# Understanding Accelerated Gradient Methods: Lyapunov Analyses and Hamiltonian Assisted Interpretations

arXiv: https://arxiv.org/abs/2304.10063 · HTML: https://mlsanity.com/paper/2304.10063/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 22 references that are in the map.
- First posted on arXiv 20 Apr 2023.

## Builds on (most cited first, 11 of 11)

- [Katyusha: The First Direct Acceleration of Stochastic Gradient Methods](https://mlsanity.com/paper/1603.05953/) (2016)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent](https://mlsanity.com/paper/1407.1537/) (2014)
- [Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization](https://mlsanity.com/paper/1409.3257/) (2014)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [A geometric alternative to Nesterov's accelerated gradient descent](https://mlsanity.com/paper/1506.08187/) (2015)
- [Direct Runge-Kutta Discretization Achieves Acceleration](https://mlsanity.com/paper/1805.00521/) (2018)
- [On the oracle complexity of smooth strongly convex minimization](https://mlsanity.com/paper/2101.09740/) (2021)
- [An optimal gradient method for smooth strongly convex minimization](https://mlsanity.com/paper/2101.09741/) (2021)
- [An optimal first order method based on optimal quadratic averaging](https://mlsanity.com/paper/1604.06543/) (2016)
- [The Physical Systems Behind Optimization Algorithms](https://mlsanity.com/paper/1612.02803/) (2016)

## Abstract (arXiv)

We formulate two classes of first-order algorithms more general than previously studied for minimizing smooth and strongly convex or, respectively, smooth and convex functions. We establish sufficient conditions, via new discrete Lyapunov analyses, for achieving accelerated convergence rates which match Nesterov's methods in the strongly and general convex settings. Next, we study the convergence of limiting ordinary differential equations (ODEs) and point out currently notable gaps between the convergence properties of the corresponding algorithms and ODEs. Finally, we propose a novel class of discrete algorithms, called the Hamiltonian assisted gradient method, directly based on a Hamiltonian function and several interpretable operations, and then demonstrate meaningful and unified interpretations of our acceleration conditions.