On Underdamped Nesterov’s Acceleration

The high-resolution ordinary differential equation (ODE) framework has proven to be particularly effective for analyzing Nesterov's accelerated gradient (NAG) method and its proximal counterpart, the faster iterative shrinkage-thresholding algorithms (FISTA). However, the theoretical framework remains incomplete, as the underdamped regime ($r < 2$) has not yet been fully incorporated. In this paper, we extend the high-resolution ODE framework to fill this gap. By introducing a temporal scaling factor $t^{\frac{r+1}{3}}$ (or $k^{\frac{r+1}{3}}$ in the discrete setting) into the mixed term, we construct new Lyapunov functions specifically tailored for the underdamped case. Notably, when $r = 2$, our construction reduces exactly to the classical Lyapunov functions known from prior analyses. The proposed approach not only characterizes the convergence rate of the minimal squared gradient norm, but also recovers the objective-value convergence rate derived from the low-resolution ODE framework. Moreover, the convergence rates obtained for the underdamped regime depend continuously with respect to the damping parameter $r$. Finally, we demonstrate that the high-resolution ODE continues to faithfully capture the convergence behavior of NAG even in the critically damped regime $r = -1$, whereas the low-resolution ODE degenerates into the conservative Newtonian system and fails to describe the dissipative dynamics. In contrast, the high-resolution ODE framework consistently characterizes the convergence rates in a manner coherent with those obtained for the underdamped case as $r = -1$.