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Kai Shao

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5 papers
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5

RLJ Journal 2025 Journal Article

Reinforcement Learning for Finite Space Mean-Field Type Game

  • Kai Shao
  • Jiacheng Shen
  • Mathieu Lauriere

Mean field type games (MFTGs) describe Nash equilibria between large coalitions: each coalition consists of a continuum of cooperative agents who maximize the average reward of their coalition while interacting non-cooperatively with a finite number of other coalitions. Although the theory has been extensively developed, we are still lacking efficient and scalable computational methods. Here, we develop reinforcement learning methods for such games in a finite space setting with general dynamics and reward functions. We start by proving that MFTG solution yields approximate Nash equilibria in finite-size coalition games. We then propose two algorithms. The first is based on quantization of mean-field spaces and Nash Q-learning. We provide convergence and stability analysis. We then propose a deep reinforcement learning algorithm, which can scale to larger spaces. Numerical experiments in 4 environments with mean-field distributions of dimension up to $200$ show the scalability and efficiency of the proposed method.

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RLC Conference 2025 Conference Paper

Reinforcement Learning for Finite Space Mean-Field Type Game

  • Kai Shao
  • Jiacheng Shen
  • Mathieu Lauriere

Mean field type games (MFTGs) describe Nash equilibria between large coalitions: each coalition consists of a continuum of cooperative agents who maximize the average reward of their coalition while interacting non-cooperatively with a finite number of other coalitions. Although the theory has been extensively developed, we are still lacking efficient and scalable computational methods. Here, we develop reinforcement learning methods for such games in a finite space setting with general dynamics and reward functions. We start by proving that MFTG solution yields approximate Nash equilibria in finite-size coalition games. We then propose two algorithms. The first is based on quantization of mean-field spaces and Nash Q-learning. We provide convergence and stability analysis. We then propose a deep reinforcement learning algorithm, which can scale to larger spaces. Numerical experiments in 4 environments with mean-field distributions of dimension up to $200$ show the scalability and efficiency of the proposed method.

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NeurIPS Conference 2025 Conference Paper

Solving Continuous Mean Field Games: Deep Reinforcement Learning for Non-Stationary Dynamics

  • Lorenzo Magnino
  • Kai Shao
  • Zida Wu
  • Jiacheng Shen
  • Mathieu Lauriere

Mean field games (MFGs) have emerged as a powerful framework for modeling interactions in large-scale multi-agent systems. Despite recent advancements in reinforcement learning (RL) for MFGs, existing methods are typically limited to finite spaces or stationary models, hindering their applicability to real-world problems. This paper introduces a novel deep reinforcement learning (DRL) algorithm specifically designed for non-stationary continuous MFGs. The proposed approach builds upon a Fictitious Play (FP) methodology, leveraging DRL for best-response computation and supervised learning for average policy representation. Furthermore, it learns a representation of the time-dependent population distribution using a Conditional Normalizing Flow. To validate the effectiveness of our method, we evaluate it on three different examples of increasing complexity. By addressing critical limitations in scalability and density approximation, this work represents a significant advancement in applying DRL techniques to complex MFG problems, bringing the field closer to real-world multi-agent systems.

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ICLR Conference 2022 Conference Paper

On the Convergence of the Monte Carlo Exploring Starts Algorithm for Reinforcement Learning

  • Che Wang
  • Shuhan Yuan
  • Kai Shao
  • Keith W. Ross

A simple and natural algorithm for reinforcement learning (RL) is Monte Carlo Exploring Starts (MCES), where the Q-function is estimated by averaging the Monte Carlo returns, and the policy is improved by choosing actions that maximize the current estimate of the Q-function. Exploration is performed by "exploring starts", that is, each episode begins with a randomly chosen state and action, and then follows the current policy to the terminal state. In the classic book on RL by Sutton & Barto (2018), it is stated that establishing convergence for the MCES algorithm is one of the most important remaining open theoretical problems in RL. However, the convergence question for MCES turns out to be quite nuanced. Bertsekas & Tsitsiklis (1996) provide a counter-example showing that the MCES algorithm does not necessarily converge. Tsitsiklis (2002) further shows that if the original MCES algorithm is modified so that the Q-function estimates are updated at the same rate for all state-action pairs, and the discount factor is strictly less than one, then the MCES algorithm converges. In this paper we make headway with the original and more efficient MCES algorithm given in Sutton et al. (1998), establishing almost sure convergence for Optimal Policy Feed-Forward MDPs, which are MDPs whose states are not revisited within any episode when using an optimal policy. Such MDPs include a large class of environments such as all deterministic environments and all episodic environments with a timestep or any monotonically changing values as part of the state. Different from the previous proofs using stochastic approximations, we introduce a novel inductive approach, which is very simple and only makes use of the strong law of large numbers.

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YNICL Journal 2021 Journal Article

Hippocampal subfield and anterior-posterior segment volumes in patients with sporadic amyotrophic lateral sclerosis

  • Shuangwu Liu
  • Qingguo Ren
  • Gaolang Gong
  • Yuan Sun
  • Bing Zhao
  • Xiaotian Ma
  • Na Zhang
  • Suyu Zhong

Neuroimaging studies of hippocampal volumes in patients with amyotrophic lateral sclerosis (ALS) have reported inconsistent results. Our aims were to demonstrate that such discrepancies are largely due to atrophy of different regions of the hippocampus that emerge in different disease stages of ALS and to explore the existence of co-pathology in ALS patients. We used the well-validated King's clinical staging system for ALS to classify patients into different disease stages. We investigated in vivo hippocampal atrophy patterns across subfields and anterior-posterior segments in different King's stages using structural MRI in 76 ALS patients and 94 health controls (HCs). The thalamus, corticostriatal tract and perforant path were used as structural controls to compare the sequence of alterations between these structures and the hippocampal subfields. Compared with HCs, ALS patients at King's stage 1 had lower volumes in the bilateral posterior subiculum and presubiculum; ALS patients at King's stage 2 exhibited lower volumes in the bilateral posterior subiculum, left anterior presubiculum and left global hippocampus; ALS patients at King's stage 3 showed significantly lower volumes in the bilateral posterior subiculum, dentate gyrus and global hippocampus. Thalamic atrophy emerged at King's stage 3. White matter tracts remained normal in a subset of ALS patients. Our study demonstrated that the pattern of hippocampal atrophy in ALS patients varies greatly across King's stages. Future studies in ALS patients that focus on the hippocampus may help to further clarify possible co-pathologies in ALS.

Details DOI
v2026.09.13