# Accelerating Rescaled Gradient Descent: Fast Optimization of Smooth Functions

arXiv: https://arxiv.org/abs/1902.08825 · HTML: https://mlsanity.com/paper/1902.08825/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2019.

## Built on by (most cited first, 8 of 8)

- [Leveraging Non-uniformity in First-order Non-convex Optimization](https://mlsanity.com/paper/2105.06072/) (2021)
- [Almost sure convergence rates of stochastic gradient methods under gradient domination](https://mlsanity.com/paper/2405.13592/) (2024)
- [ODE-based Learning to Optimize](https://mlsanity.com/paper/2406.02006/) (2024)
- [Armijo Line-search Can Make (Stochastic) Gradient Descent Provably Faster](https://mlsanity.com/paper/2503.00229/) (2025)
- [Fast symplectic integrator for Nesterov-type acceleration method](https://mlsanity.com/paper/2106.07620/) (2021)
- [AGGLIO: Global Optimization for Locally Convex Functions](https://mlsanity.com/paper/2111.03932/) (2021)
- [Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points](https://mlsanity.com/paper/2212.03765/) (2022)
- [Convergence Guarantees of Gradient Descent for Neural Networks via Generalized Lipschitz Smoothness](https://mlsanity.com/paper/2608.11479/) (2026)

## Abstract (arXiv)

We present a family of algorithms, called descent algorithms, for optimizing convex and non-convex functions. We also introduce a new first-order algorithm, called rescaled gradient descent (RGD), and show that RGD achieves a faster convergence rate than gradient descent provided the function is strongly smooth -- a natural generalization of the standard smoothness assumption on the objective function. When the objective function is convex, we present two novel frameworks for "accelerating" descent methods, one in the style of Nesterov and the other in the style of Monteiro and Svaiter, using a single Lyapunov. Rescaled gradient descent can be accelerated under the same strong smoothness assumption using both frameworks. We provide several examples of strongly smooth loss functions in machine learning and numerical experiments that verify our theoretical findings. We also present several extensions of our novel Lyapunov framework, including deriving optimal universal tensor methods and extending our framework to the coordinate setting.