  We study gradient-based optimization methods obtained by direct Runge-Kutta
discretization of the ordinary differential equation (ODE) describing the
movement of a heavy-ball under constant friction coefficient. When the function
is high order smooth and strongly convex, we show that directly simulating the
ODE with known numerical integrators achieve acceleration in a nontrivial
neighborhood of the optimal solution. In particular, the neighborhood can grow
larger as the condition number of the function increases. Furthermore, our
results also hold for nonconvex but quasi-strongly convex objectives. We
provide numerical experiments that verify the theoretical rates predicted by
our results.
