# Understanding Lookahead Dynamics Through Laplace Transform

arXiv: https://arxiv.org/abs/2506.13712 · HTML: https://mlsanity.com/paper/2506.13712/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 16 Jun 2025.

## Builds on (most cited first, 7 of 7)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Taming GANs with Lookahead-Minmax](https://mlsanity.com/paper/2006.14567/) (2020)
- [Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations](https://mlsanity.com/paper/2112.13826/) (2021)
- [Frequency-Domain Representation of First-Order Methods: A Simple and Robust Framework of Analysis](https://mlsanity.com/paper/2109.04603/) (2021)

## Abstract (arXiv)

We introduce a frequency-domain framework for convergence analysis of hyperparameters in game optimization, leveraging High-Resolution Differential Equations (HRDEs) and Laplace transforms. Focusing on the Lookahead algorithm--characterized by gradient steps k and averaging coefficient α--we transform the discrete-time oscillatory dynamics of bilinear games into the frequency domain to derive precise convergence criteria. Our higher-precision O(γ2)-HRDE models yield tighter criteria, while our first-order O(γ)-HRDE models offer practical guidance by prioritizing actionable hyperparameter tuning over complex closed-form solutions. Empirical validation in discrete-time settings demonstrates the effectiveness of our approach, which may further extend to locally linear operators, offering a scalable framework for selecting hyperparameters for learning in games.