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

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.