# On the Suboptimality of Negative Momentum for Minimax Optimization

arXiv: https://arxiv.org/abs/2008.07459 · HTML: https://mlsanity.com/paper/2008.07459/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Aug 2020.

## Built on by (most cited first, 11 of 11)

- [Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization](https://mlsanity.com/paper/2102.09468/) (2021)
- [Proximal Gradient Descent-Ascent: Variable Convergence under KL Geometry](https://mlsanity.com/paper/2102.04653/) (2021)
- [On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization](https://mlsanity.com/paper/2406.04558/) (2024)
- [Complex Momentum for Optimization in Games](https://mlsanity.com/paper/2102.08431/) (2021)
- [Negative Stepsizes Make Gradient-Descent-Ascent Converge](https://mlsanity.com/paper/2505.01423/) (2025)
- [A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimization](https://mlsanity.com/paper/2110.07098/) (2021)
- [Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning](https://mlsanity.com/paper/2112.11663/) (2021)
- [Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games](https://mlsanity.com/paper/2505.19537/) (2025)
- [Hamiltonian of polymatrix zero-sum games](https://mlsanity.com/paper/2505.12609/) (2025)
- [GDA-AM: On the effectiveness of solving minimax optimization via Anderson Acceleration](https://mlsanity.com/paper/2110.02457/) (2021)
- [Negative Momentum for Convex-Concave Optimization](https://mlsanity.com/paper/2604.17145/) (2026)

## Abstract (arXiv)

Smooth game optimization has recently attracted great interest in machine learning as it generalizes the single-objective optimization paradigm. However, game dynamics is more complex due to the interaction between different players and is therefore fundamentally different from minimization, posing new challenges for algorithm design. Notably, it has been shown that negative momentum is preferred due to its ability to reduce oscillation in game dynamics. Nevertheless, the convergence rate of negative momentum was only established in simple bilinear games. In this paper, we extend the analysis to smooth and strongly-convex strongly-concave minimax games by taking the variational inequality formulation. By connecting momentum method with Chebyshev polynomials, we show that negative momentum accelerates convergence of game dynamics locally, though with a suboptimal rate. To the best of our knowledge, this is the first work that provides an explicit convergence rate for negative momentum in this setting.