# Complex Momentum for Optimization in Games

arXiv: https://arxiv.org/abs/2102.08431 · HTML: https://mlsanity.com/paper/2102.08431/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 104 references that are in the map.
- First posted on arXiv 16 Feb 2021.

## Builds on (most cited first, 30 of 59)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Meta-Learning for Semi-Supervised Few-Shot Classification](https://mlsanity.com/paper/1803.00676/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Adversarial Examples: Attacks and Defenses for Deep Learning](https://mlsanity.com/paper/1712.07107/) (2017)
- [Lookahead Optimizer: k steps forward, 1 step back](https://mlsanity.com/paper/1907.08610/) (2019)
- [Emergent Tool Use From Multi-Agent Autocurricula](https://mlsanity.com/paper/1909.07528/) (2019)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Learning with Opponent-Learning Awareness](https://mlsanity.com/paper/1709.04326/) (2017)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [Intrinsic Motivation and Automatic Curricula via Asymmetric Self-Play](https://mlsanity.com/paper/1703.05407/) (2017)
- [The Numerics of GANs](https://mlsanity.com/paper/1705.10461/) (2017)
- [Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions](https://mlsanity.com/paper/1903.03088/) (2019)
- [On Empirical Comparisons of Optimizers for Deep Learning](https://mlsanity.com/paper/1910.05446/) (2019)
- [Open-ended Learning in Symmetric Zero-sum Games](https://mlsanity.com/paper/1901.08106/) (2019)
- [Stochastic Hyperparameter Optimization through Hypernetworks](https://mlsanity.com/paper/1802.09419/) (2018)
- [A Game Theoretic Framework for Model Based Reinforcement Learning](https://mlsanity.com/paper/2004.07804/) (2020)
- [Real World Games Look Like Spinning Tops](https://mlsanity.com/paper/2004.09468/) (2020)
- [Reducing Noise in GAN Training with Variance Reduced Extragradient](https://mlsanity.com/paper/1904.08598/) (2019)
- [From Poincar'e Recurrence to Convergence in Imperfect Information Games: Finding Equilibrium via Regularization](https://mlsanity.com/paper/2002.08456/) (2020)
- [Negative Momentum for Improved Game Dynamics](https://mlsanity.com/paper/1807.04740/) (2018)
- [Connecting Generative Adversarial Networks and Actor-Critic Methods](https://mlsanity.com/paper/1610.01945/) (2016)
- [Finding Mixed Nash Equilibria of Generative Adversarial Networks](https://mlsanity.com/paper/1811.02002/) (2018)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)

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

- [ATT3D: Amortized Text-to-3D Object Synthesis](https://mlsanity.com/paper/2306.07349/) (2023)
- [The Optimal Noise in Noise-Contrastive Learning Is Not What You Think](https://mlsanity.com/paper/2203.01110/) (2022)
- [MomentumSMoE: Integrating Momentum into Sparse Mixture of Experts](https://mlsanity.com/paper/2410.14574/) (2024)
- [Domain Adversarial Training: A Game Perspective](https://mlsanity.com/paper/2202.05352/) (2022)
- [Task Selection for AutoML System Evaluation](https://mlsanity.com/paper/2208.12754/) (2022)
- [JacNet: Learning Functions with Structured Jacobians](https://mlsanity.com/paper/2408.13237/) (2024)
- [Score Distillation Sampling for Audio: Source Separation, Synthesis, and Beyond](https://mlsanity.com/paper/2505.04621/) (2025)
- [Motion Attribution for Video Generation](https://mlsanity.com/paper/2601.08828/) (2026)
- [Expert Merging in Sparse Mixture of Experts with Nash Bargaining](https://mlsanity.com/paper/2510.16138/) (2025)
- [When is Momentum Extragradient Optimal? A Polynomial-Based Analysis](https://mlsanity.com/paper/2211.04659/) (2022)
- [Negative Momentum for Convex-Concave Optimization](https://mlsanity.com/paper/2604.17145/) (2026)

## Abstract (arXiv)

We generalize gradient descent with momentum for optimization in differentiable games to have complex-valued momentum. We give theoretical motivation for our method by proving convergence on bilinear zero-sum games for simultaneous and alternating updates. Our method gives real-valued parameter updates, making it a drop-in replacement for standard optimizers. We empirically demonstrate that complex-valued momentum can improve convergence in realistic adversarial games - like generative adversarial networks - by showing we can find better solutions with an almost identical computational cost. We also show a practical generalization to a complex-valued Adam variant, which we use to train BigGAN to better inception scores on CIFAR-10.