Mean-field optimal control with stochastic leaders

Abstract We consider interacting agent systems with a large number of stochastic agents influenced by a fixed number of external stochastic lead agents. Such settings arise, for example, in models of opinion dynamics, where a small number of leaders 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 Itô-SDE governing the dynamics of the lead agents. For a given cost functional that the lead agents seek to minimize, we show that the unique optimal control of the finite agent system converges 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 an opinion dynamics model with lead agents, where the control objective is to drive the followers to reach consensus.