# Gradient Methods with Online Scaling Part II. Practical Aspects

arXiv: https://arxiv.org/abs/2509.11007 · HTML: https://mlsanity.com/paper/2509.11007/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 13 Sep 2025.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Online Learning Rate Adaptation with Hypergradient Descent](https://mlsanity.com/paper/1703.04782/) (2017)
- [Revisiting the Polyak step size](https://mlsanity.com/paper/1905.00313/) (2019)
- [Acceleration Methods](https://mlsanity.com/paper/2101.09545/) (2021)
- [Stochastic Polyak Stepsize with a Moving Target](https://mlsanity.com/paper/2106.11851/) (2021)
- [Lyapunov Functions for First-Order Methods: Tight Automated Convergence Guarantees](https://mlsanity.com/paper/1803.06073/) (2018)
- [Gradient Methods with Online Scaling](https://mlsanity.com/paper/2411.01803/) (2024)
- [Scalable Approximate Optimal Diagonal Preconditioning](https://mlsanity.com/paper/2312.15594/) (2023)
- [Non-monotone Behavior of the Heavy Ball Method](https://mlsanity.com/paper/1811.00658/) (2018)
- [Automated tight Lyapunov analysis for first-order methods](https://mlsanity.com/paper/2302.06713/) (2023)
- [Provable and Practical Online Learning Rate Adaptation with Hypergradient Descent](https://mlsanity.com/paper/2502.11229/) (2025)
- [Gradient Methods with Online Scaling Part I. Theoretical Foundations](https://mlsanity.com/paper/2505.23081/) (2025)
- [Gradient methods exploiting spectral properties](https://mlsanity.com/paper/1905.03870/) (2019)

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

- [Optimizing the Preconditioner: A Black-box Online-to-Nonconvex Conversion with Static Regret Minimization Oracles](https://mlsanity.com/paper/2607.17607/) (2026)

## Abstract (arXiv)

Part I of this work [Gao25] establishes online scaled gradient methods (OSGM), a framework that utilizes online convex optimization to adapt stepsizes in gradient methods. This paper focuses on the practical aspects of OSGM. We leverage the OSGM framework to design new adaptive first-order methods and provide insights into their empirical behavior. The resulting method, OSGM-Best, matches the performance of quasi-Newton variants while requiring less memory and cheaper iterations. We also extend OSGM to nonconvex optimization and outline directions that connect OSGM to existing branches of optimization theory and practice.