# Frequency-Domain Representation of First-Order Methods: A Simple and Robust Framework of Analysis

arXiv: https://arxiv.org/abs/2109.04603 · HTML: https://mlsanity.com/paper/2109.04603/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2021.

## Built on by (most cited first, 3 of 3)

- [Adaptively Perturbed Mirror Descent for Learning in Games](https://mlsanity.com/paper/2305.16610/) (2023)
- [Understanding Lookahead Dynamics Through Laplace Transform](https://mlsanity.com/paper/2506.13712/) (2025)
- [Frequency-Based Hyperparameter Selection in Games](https://mlsanity.com/paper/2601.18409/) (2026)

## Abstract (arXiv)

Motivated by recent applications in min-max optimization, we employ tools from nonlinear control theory in order to analyze a class of "historical" gradient-based methods, for which the next step lies in the span of the previously observed gradients within a time horizon. Specifically, we leverage techniques developed by Hu and Lessard (2017) to build a frequency-domain framework which reduces the analysis of such methods to numerically-solvable algebraic tasks, establishing linear convergence under a class of strongly monotone and co-coercive operators. On the applications' side, we focus on the Optimistic Gradient Descent (OGD) method, which augments the standard Gradient Descent with an additional past-gradient in the optimization step. The proposed framework leads to a simple and sharp analysis of OGD -- and generalizations thereof -- under a much broader regime of parameters. Notably, this characterization directly extends under adversarial noise in the observed value of the gradient. Moreover, our frequency-domain framework provides an exact quantitative comparison between simultaneous and alternating updates of OGD. An interesting byproduct is that OGD -- and variants thereof -- is an instance of PID control, arguably one of the most influential algorithms of the last century; this observation sheds more light to the stabilizing properties of "optimism".