Machine Learning and Mean Field Control

该项目旨在研究机器学习方法的数学技术，以增强数据隐私和公平性。该项目将侧重于平均场控制，最初为物理科学引入，并在平均场游戏的命名下适用于社会科学。随机控制理论及其增强的平均场控制理论，也允许系统地结合噪声和不确定性，同时提供令人满意的性能。平均场控制理论非常适合研究机器学习的深度神经网络，因为它被设计用来处理大量的代理、神经元或复杂的数据。此外，它提供的机器学习规则预计比其他方法更容易实现和解释。该项目将提供机器学习和数据科学的新应用，以解决传统机器学习应用成本过高的问题。该项目的主要工具自然适合研究对手的影响，因此非常适合确保隐私。该项目还将通过开发课程和教材，为该领域的几名博士生和工作人员提供培训机会。本项目旨在研究监督学习和深度学习背景下的平均场博弈和平均场控制。该项目包括三种具体情况：第一，数据的经验分布被输入和输出之间的联合概率所取代的情况；二是深度网络的大量层被无限多的层和神经元所取代的情况；第三，具有无限数量代理的联合学习。该项目将开发使用希尔伯特空间控制的新方法，而不是经典的瓦瑟斯坦度量来进行概率度量。这将允许更有效地处理渐变。研究人员将分析如何用平均场控制或平均场博弈取代随机梯度算法。该项目的另一个方向将是在数据包含空间元素时使用偏微分方程的控制而不是常微分方程的控制。最后，该项目将分析解决平均场博弈和相关控制问题的新算法。该奖项反映了美国国家科学基金会的法定使命，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。

This project aims to study mathematical techniques for machine learning methods to enhance data privacy and fairness. The project will focus on mean field control, introduced originally for physical sciences, and adapted to social sciences under the nomenclature of mean field games. Stochastic control theory and its enhancement, mean field control theory, also allow the systematic incorporation of noise and uncertainty while simultaneously providing satisfactory performance. Mean field control theory is ideally suited for studying deep neural networks of machine learning because it is designed to handle large numbers of agents, neurons, or complex data. In addition, the machine learning rules provided by it are expected to be easier to implement and interpret than would be the case with other methods. The project will provide new applications of machine learning and data science to situations wherein classical machine learning would be prohibitively expensive to apply. The main tool of the project naturally lends itself to study the impact of adversaries and is thus well suited to ensuring privacy. The project will also provide training opportunities to several Ph.D. students and the workforce in this area by developing courses and educational materials. This project aims to investigate mean field games and mean field control in the context of supervised learning and deep learning. The project includes three specific cases: first, the situation when the empirical distribution of data is replaced with a joint probability between inputs and outputs; second, the situation when a large number of layers of a deep network is replaced by an infinite number of layers and neurons; third, federated learning with an infinite number of agents. The project will develop novel methods of using the Hilbert space control instead of the classical Wasserstein metric for probability measures. This will allow handling the gradients more efficiently. The investigators will analyze how a stochastic gradient algorithm is replaced by a mean field control or a mean field game. Another direction of the project will be to use control of partial differential equations instead of control of ordinary differential equations when data contain a spatial element. Finally, the project will analyze new algorithms to solve the mean field games and related control problems.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.