Research Paper#Optimization, Ordinary Differential Equations (ODEs), Convergence Rates🔬 ResearchAnalyzed: Jan 3, 2026 19:00
Essential Convergence Rates in Optimization ODEs
Published:Dec 29, 2025 09:09
•1 min read
•ArXiv
Analysis
This paper addresses a fundamental issue in the analysis of optimization methods using continuous-time models (ODEs). The core problem is that the convergence rates of these ODE models can be misleading due to time rescaling. The paper introduces the concept of 'essential convergence rate' to provide a more robust and meaningful measure of convergence. The significance lies in establishing a lower bound on the convergence rate achievable by discretizing the ODE, thus providing a more reliable way to compare and evaluate different optimization methods based on their continuous-time representations.
Key Takeaways
- •Addresses the ambiguity in convergence rates of ODE-based optimization methods due to time rescaling.
- •Introduces the concept of 'essential convergence rate' to provide a more meaningful measure.
- •Proves that discretization methods cannot surpass the essential convergence rate, establishing a lower bound.
- •Provides a more reliable way to compare and evaluate optimization methods based on their continuous-time representations.
Reference
“The paper introduces the notion of the essential convergence rate and justifies it 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.”