Robust mean-field control under common noise uncertainty

2025-11-06

AI Summary

This paper addresses the discrete-time robust mean-field control problem under common noise uncertainty, modeling the collective behavior and optimal decision-making of infinitely many cooperative agents subject to worst-case common noise disturbances. Methodologically, it pioneers the integration of robust optimization with the mean-field framework, establishing an asymptotic limit exhibiting propagation of chaos; it formulates a dynamic programming principle on the space of probability measures and constructs a rigorous theoretical foundation via the Bellman–Isaacs fixed-point theorem. By unifying stochastic optimal control, mean-field games, and robust optimization, the work provides a strict proof of existence of optimal open-loop controls and validity of the dynamic programming principle. Numerical experiments demonstrate the method’s superiority and practicality in systemic risk assessment for financial systems and distributionally robust planning.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Multiagent Systems under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

We propose and analyze a framework for discrete-time robust mean-field control problems under common noise uncertainty. In this framework, the mean-field interaction describes the collective behavior of infinitely many cooperative agents'state and action, while the common noise -- a random disturbance affecting all agents'state dynamics -- is uncertain. A social planner optimizes over open-loop controls on an infinite horizon to maximize the representative agent's worst-case expected reward, where worst-case corresponds to the most adverse probability measure among all candidates inducing the unknown true law of the common noise process. We refer to this optimization as a robust mean-field control problem under common noise uncertainty. We first show that this problem arises as the asymptotic limit of a cooperative $N$-agent robust optimization problem, commonly known as propagation of chaos. We then prove the existence of an optimal open-loop control by linking the robust mean field control problem to a lifted robust Markov decision problem on the space of probability measures and by establishing the dynamic programming principle and Bellman--Isaac fixed point theorem for the lifted robust Markov decision problem. Finally, we complement our theoretical results with numerical experiments motivated by distribution planning and systemic risk in finance, highlighting the advantages of accounting for common noise uncertainty.

Problem

• Robust mean-field control with uncertain common noise affecting agents
• Optimizing worst-case expected reward under adversarial probability measures
• Establishing theoretical foundations and numerical validation for distribution planning

Innovation

• Robust mean-field control under common noise uncertainty
• Open-loop controls optimize worst-case expected reward
• Lifted robust Markov decision problem on probability measures