# Last-iterate Convergence Separation between Extra-gradient and Optimism in Constrained Periodic Games

arXiv: https://arxiv.org/abs/2406.10605 · HTML: https://mlsanity.com/paper/2406.10605/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 15 Jun 2024.

## Builds on (most cited first, 13 of 13)

- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Last-Iterate Convergence: Zero-Sum Games and Constrained Min-Max Optimization](https://mlsanity.com/paper/1807.04252/) (2018)
- [Interaction Matters: A Note on Non-asymptotic Local Convergence of Generative Adversarial Networks](https://mlsanity.com/paper/1802.06132/) (2018)
- [On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient](https://mlsanity.com/paper/1803.06573/) (2018)
- [No-Regret Learning in Time-Varying Zero-Sum Games](https://mlsanity.com/paper/2201.12736/) (2022)
- [Multi-agent online learning in time-varying games](https://mlsanity.com/paper/1809.03066/) (2018)
- [On the Convergence of No-Regret Learning Dynamics in Time-Varying Games](https://mlsanity.com/paper/2301.11241/) (2023)
- [Online Learning in Periodic Zero-Sum Games](https://mlsanity.com/paper/2111.03377/) (2021)
- [On the Last-iterate Convergence in Time-varying Zero-sum Games: Extra Gradient Succeeds where Optimism Fails](https://mlsanity.com/paper/2310.02604/) (2023)
- [Forward Looking Best-Response Multiplicative Weights Update Methods for Bilinear Zero-sum Games](https://mlsanity.com/paper/2106.03579/) (2021)

## Built on by (most cited first, 3 of 3)

- [Hamiltonian of polymatrix zero-sum games](https://mlsanity.com/paper/2505.12609/) (2025)
- [Learning from Delayed Feedback in Games via Extra Prediction](https://mlsanity.com/paper/2509.22426/) (2025)
- [Linear Convergence in Games with Delayed Feedback via Extra Prediction](https://mlsanity.com/paper/2602.17486/) (2026)

## Abstract (arXiv)

Last-iterate behaviors of learning algorithms in repeated two-player zero-sum games have been extensively studied due to their wide applications in machine learning and related tasks. Typical algorithms that exhibit the last-iterate convergence property include optimistic and extra-gradient methods. However, most existing results establish these properties under the assumption that the game is time-independent. Recently, (Feng et al, 2023) studied the last-iterate behaviors of optimistic and extra-gradient methods in games with a time-varying payoff matrix, and proved that in an unconstrained periodic game, extra-gradient method converges to the equilibrium while optimistic method diverges. This finding challenges the conventional wisdom that these two methods are expected to behave similarly as they do in time-independent games. However, compared to unconstrained games, games with constrains are more common both in practical and theoretical studies. In this paper, we investigate the last-iterate behaviors of optimistic and extra-gradient methods in the constrained periodic games, demonstrating that similar separation results for last-iterate convergence also hold in this setting.