Understanding Lookahead Dynamics Through Laplace Transform
This provides a scalable framework for hyperparameter selection in game learning, though it appears incremental as it builds on existing HRDE methods.
The authors tackled the problem of analyzing hyperparameter convergence in game optimization by developing a frequency-domain framework using High-Resolution Differential Equations and Laplace transforms, specifically for the Lookahead algorithm, resulting in tighter convergence criteria and practical tuning guidance.
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 $O(γ^2)$-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.