## Abstract

We study infinite horizon discounted mean field control (MFC) problems with common noise through the lens of mean field Markov decision processes (MFMDP). We allow the agents to use actions that are randomized not only at the individual level but also at the level of the population. This common randomization is introduced for the purpose of exploration from a reinforcement learning (RL) paradigm. It also allows us to establish connections between both closed-loop and open-loop policies for MFC and Markov policies for the MFMDP. In particular, we show that there exists an optimal closed-loop policy for the original MFC and we prove dynamic programming principles for the state and state-action value functions. Building on this framework and the notion of state-action value function, we then propose RL methods for such problems, by adapting existing tabular and deep RL methods to the mean-field setting. The main difficulty is the treatment of the population state, which is an input of the policy and the value function. We provide convergence guarantees for the tabular Q-learning algorithm based on discretizations of the simplex. We also show that neural network based deep RL algorithms are more suitable for continuous spaces as they allow us to avoid discretizing the mean field state space. Numerical examples are provided.

Original language | English (US) |
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Pages (from-to) | 5334-5381 |

Number of pages | 48 |

Journal | Annals of Applied Probability |

Volume | 33 |

Issue number | 6 B |

DOIs | |

State | Published - Dec 2023 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- McKean- Vlasov control
- Mean field reinforcement learning
- mean field Markov decision processes