Essential Convergence Rates of Continuous-Time Models for Optimization Methods

Designing and analyzing optimization methods via continuous-time models expressed as ordinary differential equations (ODEs) is a promising approach for its intuitiveness and simplicity. A key concern, however, is that the convergence rates of such models can be arbitrarily modified by time rescaling, rendering the task of seeking ODEs with ``fast’’ convergence meaningless. To eliminate this ambiguity of the rates, we introduce the notion of the essential convergence rate. We justify this notion by proving that, under appropriate assumptions on discretization, no method obtained by discretizing an ODE can achieve a faster rate than its essential convergence rate.