Acceleration in First Order Quasi-strongly Convex Optimization by ODE Discretization

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 may 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.