# Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold

arXiv: https://arxiv.org/abs/2405.13392 · HTML: https://mlsanity.com/paper/2405.13392/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 47 references that are in the map.
- First posted on arXiv 22 May 2024.

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

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [Differentiable Game Mechanics](https://mlsanity.com/paper/1905.04926/) (2019)
- [Non-convex Min-Max Optimization: Applications, Challenges, and Recent Theoretical Advances](https://mlsanity.com/paper/2006.08141/) (2020)
- [Scaling the Scattering Transform: Deep Hybrid Networks](https://mlsanity.com/paper/1703.08961/) (2017)
- [On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach](https://mlsanity.com/paper/1910.07512/) (2019)
- [Stochastic Hamiltonian Gradient Methods for Smooth Games](https://mlsanity.com/paper/2007.04202/) (2020)
- [Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization](https://mlsanity.com/paper/2102.09468/) (2021)
- [Some Theoretical Insights into Wasserstein GANs](https://mlsanity.com/paper/2006.02682/) (2020)
- [A Closer Look at the Optimization Landscapes of Generative Adversarial Networks](https://mlsanity.com/paper/1906.04848/) (2019)
- [Gradient Descent Ascent for Minimax Problems on Riemannian Manifolds](https://mlsanity.com/paper/2010.06097/) (2020)
- [Existence, Stability and Scalability of Orthogonal Convolutional Neural Networks](https://mlsanity.com/paper/2108.05623/) (2021)
- [Global Convergence to the Equilibrium of GANs using Variational Inequalities](https://mlsanity.com/paper/1808.01531/) (2018)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [On Convergence of Gradient Descent Ascent: A Tight Local Analysis](https://mlsanity.com/paper/2207.00957/) (2022)
- [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems](https://mlsanity.com/paper/2210.09382/) (2022)
- [Riemannian Optimistic Algorithms](https://mlsanity.com/paper/2308.16004/) (2023)
- [Orthogonal Wasserstein GANs](https://mlsanity.com/paper/1911.13060/) (2019)
- [Optimistic Gradient Descent Ascent in Zero-Sum and General-Sum Bilinear Games](https://mlsanity.com/paper/2208.03085/) (2022)

## Built on by (most cited first, 1 of 1)

- [Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach](https://mlsanity.com/paper/2605.19392/) (2026)

## Abstract (arXiv)

We study min-max algorithms to solve zero-sum differential games on Riemannian manifold. Based on the notions of differential Stackelberg equilibrium and differential Nash equilibrium on Riemannian manifold, we analyze the local convergence of two representative deterministic simultaneous algorithms τ-GDA and τ-SGA to such equilibria. Sufficient conditions are obtained to establish the linear convergence rate of τ-GDA based on the Ostrowski theorem on manifold and spectral analysis. To avoid strong rotational dynamics in τ-GDA, τ-SGA is extended from the symplectic gradient-adjustment method in Euclidean space. We analyze an asymptotic approximation of τ-SGA when the learning rate ratio τ is big. In some cases, it can achieve a faster convergence rate to differential Stackelberg equilibrium compared to τ-GDA. We show numerically how the insights obtained from the convergence analysis may improve the training of orthogonal Wasserstein GANs using stochastic τ-GDA and τ-SGA on simple benchmarks.