到达和离开的情况下最终解决方案的准确性，从而暴露了准确性和灵敏度之间的潜在权衡。我们的分析依赖于将 DGD 公式化为辅助函数的梯度下降。使用 PESTO 工具箱分析我们结果的紧密度。