A Unified Model for High-Resolution ODEs: New Insights on Accelerated Methods
Abstract
Recent work on high-resolution ordinary differential equations (HR-ODEs) captures fine nuances among different momentum-based optimization methods, leading to accurate theoretical insights. However, these HR-ODEs often appear disconnected, each targeting a specific algorithm and derived with different assumptions and techniques. We present a unifying framework by showing that these diverse HR-ODEs emerge as special cases of a general HR-ODE derived using the Forced Euler-Lagrange equation. Discretizing this model recovers a wide range of optimization algorithms through different parameter choices. Using integral quadratic constraints, we also introduce a general Lyapunov function to analyze the convergence of the proposed HR-ODE and its discretizations, achieving significant improvements across various cases, including new guarantees for the triple momentum methods HR-ODE and the quasi-hyperbolic momentum method, as well as faster gradient norm minimization rates for Nesterovs accelerated gradient algorithm, among other advances.
Cite
@article{arxiv.2503.15136,
title = {A Unified Model for High-Resolution ODEs: New Insights on Accelerated Methods},
author = {Hoomaan Maskan and Konstantinos C. Zygalakis and Armin Eftekhari and Alp Yurtsever},
journal= {arXiv preprint arXiv:2503.15136},
year = {2025}
}
Comments
41 pages, 5 figures, 3 tables