# Tight Regret Upper and Lower Bounds for Optimistic Hedge in Two-Player Zero-Sum Games

arXiv: https://arxiv.org/abs/2510.11691 · HTML: https://mlsanity.com/paper/2510.11691/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 33 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Builds on (most cited first, 20 of 20)

- [DeepStack: Expert-Level Artificial Intelligence in No-Limit Poker](https://mlsanity.com/paper/1701.01724/) (2017)
- [Nash Learning from Human Feedback](https://mlsanity.com/paper/2312.00886/) (2023)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [A Minimaximalist Approach to Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2401.04056/) (2024)
- [Mastering the Game of Stratego with Model-Free Multiagent Reinforcement Learning](https://mlsanity.com/paper/2206.15378/) (2022)
- [More Adaptive Algorithms for Adversarial Bandits](https://mlsanity.com/paper/1801.03265/) (2018)
- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
- [Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems](https://mlsanity.com/paper/1808.02901/) (2018)
- [Accelerated Algorithms for Smooth Convex-Concave Minimax Problems with O(1/k2) Rate on Squared Gradient Norm](https://mlsanity.com/paper/2102.07922/) (2021)
- [Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2002.00057/) (2020)
- [Hedging in games: Faster convergence of external and swap regrets](https://mlsanity.com/paper/2006.04953/) (2020)
- [Adaptive Learning in Continuous Games: Optimal Regret Bounds and Convergence to Nash Equilibrium](https://mlsanity.com/paper/2104.12761/) (2021)
- [Tight last-iterate convergence rates for no-regret learning in multi-player games](https://mlsanity.com/paper/2010.13724/) (2020)
- [Near-Optimal No-Regret Learning for Correlated Equilibria in Multi-Player General-Sum Games](https://mlsanity.com/paper/2111.06008/) (2021)
- [Uncoupled Learning Dynamics with O(log T) Swap Regret in Multiplayer Games](https://mlsanity.com/paper/2204.11417/) (2022)
- [Instance-Dependent Regret Bounds for Learning Two-Player Zero-Sum Games with Bandit Feedback](https://mlsanity.com/paper/2502.17625/) (2025)
- [From Average-Iterate to Last-Iterate Convergence in Games: A Reduction and Its Applications](https://mlsanity.com/paper/2506.03464/) (2025)
- [Corrupted Learning Dynamics in Games](https://mlsanity.com/paper/2412.07120/) (2024)
- [Can Reinforcement Learning Solve Asymmetric Combinatorial-Continuous Zero-Sum Games?](https://mlsanity.com/paper/2502.01252/) (2025)

## Built on by (most cited first, 2 of 2)

- [Constant regret in general games via higher-order optimism](https://mlsanity.com/paper/2609.04113/) (2026)
- [Scale-Invariant Fast Convergence in Games](https://mlsanity.com/paper/2602.11857/) (2026)

## Abstract (arXiv)

In two-player zero-sum games, the learning dynamic based on optimistic Hedge achieves one of the best-known regret upper bounds among strongly-uncoupled learning dynamics. With an appropriately chosen learning rate, the social and individual regrets can be bounded by O(log(mn)) in terms of the numbers of actions m and n of the two players. This study investigates the optimality of the dependence on m and n in the regret of optimistic Hedge. To this end, we begin by refining existing regret analysis and show that, in the strongly-uncoupled setting where the opponent's number of actions is known, both the social and individual regret bounds can be improved to O(√log m log n). In this analysis, we express the regret upper bound as an optimization problem with respect to the learning rates and the coefficients of certain negative terms, enabling refined analysis of the leading constants. We then show that the existing social regret bound as well as these new social and individual regret upper bounds cannot be further improved for optimistic Hedge by providing algorithm-dependent individual regret lower bounds. Importantly, these social regret upper and lower bounds match exactly including the constant factor in the leading term. Finally, building on these results, we improve the last-iterate convergence rate and the dynamic regret of a learning dynamic based on optimistic Hedge, and complement these bounds with algorithm-dependent dynamic regret lower bounds that match the improved bounds.