# Lyapunov Exponents for Diversity in Differentiable Games

arXiv: https://arxiv.org/abs/2112.14570 · HTML: https://mlsanity.com/paper/2112.14570/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 62 of its 122 references that are in the map.
- First posted on arXiv 24 Dec 2021.

## Builds on (most cited first, 30 of 62)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [Regularized Evolution for Image Classifier Architecture Search](https://mlsanity.com/paper/1802.01548/) (2018)
- [Diversity is All You Need: Learning Skills without a Reward Function](https://mlsanity.com/paper/1802.06070/) (2018)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [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)
- [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)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Learning with Opponent-Learning Awareness](https://mlsanity.com/paper/1709.04326/) (2017)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Improving Adversarial Robustness via Promoting Ensemble Diversity](https://mlsanity.com/paper/1901.08846/) (2019)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)
- [Robots that can adapt like animals](https://mlsanity.com/paper/1407.3501/) (2014)
- ["Other-Play" for Zero-Shot Coordination](https://mlsanity.com/paper/2003.02979/) (2020)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [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)
- [Effective Diversity in Population Based Reinforcement Learning](https://mlsanity.com/paper/2002.00632/) (2020)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [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)
- [Open-ended Learning in Symmetric Zero-sum Games](https://mlsanity.com/paper/1901.08106/) (2019)

## Built on by (most cited first, 8 of 8)

- [Task Discovery: Finding the Tasks that Neural Networks Generalize on](https://mlsanity.com/paper/2212.00261/) (2022)
- [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)
- [Neural Payoff Machines: Predicting Fair and Stable Payoff Allocations Among Team Members](https://mlsanity.com/paper/2208.08798/) (2022)
- [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)
- [Using Cooperative Game Theory to Prune Neural Networks](https://mlsanity.com/paper/2311.10468/) (2023)

## Abstract (arXiv)

Ridge Rider (RR) is an algorithm for finding diverse solutions to optimization problems by following eigenvectors of the Hessian ("ridges"). RR is designed for conservative gradient systems (i.e., settings involving a single loss function), where it branches at saddles - easy-to-find bifurcation points. We generalize this idea to non-conservative, multi-agent gradient systems by proposing a method - denoted Generalized Ridge Rider (GRR) - for finding arbitrary bifurcation points. We give theoretical motivation for our method by leveraging machinery from the field of dynamical systems. We construct novel toy problems where we can visualize new phenomena while giving insight into high-dimensional problems of interest. Finally, we empirically evaluate our method by finding diverse solutions in the iterated prisoners' dilemma and relevant machine learning problems including generative adversarial networks.