Non-geodesically-convex optimization in the Wasserstein space
Optimization and Control
2025-01-08 v3 Machine Learning
Abstract
We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is nonconvex along generalized geodesics. Specifically, the objective exhibits some difference-of-convex structure along these geodesics. The setting also encompasses sampling problems where the logarithm of the target distribution is difference-of-convex. We derive multiple convergence insights for a novel semi Forward-Backward Euler scheme under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is -- to our knowledge -- still unknown in our very general non-geodesically-convex setting.
Cite
@article{arxiv.2406.00502,
title = {Non-geodesically-convex optimization in the Wasserstein space},
author = {Hoang Phuc Hau Luu and Hanlin Yu and Bernardo Williams and Petrus Mikkola and Marcelo Hartmann and Kai Puolamäki and Arto Klami},
journal= {arXiv preprint arXiv:2406.00502},
year = {2025}
}