Collaborative Research: CIF: Medium: An Algorithmic Theory of Hamiltonian Dynamics for Sampling, Optimization, and Games

Modern artificial intelligence (AI) and machine learning (ML) systems are trained using massive datasets and complex models combined with optimization algorithms. Traditional "greedy" methods, which make incremental improvements at each step, often fall short in both efficiency and adaptability when faced with problems at this scale. This project proposes a novel framework for algorithm design based on the Hamiltonian dynamics, a fundamental concept in physics and mathematics that uses the conservation principles to describe the interaction of multiple objects. Such dynamics appear naturally in many branches of computational sciences but are rarely used as a fundamental principle in algorithm design. Motivated by emerging challenges in ML, this project aims to develop a systematic methodology that leverages Hamiltonian conservation to solve problems in optimization, random sampling, and game theory. This project has the potential to revolutionize our understanding of computational and statistical problems by introducing a new class of algorithmic principles for training modern ML systems. This project will also advance the curricula for algorithms in computer science and electrical engineering, with unique training opportunities for undergraduates and graduate students, the development of open-source software, and a dissemination of ideas via joint workshops. This project will explore a framework called “the LCP scheme”, which stands for Lift, Conserve, and Project. This proceeds by taking a parameterized decision space, appropriately lifting the problem to incorporate additional variables, applying the conservation property of the Hamiltonian dynamics to update the problem state in the augmented parameter space, and finally projecting the state back into the original space. This scheme provides a fresh perspective for analyzing several known algorithms, and developing new ones, in the domains of optimization and random sampling, as well as to understand the behavior of players in multi-agent systems. This project will develop a robust algorithmic complexity theory for implementing the continuous-time Hamiltonian dynamics as discrete-time sequential procedures with an emphasis on the large-scale modern applications. By introducing concepts such as invariance, conservation, and the principle of least action, this project will provide a more nuanced view of the state evolution of computational objects that can help overcome many limitations of the standard algorithm design paradigm. 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.