Mean-field optimal control with stochastic leaders

TL;DR

This paper analyzes a mean-field optimal control problem involving stochastic followers influenced by a fixed number of stochastic leaders, establishing convergence of controls and proposing an efficient gradient descent algorithm.

Contribution

It introduces a partial mean-field limit for systems with stochastic leaders and followers, proving control convergence and developing a stochastic gradient descent method.

Findings

Optimal control of finite systems converges to the mean-field limit control.

The proposed algorithm efficiently approximates the mean-field control.

Application to opinion dynamics demonstrates effective leader influence.

Abstract

We consider interacting agent systems with a large number of stochastic agents (or particles) influenced by a fixed number of external stochastic lead agents. Such examples arise, for example in models of opinion dynamics, where a small number of leaders (influencers) can steer the behaviour of a large population of followers. In this context, we study a partial mean-field limit where the number of followers tends to infinity, while the number of leaders stays constant. The partial mean-field limit dynamics is then given by a McKean-Vlasov stochastic differential equation (SDE) for the followers, coupled to a controlled It\^o-SDE governing the dynamics of the lead agents. For a given cost functional that the lead agents seek to minimise, we show that the unique optimal control of the finite agent system convergences to the optimal control of the limiting system. This establishes that the low-dimensional control of the partial (mean-field) system provides an effective approximation for controlling the high-dimensional finite agent system. In addition, we propose a stochastic gradient descent algorithm that can efficiently approximate the mean-field control. Our theoretical results are illustrated on opinion dynamics model with lead agents, where the control objective is to drive the followers to reach consensus in finite time.