← Back to all news
arXiv Machine Learning By Rishabh Dixit, Pranav Upadrashta, Alex Cloninger

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

Read the original on arXiv Machine Learning →

The paper introduces a new convergence framework for solving distributionally robust optimization problems formulated as nonconvex, nonconcave minimax problems over a Euclidean space and a Riemannian manifold. It defines a "basin saddle point"—a locally defined Nash equilibrium—and proves that a Riemannian gradient ascent–descent algorithm converges to such points under a local Łojasiewicz growth condition. The authors apply this theory to a statistical risk DRO problem over Gaussian measures, deriving explicit convergence rates and constants in terms of data dimension, loss moments, and reference covariance.

Machine-generated by The Flow from the publisher's headline and feed description — not written or checked by a human. The full article lives at arXiv Machine Learning.

Related stories

arXiv Statistics ML
5d ago

Second-order Sinkhorn Geometry and $\sqrt{n}$-Inference

The paper develops a formal Riemannian framework for the Sinkhorn divergence, a debiased entropic optimal transport metric. It proves Hadamard differentiability of the Sinkhorn metric tensor, derives the Levi‑Civita covariant derivative, and obtains geodesic and parallel transport equations as ODEs on a fixed RKHS. Using Banach‑space ODE theory, the authors linearize these equations to establish √ n‑consistent Gaussian limits for plug‑in estimators of the exponential map, logarithmic map, parallel transport, and the induced Sinkhorn distance, and show bootstrap consistency for all these objects.

By Tristan Luca Saidi, Florian Gunsilius, Gonzalo Mena
More like this →
arXiv Statistics ML
Oct 2

Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity

The paper presents a convergence analysis of the Wasserstein proximal algorithm without requiring geodesic convexity of the objective functional. By assuming a Wasserstein analog of the Euclidean Polyak–Łojasiewicz inequality, the authors prove that the algorithm achieves unbiased, linear convergence, improving on previous rates that required strong geodesic convexity. The analysis is extended to inexact proximal steps for geodesically semiconvex objectives, and numerical experiments show faster convergence than noisy gradient descent on mean‑field neural networks.

By Shuailong Zhu, Xiaohui Chen
More like this →
arXiv Machine Learning
Jul 16

Smooth Quasar-Convex Optimization with Constraints

arXiv:2510. 01943v3 Announce Type: replace-cross Abstract: Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others.

By David Mart\'inez-Rubio
More like this →