# Continuous-Time Analysis of Accelerated Gradient Methods via Conservation Laws in Dilated Coordinate Systems

arXiv: https://arxiv.org/abs/2202.05501 · HTML: https://mlsanity.com/paper/2202.05501/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2022.

## Built on by (most cited first, 4 of 4)

- [Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points](https://mlsanity.com/paper/2212.03765/) (2022)
- [A Passivity-Based Method for Accelerated Convex Optimisation](https://mlsanity.com/paper/2306.11474/) (2023)
- [Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods](https://mlsanity.com/paper/2409.00913/) (2024)
- [Distributed Optimization via Energy Conservation Laws in Dilated Coordinates](https://mlsanity.com/paper/2409.19279/) (2024)

## Abstract (arXiv)

We analyze continuous-time models of accelerated gradient methods through deriving conservation laws in dilated coordinate systems. Namely, instead of analyzing the dynamics of X(t), we analyze the dynamics of W(t)=tα(X(t)-Xc) for some α and Xc and derive a conserved quantity, analogous to physical energy, in this dilated coordinate system. Through this methodology, we recover many known continuous-time analyses in a streamlined manner and obtain novel continuous-time analyses for OGM-G, an acceleration mechanism for efficiently reducing gradient magnitude that is distinct from that of Nesterov. Finally, we show that a semi-second-order symplectic Euler discretization in the dilated coordinate system leads to an O(1/k2) rate on the standard setup of smooth convex minimization, without any further assumptions such as infinite differentiability.