Learning Sparse Graphon Mean Field Games

TL;DR

This paper introduces LPGMFGs, a new formulation of graphon mean field games using $L^p$ graphons, enabling scalable analysis and solution approximation for sparse and power law networks in multi-agent systems.

Contribution

It extends GMFGs to sparse networks with $L^p$ graphons, providing theoretical guarantees and an efficient learning algorithm for large-scale multi-agent problems.

Findings

LPGMFGs accurately approximate solutions for sparse networks.

The extended OMD algorithm accelerates learning in large systems.

Empirical results confirm the approach's effectiveness on power law networks.

Abstract

Although the field of multi-agent reinforcement learning (MARL) has made considerable progress in the last years, solving systems with a large number of agents remains a hard challenge. Graphon mean field games (GMFGs) enable the scalable analysis of MARL problems that are otherwise intractable. By the mathematical structure of graphons, this approach is limited to dense graphs which are insufficient to describe many real-world networks such as power law graphs. Our paper introduces a novel formulation of GMFGs, called LPGMFGs, which leverages the graph theoretical concept of $L^p$ graphons and provides a machine learning tool to efficiently and accurately approximate solutions for sparse network problems. This especially includes power law networks which are empirically observed in various application areas and cannot be captured by standard graphons. We derive theoretical existence and convergence guarantees and give empirical examples that demonstrate the accuracy of our learning approach for systems with many agents. Furthermore, we extend the Online Mirror Descent (OMD) learning algorithm to our setup to accelerate learning speed, empirically show its capabilities, and conduct a theoretical analysis using the novel concept of smoothed step graphons. In general, we provide a scalable, mathematically well-founded machine learning approach to a large class of otherwise intractable problems of great relevance in numerous research fields.