# On the stability of gradient descent with second order dynamics for time-varying cost functions

arXiv: https://arxiv.org/abs/2405.13765 · HTML: https://mlsanity.com/paper/2405.13765/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 44 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 20 of 20)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [A Unified Analysis of Stochastic Momentum Methods for Deep Learning](https://mlsanity.com/paper/1808.10396/) (2018)
- [Stability and Convergence Trade-off of Iterative Optimization Algorithms](https://mlsanity.com/paper/1804.01619/) (2018)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Introduction to Online Control](https://mlsanity.com/paper/2211.09619/) (2022)
- [Algorithmic Instabilities of Accelerated Gradient Descent](https://mlsanity.com/paper/2102.02167/) (2021)
- [Connections Between Adaptive Control and Optimization in Machine Learning](https://mlsanity.com/paper/1904.05856/) (2019)
- [Understanding Nesterov's Acceleration via Proximal Point Method](https://mlsanity.com/paper/2005.08304/) (2020)
- [A Stable High-order Tuner for General Convex Functions](https://mlsanity.com/paper/2011.09996/) (2020)
- [Accelerated Learning with Robustness to Adversarial Regressors](https://mlsanity.com/paper/2005.01529/) (2020)
- [Variations on Barbalat's Lemma](https://mlsanity.com/paper/1411.1611/) (2014)

## Abstract (arXiv)

Gradient based optimization algorithms deployed in Machine Learning (ML) applications are often analyzed and compared by their convergence rates or regret bounds. While these rates and bounds convey valuable information they don't always directly translate to stability guarantees. Stability and similar concepts, like robustness, will become ever more important as we move towards deploying models in real-time and safety critical systems. In this work we build upon the results in Gaudio et al. 2021 and Moreu & Annaswamy 2022 for gradient descent with second order dynamics when applied to explicitly time varying cost functions and provide more general stability guarantees. These more general results can aid in the design and certification of these optimization schemes so as to help ensure safe and reliable deployment for real-time learning applications. We also hope that the techniques provided here will stimulate and cross-fertilize the analysis that occurs on the same algorithms from the online learning and stochastic optimization communities.