# Discretization Drift in Two-Player Games

arXiv: https://arxiv.org/abs/2105.13922 · HTML: https://mlsanity.com/paper/2105.13922/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 41 references that are in the map.
- First posted on arXiv 28 May 2021.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Least Squares Generative Adversarial Networks](https://mlsanity.com/paper/1611.04076/) (2016)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [Implicit Gradient Regularization](https://mlsanity.com/paper/2009.11162/) (2020)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [The Numerics of GANs](https://mlsanity.com/paper/1705.10461/) (2017)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [A Game Theoretic Framework for Model Based Reinforcement Learning](https://mlsanity.com/paper/2004.07804/) (2020)
- [Gradient descent GAN optimization is locally stable](https://mlsanity.com/paper/1706.04156/) (2017)
- [Catastrophic Fisher Explosion: Early Phase Fisher Matrix Impacts Generalization](https://mlsanity.com/paper/2012.14193/) (2020)
- [On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Games](https://mlsanity.com/paper/1901.00838/) (2019)
- [On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach](https://mlsanity.com/paper/1910.07512/) (2019)
- [Competitive Gradient Descent](https://mlsanity.com/paper/1905.12103/) (2019)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [On dissipative symplectic integration with applications to gradient-based optimization](https://mlsanity.com/paper/2004.06840/) (2020)
- [Training Generative Adversarial Networks by Solving Ordinary Differential Equations](https://mlsanity.com/paper/2010.15040/) (2020)
- [Nash Convergence of Gradient Dynamics in Iterated General-Sum Games](https://mlsanity.com/paper/1301.3892/) (2013)
- [Implicit competitive regularization in GANs](https://mlsanity.com/paper/1910.05852/) (2019)

## Built on by (most cited first, 14 of 14)

- [Why neural networks find simple solutions: the many regularizers of geometric complexity](https://mlsanity.com/paper/2209.13083/) (2022)
- [On the Implicit Bias of Adam](https://mlsanity.com/paper/2309.00079/) (2023)
- [On a continuous time model of gradient descent dynamics and instability in deep learning](https://mlsanity.com/paper/2302.01952/) (2023)
- [Deep Fusion: Efficient Network Training via Pre-trained Initializations](https://mlsanity.com/paper/2306.11903/) (2023)
- [Implicit biases in multitask and continual learning from a backward error analysis perspective](https://mlsanity.com/paper/2311.00235/) (2023)
- [First-Order Predictable but Pairwise Fragile: Local Task Adaptation in Trained Transformers](https://mlsanity.com/paper/2607.16821/) (2026)
- [Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games](https://mlsanity.com/paper/2505.19537/) (2025)
- [Implicit regularisation in stochastic gradient descent: from single-objective to two-player games](https://mlsanity.com/paper/2307.05789/) (2023)
- [Continuous-time Analysis for Variational Inequalities: An Overview and Desiderata](https://mlsanity.com/paper/2207.07105/) (2022)
- [Learning by solving differential equations](https://mlsanity.com/paper/2505.13397/) (2025)
- [Majority Kernels: An Approach to Leverage Big Model Dynamics for Efficient Small Model Training](https://mlsanity.com/paper/2402.05033/) (2024)
- [Corridor Geometry in Gradient-Based Optimization](https://mlsanity.com/paper/2402.08818/) (2024)
- [Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach](https://mlsanity.com/paper/2605.19392/) (2026)
- [Predicting Block-Coordinate Performance via Cross-Curvature](https://mlsanity.com/paper/2609.33489/) (2026)

## Abstract (arXiv)

Gradient-based methods for two-player games produce rich dynamics that can solve challenging problems, yet can be difficult to stabilize and understand. Part of this complexity originates from the discrete update steps given by simultaneous or alternating gradient descent, which causes each player to drift away from the continuous gradient flow -- a phenomenon we call discretization drift. Using backward error analysis, we derive modified continuous dynamical systems that closely follow the discrete dynamics. These modified dynamics provide an insight into the notorious challenges associated with zero-sum games, including Generative Adversarial Networks. In particular, we identify distinct components of the discretization drift that can alter performance and in some cases destabilize the game. Finally, quantifying discretization drift allows us to identify regularizers that explicitly cancel harmful forms of drift or strengthen beneficial forms of drift, and thus improve performance of GAN training.