CAREER: Learning Theory for Large-scale Stochastic Games

在人口众多的现代金融市场和经济系统中，决策已经演变成一个多方面的过程，涉及人口异质性、多样化的信息结构和人机交互等各个方面。该项目旨在开发新的学习框架和数学基础，加强我们对人口众多的社会系统的稳定性，效率和公平性的理解。在这项研究中开发的新框架的设计具有灵活的模型假设，能够从不完全信息中学习，并适应异质风险偏好以及信息不对称。这项研究将涉及本科生和研究生，强调数学和机器学习的跨学科培训。此外，还将建立一个外展计划，以吸引STEM中代表性不足的群体。该项目的核心是机器学习理论的数学进步，用于具有许多交互代理的随机系统，称为“平均场游戏”。第一个目标是开发新的数学模型和学习算法的平均场游戏下的结构属性，如graphon相互作用或人口分布的额外的汇总统计。这种发展依赖于新的近似方案和稳定性分析的基础上，本地传播的流量。第二个目标侧重于委托代理问题，代理人有不同的风险偏好或获取新信息的能力。这些主题在动态环境中提出了重大挑战，导致一类新的随机偏微分方程，需要新的发展良好的definedness和正则性理论。最终目标的重点是构建生成模型（模拟器）与交互式平均场代理，解决基于代理的模拟器文献的可扩展性问题。为了利用神经网络的计算能力，一个关键目标是建立一个分布意义上的通用近似定理和迭代深度学习方案的收敛性，以训练模拟器。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In modern financial markets and economic systems with large populations, decision-making has evolved into a multifaceted process involving various aspects such as population heterogeneity, diverse information structures, and human-AI interactions. This project aims to develop new learning frameworks and mathematical foundations that strengthen our understanding of the stability, efficiency, and fairness of societal systems with large populations. Novel frameworks developed in this research are designed to have flexible model assumptions, be able to learn from incomplete information, and accommodate heterogeneous risk preferences as well as information asymmetry. This research will involve both undergraduate and graduate students, emphasizing cross-disciplinary training in mathematics and machine learning. Additionally, an outreach program will be established to engage underrepresented groups in STEM.This project places at its core the mathematical advancement of machine learning theory for stochastic systems with many interacting agents, known as “mean-field games”. The first goal is to develop new mathematical models and learning algorithms for mean-field games under structural properties such as graphon interactions or additional summary statistics of the population distribution. This development relies on new approximation schemes and stability analyses based on the local propagation of flows. The second goal focuses on principal-agent problems, where agents have diverse risk preferences or the capability to acquire new information. These topics pose significant challenges in a dynamic setting, leading to a novel class of stochastic partial differential equations that require new developments for well-definedness and regularity theory. The final goal focuses on constructing generative models (simulators) with interactive mean-field agents, addressing the scalability issue in agent-based simulator literature. To leverage the computational power of neural networks, a key objective is to establish a universal approximation theorem in the distributional sense and the convergence of an iterative deep-learning scheme to train the simulator.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.