CAREER: Adversarial Robustness through the Lens of Mathematical Analysis and Geometry

The past few years have witnessed an expansion of artificial intelligence and machine learning in several domains of applications at lightning speed. This unprecedented technological development has motivated new paradigms, essential in safety critical applications, for judging the success of data analysis methodologies, displacing high predictive power as the sole criterion for training models and giving more relevance to a more nuanced learning that can factor in reliability, privacy, and fairness criteria. This project will explore foundational questions, theoretical and algorithmic, in line with this expansion of paradigms, and provide research opportunities for undergraduate and graduate students. This project is distinctive in its analytical and geometric character and aims at providing an overarching perspective on timely problems in machine learning that will trigger a variety of connections between adversarial robustness and privacy and fields like optimal transportation, inverse problems, interacting particle systems and mean field PDEs, analysis of PDEs of geometric type, and geometric variational problems, among others. Some concrete goals that will be pursued in this project include to: analyze the geometric structure of adversarial attacks in a variety of learning settings, understand the regularization capabilities of adversarial training, study the connections between adversarial training and different forms of explicit regularization methods, develop algorithmic strategies, by exploiting different geometric structures in spaces of measures, to enforce robustness and privacy, and design and study of new geometric frameworks for adversarial training. 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.