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

Dena Firoozi, Anastasis Kratsios, Xuwei Yang

TL;DR

The paper develops a principled framework for solving infinite-dimensional LQ mean field games by learning the rules-to-equilibrium operator with neural operators. It establishes that regularized rules-to-equilibrium maps are locally Lipschitz and PAC-learnable by Lipschitz Residually-Guided Neural Operators in Hilbert spaces, with explicit finite-sample guarantees. A universal Lipschitz NO approximation theorem, together with optimal-transport–based generalization bounds, yields practical guarantees on small NOs (depth O(1), width O(ε^{-c}) up to log factors) to approximate the operator across a broad class of problem variations. Under structured input sets (exponentially ellipsoidal, tempered sampling), the results provide faster rates and scalable learning for infinite families of LQ MFGs, enabling robust, data-driven solution descriptions across perturbations and continuum-parameterized agents.

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).

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

TL;DR

The paper develops a principled framework for solving infinite-dimensional LQ mean field games by learning the rules-to-equilibrium operator with neural operators. It establishes that regularized rules-to-equilibrium maps are locally Lipschitz and PAC-learnable by Lipschitz Residually-Guided Neural Operators in Hilbert spaces, with explicit finite-sample guarantees. A universal Lipschitz NO approximation theorem, together with optimal-transport–based generalization bounds, yields practical guarantees on small NOs (depth O(1), width O(ε^{-c}) up to log factors) to approximate the operator across a broad class of problem variations. Under structured input sets (exponentially ellipsoidal, tempered sampling), the results provide faster rates and scalable learning for infinite families of LQ MFGs, enabling robust, data-driven solution descriptions across perturbations and continuum-parameterized agents.

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 -Lipschitz learners in infinite dimensions, directly applicable as the Lipschitz constants of our approximating NOs are controlled in (ii).
Paper Structure (45 sections, 30 theorems, 223 equations)

This paper contains 45 sections, 30 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Contributions
  3. Technical Contributions
  4. Organization of Paper
  5. Preliminaries
  6. Infinite-Dimensional LQ MFGs
  7. Space of Variations on Rules of LQ MFGs
  8. Rules-to-Equilibrium Operator
  9. Statistics in Infinite Dimensions
  10. Infinite-Dimensional Deep Learning: Neural Operator
  11. Main Results
  12. Explanation of Proofs via Supporting Results
  13. Unlocking Favourable Rates
  14. Proofs of Stability Estimates
  15. Stability of the equilibrium with respect to operator A
  16. ...and 30 more sections

Key Result

Proposition 3.1

Fix a reference model $(A^{\dagger},B^{\dagger},F_2^{\dagger})$ and radii $\rho_A,\rho_B,\rho_{F_2}>0$. There exists a time $T^{\star}>0$, depending only on the radii $(\rho_A,\rho_B,\rho_{F_2})$ and on the $C_0$-semi-group of $A^{\dagger}$, such that the operator $\mathfrak{R}$ is well-defined and, where the Lipschitz constant $L> 0$ depends onlyAn explicit dependence of $L$ on $\mathbb{B}_{\math

Theorems & Definitions (75)

  • Example 1: Centred Gaussian with Diagonal Covariance
  • Definition 2.1: Residually-Guided Neural Operator (RNO)
  • Remark 1
  • Example 2
  • Proposition 3.1: Local Well-Posedness of the Rules-to-Equilibrium Operator
  • Proposition 3.2: Global Lipschitzness of the Regularized Rules-to-equilibrium Map
  • Theorem 3.3: Regularized Rules-to-Equilibria Operators are PAC-Learnable by RNOs
  • Theorem 3.4: Small Empirical Risk Minimizing RNOs Exist
  • Theorem 3.5
  • proof
  • ...and 65 more