Understanding Accelerated Gradient Methods: Lyapunov Analyses and Hamiltonian Assisted Interpretations
Year: 2023
Type: preprint
Abstract: We formulate two classes of first-order algorithms more general than previously studied for minimizing smooth and strongly convex or, respectively, smooth and convex functions. We establish sufficient conditions, via new discrete Lyapunov analyses, for achieving accelerated convergence rates which match Nesterov's methods in the strongly and general convex settings. Next, we study the convergence of limiting ordinary differential equations (ODEs) and point out currently notable gaps between the convergence properties of the corresponding algorithms and ODEs. Finally, we propose a novel class of discrete algorithms, called the Hamiltonian assisted gradient method, directly based on a Hamiltonian function and several interpretable operations, and then demonstrate meaningful and unified interpretations of our acceleration conditions.
Source: arXiv (Cornell University)
Authors: Fu, Penghui, Tan, Zhiqiang
Institution: Rutgers, The State University of New Jersey
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4366733163