# Riemannian ascent--descent for nonconvex nonconcave minimax landscapes: convergence to basin saddle points and applications to distributionally robust optimization

arXiv: https://arxiv.org/abs/2609.14141 · HTML: https://mlsanity.com/paper/2609.14141/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 75 references that are in the map.
- First posted on arXiv 12 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Solving a Class of Non-Convex Min-Max Games Using Iterative First Order Methods](https://mlsanity.com/paper/1902.08297/) (2019)
- [Global Convergence and Variance-Reduced Optimization for a Class of Nonconvex-Nonconcave Minimax Problems](https://mlsanity.com/paper/2002.09621/) (2020)
- [What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?](https://mlsanity.com/paper/1902.00618/) (2019)
- [The global convergence time of stochastic gradient descent in non-convex landscapes: Sharp estimates via large deviations](https://mlsanity.com/paper/2503.16398/) (2025)
- [Solving Non-Convex Non-Concave Min-Max Games Under Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1812.02878/) (2018)
- [Worst-case generation via minimax optimization in Wasserstein space](https://mlsanity.com/paper/2512.08176/) (2025)
- [Optimality Conditions for Constrained Minimax Optimization](https://mlsanity.com/paper/2004.09730/) (2020)
- [Resistance Distance and Linearized Optimal Transport on Graphs](https://mlsanity.com/paper/2404.15261/) (2024)
- [Distributionally Robust Optimization with Adversarial Data Contamination](https://mlsanity.com/paper/2507.10718/) (2025)
- [Loss Gradient Gaussian Width based Generalization and Optimization Guarantees](https://mlsanity.com/paper/2406.07712/) (2024)

## Abstract (arXiv)

We study a class of distributionally robust optimization (DRO) problems for the statistical risk problem, formulated as minimax problems over the product of a Euclidean space and a Riemannian manifold. Because the resulting minimax landscape is nonconvex nonconcave in general, no globally convergent first order method is known to be available. We instead introduce the notion of a basin saddle point, a Nash equilibrium defined locally on the Cartesian product of a δ basin around a connected component of the local minima critical set and a geodesic ball on the measure manifold. We develop an abstract convergence framework for a Riemannian gradient ascent multistep descent iteration to a basin saddle point under a local Lojasiewicz type growth condition, with exponent β ∈ (1,2], in the δ basin around connected components of the local minima critical sets. Under Lipschitz regularity of critical sets we establish linear convergence for β = 2 and polynomial convergence for β ∈ (1,2) to a basin saddle point, with explicit dependence on the sectional curvature of the manifold. We then instantiate this framework for the statistical risk DRO problem over Gaussian measures, where the ambiguity set is naturally modeled as the product of Euclidean space and the Bures Wasserstein manifold of covariance matrices, which we relax to a penalized DRO formulation. We derive nonasymptotic Hessian estimates for the resulting Lagrangian, establish existence and local uniqueness of its maximizer, and prove that an alternating Riemannian gradient scheme converges to a basin saddle point of the penalized DRO problem, recovering the linear and polynomial rates of the abstract theory with all constants explicit in terms of data dimension, loss moments, and the reference covariance.