mlsanityData to 14 Oct 2026

Discretization Drift in Two-Player Games

arXiv 2105.13922 ↗First posted 28 May 2021stat.ML cs.LGcited by 14

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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.

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

28 of 41 references.

References · 28

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2016Improved Techniques for Training GANscited by 2,970
  4. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,626
  5. 2017Improved Training of Wasserstein GANscited by 2,235
  6. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,636
  7. 2016Least Squares Generative Adversarial Networkscited by 1,001
  8. 2018Which Training Methods for GANs do actually Converge?cited by 679
  9. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 594
  10. 2015A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insightscited by 206
  11. 2021On the Origin of Implicit Regularization in Stochastic Gradient Descentcited by 184
  12. 2020Implicit Gradient Regularizationcited by 146
16 more
  1. 2015Stochastic modified equations and adaptive stochastic gradient algorithmscited by 142
  2. 2018The Mechanics of n-Player Differentiable Gamescited by 118
  3. 2016A Variational Perspective on Accelerated Methods in Optimizationcited by 110
  4. 2017The Numerics of GANscited by 105
  5. 2018The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimizationcited by 101
  6. 2020A Game Theoretic Framework for Model Based Reinforcement Learningcited by 73
  7. 2017Gradient descent GAN optimization is locally stablecited by 67
  8. 2020Catastrophic Fisher Explosion: Early Phase Fisher Matrix Impacts Generalizationcited by 62
  9. 2019On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Gamescited by 35
  10. 2019On Solving Minimax Optimization Locally: A Follow-the-Ridge Approachcited by 34
  11. 2019Competitive Gradient Descentcited by 32
  12. 2019Conformal Symplectic and Relativistic Optimizationcited by 29
  13. 2020On dissipative symplectic integration with applications to gradient-based optimizationcited by 23
  14. 2020Training Generative Adversarial Networks by Solving Ordinary Differential Equationscited by 21
  15. 2013Nash Convergence of Gradient Dynamics in Iterated General-Sum Gamescited by 18
  16. 2019Implicit competitive regularization in GANscited by 14

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Citing papers per year: 2022: 2; 2023: 5; 2024: 2; 2025: 2; 2026: 3.

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  1. 2022Why neural networks find simple solutions: the many regularizers of geometric complexitycited by 39
  2. 2023On the Implicit Bias of Adamcited by 27
  3. 2023On a continuous time model of gradient descent dynamics and instability in deep learningcited by 11
  4. 2023Deep Fusion: Efficient Network Training via Pre-trained Initializationscited by 4
  5. 2023Implicit biases in multitask and continual learning from a backward error analysis perspectivecited by 4
  6. 2026First-Order Predictable but Pairwise Fragile: Local Task Adaptation in Trained Transformerscited by 4
  7. 2025Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Gamescited by 3
  8. 2023Implicit regularisation in stochastic gradient descent: from single-objective to two-player gamescited by 2
  9. 2022Continuous-time Analysis for Variational Inequalities: An Overview and Desideratacited by 1
  10. 2025Learning by solving differential equationscited by 1
  11. 2024Majority Kernels: An Approach to Leverage Big Model Dynamics for Efficient Small Model Trainingcited by 0
  12. 2024Corridor Geometry in Gradient-Based Optimizationcited by 0
2 more
  1. 2026Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approachcited by 0
  2. 2026Predicting Block-Coordinate Performance via Cross-Curvaturecited by 0