# Negative Momentum for Improved Game Dynamics

arXiv: https://arxiv.org/abs/1807.04740 · HTML: https://mlsanity.com/paper/1807.04740/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2018.

## Built on by (most cited first, 30 of 60)

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- [Tight Last-Iterate Convergence of the Extragradient and the Optimistic Gradient Descent-Ascent Algorithm for Constrained Monotone Variational Inequalities](https://mlsanity.com/paper/2204.09228/) (2022)
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- [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)
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- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Sampling without Replacement Leads to Faster Rates in Finite-Sum Minimax Optimization](https://mlsanity.com/paper/2206.02953/) (2022)
- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [On the One-sided Convergence of Adam-type Algorithms in Non-convex Non-concave Min-max Optimization](https://mlsanity.com/paper/2109.14213/) (2021)
- [Learning (Very) Simple Generative Models Is Hard](https://mlsanity.com/paper/2205.16003/) (2022)
- [MomentumSMoE: Integrating Momentum into Sparse Mixture of Experts](https://mlsanity.com/paper/2410.14574/) (2024)
- [Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations](https://mlsanity.com/paper/2112.13826/) (2021)
- [Lyapunov Exponents for Diversity in Differentiable Games](https://mlsanity.com/paper/2112.14570/) (2021)
- [Direct-Search for a Class of Stochastic Min-Max Problems](https://mlsanity.com/paper/2102.11386/) (2021)
- [Domain Adversarial Training: A Game Perspective](https://mlsanity.com/paper/2202.05352/) (2022)
- [Minimax Optimality (Probably) Doesn't Imply Distribution Learning for GANs](https://mlsanity.com/paper/2201.07206/) (2022)
- [Score Distillation Sampling for Audio: Source Separation, Synthesis, and Beyond](https://mlsanity.com/paper/2505.04621/) (2025)
- [On the Initialization for Convex-Concave Min-max Problems](https://mlsanity.com/paper/2103.00284/) (2021)
- [Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning](https://mlsanity.com/paper/2112.11663/) (2021)
- [Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics](https://mlsanity.com/paper/2203.14129/) (2022)
- [Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization](https://mlsanity.com/paper/2206.08573/) (2022)
- [STay-ON-the-Ridge: Guaranteed Convergence to Local Minimax Equilibrium in Nonconvex-Nonconcave Games](https://mlsanity.com/paper/2210.09769/) (2022)

## Abstract (arXiv)

Games generalize the single-objective optimization paradigm by introducing different objective functions for different players. Differentiable games often proceed by simultaneous or alternating gradient updates. In machine learning, games are gaining new importance through formulations like generative adversarial networks (GANs) and actor-critic systems. However, compared to single-objective optimization, game dynamics are more complex and less understood. In this paper, we analyze gradient-based methods with momentum on simple games. We prove that alternating updates are more stable than simultaneous updates. Next, we show both theoretically and empirically that alternating gradient updates with a negative momentum term achieves convergence in a difficult toy adversarial problem, but also on the notoriously difficult to train saturating GANs.