Essential convergence rate of ordinary differential equations appearing in optimization

Some continuous optimization methods can be connected to ordinary differential equations (ODEs) by taking continuous limits, and their convergence rates can be explained by the ODEs. However, since such ODEs can achieve any convergence rate by time scaling, the correspondence is not as straightforward as usually expected, and deriving new methods through ODEs is not quite direct. In this letter, we pay attention to stability restriction in discretizing ODEs and show that acceleration by time scaling always implies deceleration in discretization; they balance out so that we can define an attainable unique convergence rate which we call an "essential convergence rate".