An $O(s^r)$-Resolution ODE Framework for Understanding Discrete-Time Algorithms and Applications to the Linear Convergence of Minimax Problems

Understand

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs).

• Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a \emph{suitable} ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a DTA and its corresponding ODEs.
• In this paper, we propose a new machinery -- an $O(s^r)$-resolution ODE framework -- for analyzing the behavior of a generic DTA, which (partially) answers the above two questions.
• The framework contains three steps: 1.