Frequency-Based Hyperparameter Selection in Games

TL;DR

This work addresses hyperparameter tuning for LookAhead in games, where rotational dynamics impede standard optimization. By formulating a frequency-domain, modal stability framework, the authors derive convergence guarantees for LookAhead and introduce Modal LookAhead (MoLA), which adaptively selects the horizon $k$ and averaging weight $\alpha$ to maximize stability based on the problem’s dominant modes. MoLA achieves provable $O(1/T)$ convergence for the averaged iterate while consistently outperforming baselines in bilinear and strongly convex–strongly concave settings, with only lightweight spectral computations per LookAhead cycle. The approach provides practical hyperparameter rules that link phase alignment and amplitude damping of oscillatory modes to stability margins, enabling faster, more robust training in rotational game dynamics with minimal computational overhead. The results suggest broad applicability to GANs and multi-agent RL, and point to future work on scaling spectral estimation and periodic re-tuning in large-scale problems.

Abstract

Learning in smooth games fundamentally differs from standard minimization due to rotational dynamics, which invalidate classical hyperparameter tuning strategies. Despite their practical importance, effective methods for tuning in games remain underexplored. A notable example is LookAhead (LA), which achieves strong empirical performance but introduces additional parameters that critically influence performance. We propose a principled approach to hyperparameter selection in games by leveraging frequency estimation of oscillatory dynamics. Specifically, we analyze oscillations both in continuous-time trajectories and through the spectrum of the discrete dynamics in the associated frequency-based space. Building on this analysis, we introduce \emph{Modal LookAhead (MoLA)}, an extension of LA that selects the hyperparameters adaptively to a given problem. We provide convergence guarantees and demonstrate in experiments that MoLA accelerates training in both purely rotational games and mixed regimes, all with minimal computational overhead.

Figures (11)

• Figure 1: MoLA selects $(k,\alpha)$ so the average iterate is closer to the origin than for random choices. It sets $k$ so GD accumulates roughly a $\pi$ phase shift, then picks $\alpha$ to minimize the LA iterate’s distance to the origin.
• Figure 2: Convergence to equilibrium of EG, OGDA, LA, and MoLA in BG with $d=100, \gamma=0.01$. GD is omitted since it diverges away.
• Figure 3: Convergence against CPU time of EG, OGDA, LA, and MoLA in BG with $d=100, \gamma=0.01$.
• Figure 4: Distance to equilibrium vs. iterations for GD, EG, OGDA, LA, and MoLA in a more rotational setting of SC-SC game with $d=100, \gamma=0.01$. The x-axis reports iteration count; the y-axis reports the Euclidean distance to the Nash equilibrium.
• Figure 5: Optimal LA horizon ($k$ vs. rotation factor $\beta$ in Quadratic Game. Lower $\beta$ (more potential/convex–concave) favors larger horizons, while higher $\beta$ (more rotational; $\beta \rightarrow 1$) favors smaller $k$.

Theorems & Definitions (42)

• Definition 2.1: Variational Inequality (VI)
• Definition 2.2: Monotone Operator
• Definition 2.3: Lipschitz Operator
• Definition 2.4: Nonexpansive operator
• Definition 2.5: Fejér monotonicity bauschke2011convex
• Definition 2.6: Saddle point
• Definition 2.7: Primal--dual gap of average iterate
• Definition 2.8: Real Schur decomposition
• Lemma 1
• proof : Proof sketch