# Reinforcement Learning for Discrete-time LQG Mean Field Social Control Problems with Unknown Dynamics

arXiv: https://arxiv.org/abs/2507.01420 · HTML: https://mlsanity.com/paper/2507.01420/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 13 of its 50 references that are in the map.
- First posted on arXiv 2 Jul 2025.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 13 of 13)

- [Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning](https://mlsanity.com/paper/2102.01585/) (2021)
- [Mean-Field Controls with Q-learning for Cooperative MARL: Convergence and Complexity Analysis](https://mlsanity.com/paper/2002.04131/) (2020)
- [Unified Reinforcement Q-Learning for Mean Field Game and Control Problems](https://mlsanity.com/paper/2006.13912/) (2020)
- [On the Convergence of Model Free Learning in Mean Field Games](https://mlsanity.com/paper/1907.02633/) (2019)
- [Scalable Deep Reinforcement Learning Algorithms for Mean Field Games](https://mlsanity.com/paper/2203.11973/) (2022)
- [On the convergence of closed-loop Nash equilibria to the mean field game limit](https://mlsanity.com/paper/1808.02745/) (2018)
- [Mean Field LQG Social Optimization: A Reinforcement Learning Approach](https://mlsanity.com/paper/2410.15119/) (2024)
- [Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach](https://mlsanity.com/paper/1903.08776/) (2019)
- [Robust Mean Field Social Control Problems with Applications in Analysis of Opinion Dynamics](https://mlsanity.com/paper/2002.12040/) (2020)
- [Mean Field Linear Quadratic Control: Uniform Stabilization and Social Optimality](https://mlsanity.com/paper/2002.12363/) (2020)
- [Dynamic Collective Choice: Social Optima](https://mlsanity.com/paper/1606.05272/) (2016)
- [Semi-Explicit Solutions to some Non-Linear Non-Quadratic Mean-Field-Type Games: A Direct Method](https://mlsanity.com/paper/1812.06695/) (2018)
- [LQG Mean Field Games with a Major Agent: Nash Certainty Equivalence versus Probabilistic Approach](https://mlsanity.com/paper/2012.04866/) (2020)

## Abstract (arXiv)

This paper studies the discrete-time linear-quadratic-Gaussian mean field (MF) social control problem in an infinite horizon, where the dynamics of all agents are unknown. The objective is to design a reinforcement learning (RL) algorithm to approximate the decentralized asymptotic optimal social control in terms of two algebraic Riccati equations (AREs). In this problem, a coupling term is introduced into the system dynamics to capture the interactions among agents. This causes the equivalence between model-based and model-free methods to be invalid, which makes it difficult to directly apply traditional model-free algorithms. Firstly, under the assumptions of system stabilizability and detectability, a model-based policy iteration algorithm is proposed to approximate the stabilizing solution of the AREs. The algorithm is proven to be convergent in both cases of semi-positive definite and indefinite weight matrices. Subsequently, by adopting the method of system transformation, a model-free RL algorithm is designed to solve for asymptotic optimal social control. During the iteration process, the updates are performed using data collected from any two agents and MF state. Finally, a numerical case is provided to verify the effectiveness of the proposed algorithm.