# Gradient Descent Ascent for Minimax Problems on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2010.06097 · HTML: https://mlsanity.com/paper/2010.06097/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2020.

## Built on by (most cited first, 16 of 16)

- [A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold](https://mlsanity.com/paper/2302.08210/) (2023)
- [AdaGDA: Faster Adaptive Gradient Descent Ascent Methods for Minimax Optimization](https://mlsanity.com/paper/2106.16101/) (2021)
- [Riemannian Hamiltonian methods for min-max optimization on manifolds](https://mlsanity.com/paper/2204.11418/) (2022)
- [Decentralized Stochastic Gradient Descent Ascent for Finite-Sum Minimax Problems](https://mlsanity.com/paper/2212.02724/) (2022)
- [First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces](https://mlsanity.com/paper/2206.02041/) (2022)
- [Solving a Class of Non-Convex Minimax Optimization in Federated Learning](https://mlsanity.com/paper/2310.03613/) (2023)
- [Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm](https://mlsanity.com/paper/2202.06950/) (2022)
- [A vector quantized masked autoencoder for audiovisual speech emotion recognition](https://mlsanity.com/paper/2305.03568/) (2023)
- [Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning](https://mlsanity.com/paper/2112.11663/) (2021)
- [Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penalties](https://mlsanity.com/paper/2305.16186/) (2023)
- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)
- [A Block Coordinate Descent Method for Nonsmooth Composite Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2304.03641/) (2023)
- [Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation](https://mlsanity.com/paper/2306.16617/) (2023)
- [Faster Stochastic Variance Reduction Methods for Compositional MiniMax Optimization](https://mlsanity.com/paper/2308.09604/) (2023)
- [Extragradient Type Methods for Riemannian Variational Inequality Problems](https://mlsanity.com/paper/2309.14155/) (2023)
- [Local convergence of simultaneous min-max algorithms to differential equilibrium on Riemannian manifold](https://mlsanity.com/paper/2405.13392/) (2024)

## Abstract (arXiv)

In the paper, we study a class of useful minimax problems on Riemanian manifolds and propose a class of effective Riemanian gradient-based methods to solve these minimax problems. Specifically, we propose an effective Riemannian gradient descent ascent (RGDA) algorithm for the deterministic minimax optimization. Moreover, we prove that our RGDA has a sample complexity of O(kappa2ε-2) for finding an ε-stationary solution of the Geodesically-Nonconvex Strongly-Concave (GNSC) minimax problems, where kappa denotes the condition number. At the same time, we present an effective Riemannian stochastic gradient descent ascent (RSGDA) algorithm for the stochastic minimax optimization, which has a sample complexity of O(kappa4ε-4) for finding an ε-stationary solution. To further reduce the sample complexity, we propose an accelerated Riemannian stochastic gradient descent ascent (Acc-RSGDA) algorithm based on the momentum-based variance-reduced technique. We prove that our Acc-RSGDA algorithm achieves a lower sample complexity of tildeO(kappa4ε-3) in searching for an ε-stationary solution of the GNSC minimax problems. Extensive experimental results on the robust distributional optimization and robust Deep Neural Networks (DNNs) training over Stiefel manifold demonstrate efficiency of our algorithms.