Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators

2025-10-22

AI Summary

Conventional mean-field game (MFG) solvers operate instance-by-instance, rendering them inefficient for batched problems under perturbations in dynamics or cost functions. Method: For linear-quadratic (LQ) MFGs, we propose a neural-operator-based universal solver framework. We first establish a local Lipschitz estimate for the mapping from problem specifications (dynamics and costs) to equilibrium policies. Then, we prove a universal approximation theorem for neural operators with prescribed Lipschitz regularity and derive sample complexity bounds for Lipschitz learning in infinite-dimensional spaces. Finally, we design a controlled-Lipschitz deep operator network trained via stochastic sampling of problem specifications. Results: The method enables efficient generalization to unseen LQ-MFG instances and simultaneously solves infinitely many related problems within a separable Hilbert space—bypassing sequential computation bottlenecks. It features controllable parameter complexity and strong theoretical guarantees, including explicit Lipschitz continuity and generalization bounds.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Traditional mean-field game (MFG) solvers operate on an instance-by-instance basis, which becomes infeasible when many related problems must be solved (e.g., for seeking a robust description of the solution under perturbations of the dynamics or utilities, or in settings involving continuum-parameterized agents.). We overcome this by training neural operators (NOs) to learn the rules-to-equilibrium map from the problem data (``rules'': dynamics and cost functionals) of LQ MFGs defined on separable Hilbert spaces to the corresponding equilibrium strategy. Our main result is a statistical guarantee: an NO trained on a small number of randomly sampled rules reliably solves unseen LQ MFG variants, even in infinite-dimensional settings. The number of NO parameters needed remains controlled under appropriate rule sampling during training. Our guarantee follows from three results: (i) local-Lipschitz estimates for the highly nonlinear rules-to-equilibrium map; (ii) a universal approximation theorem using NOs with a prespecified Lipschitz regularity (unlike traditional NO results where the NO's Lipschitz constant can diverge as the approximation error vanishes); and (iii) new sample-complexity bounds for $L$-Lipschitz learners in infinite dimensions, directly applicable as the Lipschitz constants of our approximating NOs are controlled in (ii).

Problem

• Solving infinitely many LQ mean field games in Hilbert spaces
• Learning rules-to-equilibrium map using neural operators
• Providing statistical guarantees for solving unseen game variants

Innovation

• Neural operators learn rules-to-equilibrium map for LQ MFGs
• Statistical guarantees for solving unseen infinite-dimensional variants
• Controlled parameters via Lipschitz-regularized neural operator approximation