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Mean-field optimal control with stochastic leaders

Sebastian Zimper, Ana Djurdjevac, Carsten Hartmann, Christof Schütte, Nataša Djurdjevac Conrad

TL;DR

This work develops a rigorous partial mean-field framework for controlling a large population of stochastic agents with a fixed number of external stochastic leaders. It proves a conditional McKean–Vlasov limit for followers coupled to leader dynamics, expresses the limit via a nonlinear Fokker–Planck equation, and establishes Gamma-convergence of finite-N optimizers to the mean-field optimizer, ensuring accurate low-dimensional control. A practical gradient-descent-based numerical method is proposed and adapted to both finite and mean-field regimes, with a detailed application to the noisy Hegselmann–Krause model demonstrating finite-time consensus. The results illuminate scalable strategies to steer large populations in opinion dynamics while providing a rigorous link between finite simulations and mean-field approximations, with attention to ethical considerations in influence technologies.

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 Itô-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.

Mean-field optimal control with stochastic leaders

TL;DR

This work develops a rigorous partial mean-field framework for controlling a large population of stochastic agents with a fixed number of external stochastic leaders. It proves a conditional McKean–Vlasov limit for followers coupled to leader dynamics, expresses the limit via a nonlinear Fokker–Planck equation, and establishes Gamma-convergence of finite-N optimizers to the mean-field optimizer, ensuring accurate low-dimensional control. A practical gradient-descent-based numerical method is proposed and adapted to both finite and mean-field regimes, with a detailed application to the noisy Hegselmann–Krause model demonstrating finite-time consensus. The results illuminate scalable strategies to steer large populations in opinion dynamics while providing a rigorous link between finite simulations and mean-field approximations, with attention to ethical considerations in influence technologies.

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 Itô-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.
Paper Structure (16 sections, 15 theorems, 90 equations, 5 figures, 2 algorithms)

This paper contains 16 sections, 15 theorems, 90 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review and our contribution
  3. Outline of the paper
  4. Optimal control for the $N$ particle system with leaders
  5. Derivation of the mean-field limit
  6. Optimal control for the McKean-Vlasov system
  7. Numerical method for approximating the optimal control
  8. Approximating the $N$-particle control
  9. Approximating the mean-field control
  10. Choice of permissible controls
  11. Application to the noisy Hegselmann-Krause model
  12. Conclusion and outlook
  13. Appendix
  14. Proofs for Section \ref{['sec:MFLderivation']}
  15. Proofs for Section \ref{['sec:MFLcontrol']}
  16. ...and 1 more sections

Key Result

Proposition 3.3

If $\mathbb{E}[|\bar{X}_0 |^q + |\bar{Y}_0 |^q] < \infty$ for some $q \geq 2$ and Assumptions ass:Lipschitz and ass:Coeff_square_bound hold, then eq:BasicMcKeanSDE has a unique strong solution $(\bar{X},\bar{Y})$ in the sense of Definition def:sol such that where C is some constant depending on $T$, $q$ and the Lipschitz constants from the assumptions.

Figures (5)

  • Figure 1: Left: Simulation of the noisy Hegselmann-Krause model for $N=99$ followers (dark-blue lines) without a leader present. Right: Histograms of the follower's opinions at $t=0$ (red) and $t=1$ (black).
  • Figure 2: Results obtained by implementing the gradient descent based on Algorithm \ref{['alg:one']} for the noisy Hegselmann-Krause model with one leader and $99$ followers. The optimal control is approximated by functions which are piecewise constant over $m=5$ evenly space intervals \ref{['eq:FinColl']} on the time interval $t \in [0,1]$. Left: The five components of the control $u^i$ for each iteration $i$ of the minimisation scheme. The subscript $u_k^i$ denotes the $k$-th component of $u^i$. Right: The total cost $J^N(u^i)$ for each control shown in left panel.
  • Figure 3: Left: System dynamics under the optimal discrete-time Markov control computed using Algorithm \ref{['alg:two']}, for $N=99$ followers (dark-blue lines) and one leader (red line). The black, dashed, vertical lines indicate when the piecewise constant control changes. Right: Histograms of the follower's opinions at $t=0$ (red) and $t=1$ (black).
  • Figure 4: Similar to Fig. \ref{['fig:m5_results']}, but for $\sigma = 0.1$ (top-row) and $\sigma=0.2$ (bottom-row).
  • Figure 5: Left: System dynamics of the limiting non-linear Fokker-Planck equation coupled to the SDE of the leader when the optimal discrete-time Markov control is applied. The position of the leader is indicated by the red line and the density of followers $g_t(x)$ coloured according to the heatmap on the right of the plots. Right: The profile of $g_t(x)$ at the initial (red) and final (black) times.

Theorems & Definitions (30)

  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.4
  • Corollary 4.5
  • proof
  • ...and 20 more