Machine Learning and Mean Field Control

University Of Texas At Dallas, Richardson TX

Investigators

Abstract

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.