Stability of Decentralized Gradient Descent in Open Multi-Agent Systems

TL;DR

This paper analyzes the stability of decentralized gradient descent in open multi-agent systems where agents can join or leave, providing bounds on the sensitivity of the global minimizer and agent states.

Contribution

It characterizes the sensitivity of the global minimizer to agent arrivals and departures and bounds the agents' states independently of the sequence of changes.

Findings

Sensitivity bounds depend on condition number and number of agents

Agent states can be bounded independently of arrival/departure sequence

Trade-off identified between solution accuracy and system sensitivity

Abstract

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.