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Finite Approximations for Mean Field Type Multi-Agent Control and Their Near Optimality

Erhan Bayraktar, Nicole Bauerle, Ali Devran Kara

TL;DR

This work develops finite-state and finite-action approximations for discrete-time mean-field type multi-agent control with continuous state and action spaces, formulating the problem as a centralized measure-valued MDP and then linking the finite/continuous problems via rigorous Lipschitz and Wasserstein-based error bounds. By discretizing the state and action spaces and leveraging the infinite-population limit, the authors obtain symmetric policies that eliminate coordination bottlenecks, while preserving near-optimal performance through explicit regret bounds that scale with discretization errors. They further propose two scalable strategies for large populations—direct aggregation of measures and population sampling—each accompanied by regret guarantees, and they discuss the trade-offs between requiring perfect mean-field feedback and relying on partial observations. Overall, the paper provides a principled, quantifiable pathway from intractable N-agent problems to practical, provably near-optimal policies suitable for large-scale mean-field control applications. The results advance the practical deployment of mean-field control by delivering concrete approximation schemes with rigorous performance guarantees under general non-linear dynamics and common randomness.

Abstract

We study a multi-agent mean field type control problem in discrete time where the agents aim to find a socially optimal strategy and where the state and action spaces for the agents are assumed to be continuous. The agents are only weakly coupled through the distribution of their state variables. The problem in its original form can be formulated as a classical Markov decision process (MDP), however, this formulation suffers from several practical difficulties. In this work, we attempt to overcome the curse of dimensionality, coordination complexity between the agents, and the necessity of perfect feedback collection from all the agents (which might be hard to do for large populations.) We provide several approximations: we establish the near optimality of the action and state space discretization of the agents under standard regularity assumptions for the considered formulation by constructing and studying the measure valued MDP counterpart for finite and infinite population settings. It is a well known approach to consider the infinite population problem for mean-field type models, since it provides symmetric policies for the agents which simplifies the coordination between the agents. However, the optimality analysis is harder as the state space of the measure valued infinite population MDP is continuous (even after space discretization of the agents). Therefore, as a final step, we provide further approximations for the infinite population problem by focusing on smaller sized sub-population distributions.

Finite Approximations for Mean Field Type Multi-Agent Control and Their Near Optimality

TL;DR

This work develops finite-state and finite-action approximations for discrete-time mean-field type multi-agent control with continuous state and action spaces, formulating the problem as a centralized measure-valued MDP and then linking the finite/continuous problems via rigorous Lipschitz and Wasserstein-based error bounds. By discretizing the state and action spaces and leveraging the infinite-population limit, the authors obtain symmetric policies that eliminate coordination bottlenecks, while preserving near-optimal performance through explicit regret bounds that scale with discretization errors. They further propose two scalable strategies for large populations—direct aggregation of measures and population sampling—each accompanied by regret guarantees, and they discuss the trade-offs between requiring perfect mean-field feedback and relying on partial observations. Overall, the paper provides a principled, quantifiable pathway from intractable N-agent problems to practical, provably near-optimal policies suitable for large-scale mean-field control applications. The results advance the practical deployment of mean-field control by delivering concrete approximation schemes with rigorous performance guarantees under general non-linear dynamics and common randomness.

Abstract

We study a multi-agent mean field type control problem in discrete time where the agents aim to find a socially optimal strategy and where the state and action spaces for the agents are assumed to be continuous. The agents are only weakly coupled through the distribution of their state variables. The problem in its original form can be formulated as a classical Markov decision process (MDP), however, this formulation suffers from several practical difficulties. In this work, we attempt to overcome the curse of dimensionality, coordination complexity between the agents, and the necessity of perfect feedback collection from all the agents (which might be hard to do for large populations.) We provide several approximations: we establish the near optimality of the action and state space discretization of the agents under standard regularity assumptions for the considered formulation by constructing and studying the measure valued MDP counterpart for finite and infinite population settings. It is a well known approach to consider the infinite population problem for mean-field type models, since it provides symmetric policies for the agents which simplifies the coordination between the agents. However, the optimality analysis is harder as the state space of the measure valued infinite population MDP is continuous (even after space discretization of the agents). Therefore, as a final step, we provide further approximations for the infinite population problem by focusing on smaller sized sub-population distributions.
Paper Structure (32 sections, 14 theorems, 146 equations)

This paper contains 32 sections, 14 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Problem Motivation
  3. Literature Review and Contributions
  4. Measure Valued Centralized MDP Construction
  5. Equivalence between the Team Problem and the Measure Valued MDP
  6. Finite Model Construction
  7. Action Space Discretization
  8. State Space Discretization
  9. Measure Valued Finite MDP Construction
  10. Near Optimality of the Policy Constructed From the Finite Model
  11. Infinite Population Limit
  12. Measure Valued MDP for the Infinite Population on $\mathds{X}$
  13. Measure Valued MDP for the Infinite Population on $\hat{\mathds{X}}$
  14. Effect of Discretization on the Value Function of the Infinite Population MDP
  15. Further Approximations for Large N
  16. ...and 17 more sections

Key Result

Lemma 1

Under Assumption main_assmp, we have that the optimal value function is constant over the initial states with the same distribution, i.e. if ${\bf{x_0,x_0'}}\in \mathds{X}^N$ have the same distribution, $\mu_{\bf x_0}=\mu_{\bf x_0'}$, then

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 21 more