Optimism Without Regularization: Constant Regret in Zero-Sum Games

This paper studies the optimistic variant of Fictitious Play for learning in two-player zero-sum games. While it is known that Optimistic FTRL -- a regularized algorithm with a bounded stepsize parameter -- obtains constant regret in this setting, we show for the first time that similar, optimal rates are also achievable without regularization: we prove for two-strategy games that Optimistic Fictitious Play (using any tiebreaking rule) obtains only constant regret, providing surprising new evidence on the ability of non-no-regret algorithms for fast learning in games. Our proof technique leverages a geometric view of Optimistic Fictitious Play in the dual space of payoff vectors, where we show a certain energy function of the iterates remains bounded over time. Additionally, we also prove a regret lower bound of $\Omega(\sqrt{T})$ for Alternating Fictitious Play. In the unregularized regime, this separates the ability of optimism and alternation in achieving $o(\sqrt{T})$ regret.

@article{lazarsfeld2025_2506.16736,
  title={ Optimism Without Regularization: Constant Regret in Zero-Sum Games },
  author={ John Lazarsfeld and Georgios Piliouras and Ryann Sim and Stratis Skoulakis },
  journal={arXiv preprint arXiv:2506.16736},
  year={ 2025 }
}