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

תקציר מקורי באנגליתarXiv:2609.14141v1 Announce Type: cross Abstract: 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 $\delta$ 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 \L{}ojasi