@inproceedings{
cai2024on,
title={On Tractable \${\textbackslash}Phi\$-Equilibria in Non-Concave Games},
author={Yang Cai and Constantinos Costis Daskalakis and Haipeng Luo and Chen-Yu Wei and Weiqiang Zheng},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=3CtTMF5zzM}
}
While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to a coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when utilities are non-concave -- a common scenario in machine learning applications involving strategies parameterized by deep neural networks, or when agents' utilities are computed by neural networks, or both. Non-concave games introduce significant game-theoretic and optimization challenges: (i) Nash equilibria may not exist; (ii) local Nash equilibria, though they exist, are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria generally have infinite support and are intractable. To sidestep these challenges, we revisit the classical solution concept of $\Phi$-equilibria introduced by Greenwald and Jafari [GJ03], which is guaranteed to exist for an arbitrary set of strategy modifications $\Phi$ even in non-concave games [SL07]. However, the tractability of $\Phi$-equilibria in such games remains elusive. In this paper, we initiate the study of tractable $\Phi$-equilibria in non-concave games and examine several natural families of strategy modifications. We show that when $\Phi$ is finite, there exists an efficient uncoupled learning algorithm that approximates the corresponding $\Phi$-equilibria. Additionally, we explore cases where $\Phi$ is infinite but consists of local modifications, showing that Online Gradient Descent can efficiently approximate $\Phi$-equilibria in non-trivial regimes.
On Tractable Φ-Equilibria in Non-Concave Games
Yang Cai Constantinos Costis Daskalakis Haipeng Luo Chen-Yu Wei Weiqiang Zheng