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

We propose a class of textitEuler-Lagrange equations indexed by a pair of parameters (α,r) that generalizes Nesterov's accelerated gradient methods for convex (α=1) and strongly convex (α=0) functions from a continuous-time perspective. This class of equations also serves as an interpolation between the two Nesterov's schemes. The corresponding textitHamiltonian 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<α<1) that outperform Nesterov's when time is sufficiently large for μ-strongly convex functions, without requiring a priori knowledge of μ. 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.