A class of generalized Nesterov's accelerated gradient method from dynamical perspective

TL;DR

This paper introduces a generalized class of accelerated gradient methods based on Euler-Lagrange equations, unifying and extending Nesterov's schemes with improved convergence rates for strongly convex functions.

Contribution

It develops a new continuous-time framework that interpolates between Nesterov's methods and derives enhanced convergence rates without prior knowledge of strong convexity parameters.

Findings

The proposed equations outperform classical Nesterov's methods for large times.

A unified Lyapunov framework is established for convex and strongly convex functions.

Symplectic Euler scheme effectively integrates the Hamiltonian systems.

Abstract

We propose a class of \textit{Euler-Lagrange} equations indexed by a pair of parameters ($\alpha,r$) that generalizes Nesterov's accelerated gradient methods for convex ($\alpha=1$) and strongly convex ($\alpha=0$) functions from a continuous-time perspective. This class of equations also serves as an interpolation between the two Nesterov's schemes. The corresponding \textit{Hamiltonian} systems can be integrated via the symplectic Euler scheme with a fixed step-size. Furthermore, we can obtain the convergence rates for these equations ($0<\alpha<1$) that outperform Nesterov's when time is sufficiently large for $\mu$-strongly convex functions, without requiring a priori knowledge of $\mu$. We demonstrate this by constructing a class of Lyapunov functions that also provide a unified framework for Nesterov's schemes for convex and strongly convex functions.