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

arXiv · AI, language, vision and robotics · article · Sep 12, 2026 · UTC

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 \emph{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 devel

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