An $$O(s^r)$$-resolution ODE framework for understanding discrete-time algorithms and applications to the linear convergence of minimax problems

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how obtain suitable ODE from given DTA, (ii) connection between convergence DTA its corresponding ODEs. In this paper, we propose new machinery—an $$O(s^r)$$ -resolution framework—for analyzing behavior generic which (partially) answers above questions. The framework contains three steps: 1. To define hierarchy ODEs parameterized by degree r, where s step-size DTA. We present principal approach construct unique DTA; 2. analyze resulting ODE, -linear-convergence condition with respect an energy function, under converges linearly optimal solution; 3. bridge properties ODEs, properness function show that linear proper can automatically guarantee better illustrate machinery, utilize study classic algorithms—gradient descent ascent (GDA), proximal point method (PPM) extra-gradient (EGM)—for solving unconstrained minimax problem $$\min _{x\in \mathbb {R}^n} \max _{y\in {R}^m} L(x,y)$$ . Their O(s)-resolution explain puzzling convergent/divergent behaviors GDA, PPM EGM when L(x, y) bilinear showcase interaction terms help PPM/EGM but hurts GDA. Furthermore, their O(s)-linear-convergence conditions not only unify known scenarios have convergence, also these exhibit in much broader contexts, including class nonconvex-nonconcave problems. Finally, design optimization for problems, studying difference GDA PPM/EGM.