# Stochastic Semi-Gradient Descent for Learning Mean Field Games with Population-Aware Function Approximation

arXiv: https://arxiv.org/abs/2408.08192 · HTML: https://mlsanity.com/paper/2408.08192/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 53 references that are in the map.
- First posted on arXiv 15 Aug 2024.

## Builds on (most cited first, 29 of 29)

- [Provably Efficient Reinforcement Learning with Linear Function Approximation](https://mlsanity.com/paper/1907.05388/) (2019)
- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [Is Q-Learning Minimax Optimal? A Tight Sample Complexity Analysis](https://mlsanity.com/paper/2102.06548/) (2021)
- [Finite-Sample Analysis for SARSA with Linear Function Approximation](https://mlsanity.com/paper/1902.02234/) (2019)
- [Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning](https://mlsanity.com/paper/2102.01585/) (2021)
- [Fictitious Play for Mean Field Games: Continuous Time Analysis and Applications](https://mlsanity.com/paper/2007.03458/) (2020)
- [Learning in Mean Field Games: A Survey](https://mlsanity.com/paper/2205.12944/) (2022)
- [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)
- [Q-Learning in Regularized Mean-field Games](https://mlsanity.com/paper/2003.12151/) (2020)
- [Learning in Mean Field Games: the Fictitious Play](https://mlsanity.com/paper/1507.06280/) (2015)
- [Scalable Deep Reinforcement Learning Algorithms for Mean Field Games](https://mlsanity.com/paper/2203.11973/) (2022)
- [Scaling up Mean Field Games with Online Mirror Descent](https://mlsanity.com/paper/2103.00623/) (2021)
- [Mean Field Games Flock! The Reinforcement Learning Way](https://mlsanity.com/paper/2105.07933/) (2021)
- [Finite-Time Analysis of On-Policy Heterogeneous Federated Reinforcement Learning](https://mlsanity.com/paper/2401.15273/) (2024)
- [Policy Mirror Ascent for Efficient and Independent Learning in Mean Field Games](https://mlsanity.com/paper/2212.14449/) (2022)
- [Oracle-free Reinforcement Learning in Mean-Field Games along a Single Sample Path](https://mlsanity.com/paper/2208.11639/) (2022)
- [On the Convergence of SARSA with Linear Function Approximation](https://mlsanity.com/paper/2202.06828/) (2022)
- [A Survey on Large-Population Systems and Scalable Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2209.03859/) (2022)
- [On the Statistical Efficiency of Mean-Field Reinforcement Learning with General Function Approximation](https://mlsanity.com/paper/2305.11283/) (2023)
- [Reinforcement Learning in Non-Stationary Discrete-Time Linear-Quadratic Mean-Field Games](https://mlsanity.com/paper/2009.04350/) (2020)
- [Convergence of Multi-Scale Reinforcement Q-Learning Algorithms for Mean Field Game and Control Problems](https://mlsanity.com/paper/2312.06659/) (2023)
- [Reinforcement Learning Algorithm for Mixed Mean Field Control Games](https://mlsanity.com/paper/2205.02330/) (2022)
- [Dynamic driving and routing games for autonomous vehicles on networks: A mean field game approach](https://mlsanity.com/paper/2012.08388/) (2020)
- [Graphon Mean Field Games with a Representative Player: Analysis and Learning Algorithm](https://mlsanity.com/paper/2405.08005/) (2024)
- [Mean field games with nonlinear mobilities in pedestrian dynamics](https://mlsanity.com/paper/1304.5201/) (2013)
- [Learning in Herding Mean Field Games: Single-Loop Algorithm with Finite-Time Convergence Analysis](https://mlsanity.com/paper/2408.04780/) (2024)
- [A Single Online Agent Can Efficiently Learn Mean Field Games](https://mlsanity.com/paper/2405.03718/) (2024)

## Built on by (most cited first, 5 of 5)

- [Operator Learning for Families of Finite-State Mean-Field Games](https://mlsanity.com/paper/2602.13169/) (2026)
- [Personalized Collaborative Learning with Affinity-Based Variance Reduction](https://mlsanity.com/paper/2510.16232/) (2025)
- [Networked Communication for Mean-Field Games with Function Approximation and Empirical Mean-Field Estimation](https://mlsanity.com/paper/2408.11607/) (2024)
- [Last Iterate Convergence in Monotone Mean Field Games](https://mlsanity.com/paper/2410.05127/) (2024)
- [All in One: Generative Modeling as Mean-Field Game Design](https://mlsanity.com/paper/2607.23026/) (2026)

## Abstract (arXiv)

Mean field games (MFGs) model interactions in large-population multi-agent systems through population distributions. Traditional learning methods for MFGs are based on fixed-point iteration (FPI), where policy updates and induced population distributions are computed separately and sequentially. However, FPI-type methods may suffer from inefficiency and instability due to potential oscillations caused by this forward-backward procedure. In this work, we propose a novel perspective that treats the policy and population as a unified parameter controlling the game dynamics. By applying stochastic parameter approximation to this unified parameter, we develop SemiSGD, a simple stochastic gradient descent (SGD)-type method, where an agent updates its policy and population estimates simultaneously and fully asynchronously. Building on this perspective, we further apply linear function approximation (LFA) to the unified parameter, resulting in the first population-aware LFA (PA-LFA) for learning MFGs on continuous state-action spaces. A comprehensive finite-time convergence analysis is provided for SemiSGD with PA-LFA, including its convergence to the equilibrium for linear MFGs -- a class of MFGs with a linear structure concerning the population -- under the standard contractivity condition, and to a neighborhood of the equilibrium under a more practical condition. We also characterize the approximation error for non-linear MFGs. We validate our theoretical findings with six experiments on three MFGs.