# Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization

arXiv: https://arxiv.org/abs/2311.10053 · HTML: https://mlsanity.com/paper/2311.10053/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 16 Nov 2023.

## Builds on (most cited first, 14 of 14)

- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [First-order optimization algorithms via inertial systems with Hessian driven damping](https://mlsanity.com/paper/1907.10536/) (2019)
- [Fast convex optimization via inertial dynamics with Hessian driven damping](https://mlsanity.com/paper/1601.07113/) (2016)
- [Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3](https://mlsanity.com/paper/1706.05671/) (2017)
- [From the Ravine method to the Nesterov method and vice versa: a dynamical system perspective](https://mlsanity.com/paper/2201.11643/) (2022)
- [Optimal convergence rates for Nesterov acceleration](https://mlsanity.com/paper/1805.05719/) (2018)
- [An extension of the second order dynamical system that models Nesterov's convex gradient method](https://mlsanity.com/paper/1908.02574/) (2019)
- [A Control-Theoretic Perspective on Optimal High-Order Optimization](https://mlsanity.com/paper/1912.07168/) (2019)
- [Fast optimization via inertial dynamics with closed-loop damping](https://mlsanity.com/paper/2008.02261/) (2020)
- [A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert spaces, with complexity O(1/n^2)](https://mlsanity.com/paper/1502.04286/) (2015)
- [Fast convex optimization via closed-loop time scaling of gradient dynamics](https://mlsanity.com/paper/2301.00701/) (2023)
- [Asymptotic for a second order evolution equation with convex potential and vanishing damping term](https://mlsanity.com/paper/1509.05598/) (2015)

## Built on by (most cited first, 1 of 1)

- [Stochastic Adaptive Gradient Descent Without Descent](https://mlsanity.com/paper/2509.14969/) (2025)

## Abstract (arXiv)

We introduce an autonomous system with closed-loop damping for first-order convex optimization. While, to this day, optimal rates of convergence are almost exclusively achieved by non-autonomous methods via open-loop damping (e.g., Nesterov's algorithm), we show that our system, featuring a closed-loop damping, exhibits a rate arbitrarily close to the optimal one. We do so by coupling the damping and the speed of convergence of the system via a well-chosen Lyapunov function. By discretizing our system we then derive an algorithm and present numerical experiments supporting our theoretical findings.