UC Berkeley

In today's rapidly evolving technological landscape, the development and advancement of computational tools and algorithms have become paramount across a wide range of research fields. This holds particularly true in various domains of computational mathematics, encompassing areas such as machine learning, optimization, and algorithmic game theory. The computational tools serve as essential enablers, empowering researchers and practitioners by facilitating efficient modeling, analysis, and prediction. Algorithms are essential components of computational tools, which provide instructions for data processing, pattern recognition, and decision-making.

This dissertation focuses on developing computational tools and algorithms for specific applications in the interconnected fields of optimization, machine learning (ML), and mean-field games (MFGs). First, to address the absence of a comprehensive computational tool for MFGs, we present MFGLib, an open-source Python library designed to provide a user-friendly and customizable interface for solving Nash equilibria in generic MFGs. Second, we demonstrate that the search for Nash equilibria in MFGs and various ML problems can be formulated as non-convex optimization problems, where the presence of saddle points significantly impedes the effectiveness of gradient descent algorithm and its variants. To help optimization algorithms escape saddle points efficiently, we introduce a novel perturbation mechanism based on the dynamics of vertex-repelling random walk. This leads to the development of two new algorithms, perturbed gradient descent adapted to occupation time (PGDOT) and its accelerated version (PAGDOT). Theoretical guarantees for these algorithms are established, and through extensive numerical experiments, we showcase their superiority over several state-of-the-art optimization methods. Last, we explore a relatively independent machine learning task---detecting overutilization and fraud in healthcare. We focus on developing an ensemble model based on Stacked Generalization (stacking) to detect overutilization in Medicare within the field of Ophthalmology. Our results highlight the superiority of the stacking ensemble model over traditional ML models in accurately distinguishing overutilizing ophthalmologists from non-fraudulent ones.