Operator Learning for Families of Finite-State Mean-Field Games

TL;DR

This paper introduces an operator learning framework for efficiently solving parametric families of finite-state mean-field games, providing theoretical guarantees and demonstrating high accuracy on cybersecurity and high-dimensional benchmark models.

Contribution

The paper presents a novel operator learning approach that generalizes across different initial conditions and costs in finite-state MFGs, with theoretical error bounds and empirical validation.

Findings

Achieves accurate approximation of MFG solutions in cybersecurity example

Effective in high-dimensional quadratic MFG models

Provides theoretical guarantees on approximation error and generalization

Abstract

Finite-state mean-field games (MFGs) arise as limits of large interacting particle systems and are governed by an MFG system, a coupled forward-backward differential equation consisting of a forward Kolmogorov-Fokker-Planck (KFP) equation describing the population distribution and a backward Hamilton-Jacobi-Bellman (HJB) equation defining the value function. Solving MFG systems efficiently is challenging, with the structure of each system depending on an initial distribution of players and the terminal cost of the game. We propose an operator learning framework that solves parametric families of MFGs, enabling generalization without retraining for new initial distributions and terminal costs. We provide theoretical guarantees on the approximation error, parametric complexity, and generalization performance of our method, based on a novel regularity result for an appropriately defined flow map corresponding to an MFG system. We demonstrate empirically that our framework achieves accurate approximation for two representative instances of MFGs: a cybersecurity example and a high-dimensional quadratic model commonly used as a benchmark for numerical methods for MFGs.