Reinforcement Learning for Non-stationary Discrete-Time Linear–Quadratic Mean-Field Games in Multiple Populations

Muhammad Aneeq uz Zaman, Erik Miehling, Tamer Başar

Research output: Contribution to journalArticlepeer-review


Scalability of reinforcement learning algorithms to multi-agent systems is a significant bottleneck to their practical use. In this paper, we approach multi-agent reinforcement learning from a mean-field game perspective, where the number of agents tends to infinity. Our analysis focuses on the structured setting of systems with linear dynamics and quadratic costs, named linear–quadratic mean-field games, evolving over a discrete-time infinite horizon where agents are assumed to be partitioned into finitely many populations connected by a network of known structure. The functional forms of the agents’ costs and dynamics are assumed to be the same within populations, but differ between populations. We first characterize the equilibrium of the mean-field game which further prescribes an ϵ-Nash equilibrium for the finite population game. Our main focus is on the design of a learning algorithm, based on zero-order stochastic optimization, for computing mean-field equilibria. The algorithm exploits the affine structure of both the equilibrium controller and equilibrium mean-field trajectory by decomposing the learning task into first learning the linear terms and then learning the affine terms. We present a convergence proof and a finite-sample bound quantifying the estimation error as a function of the number of samples.

Original languageEnglish (US)
Pages (from-to)118-164
Number of pages47
JournalDynamic Games and Applications
Issue number1
StatePublished - Mar 2023


  • Large population games on networks
  • Mean-field games
  • Multi-agent reinforcement learning
  • Zero-order stochastic optimization

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'Reinforcement Learning for Non-stationary Discrete-Time Linear–Quadratic Mean-Field Games in Multiple Populations'. Together they form a unique fingerprint.

Cite this