Noise-to-State Exponentially Stable Distributed Convex Optimization on Weight-Balanced Digraphs

Abstract

This paper studies the robustness properties against additive persistent noise of a class of distributed continuous-time algorithms for convex optimization. A team of agents, each with its own private objective function, seeks to collectively determine the global decision vector that minimizes the sum of all the objectives. The team communicates over a weight-balanced, strongly connected digraph and both interagent communication and agent computation are corrupted by noise. Under the proposed distributed algorithm, each agent updates its estimate of the global solution using the gradient of its local objective function while, at the same time, seeking to agree with its neighbors' estimates via proportional-integral feedback on their disagreement. Under mild conditions on the local objective functions, we show that this strategy is noise-to-state exponentially stable in the second moment with respect to the optimal solution. Our technical approach combines notions and tools from graph theory, stochastic di...