Non-Geodesically-Convex Optimization in the Wasserstein Space

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.

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Cite

Text

Luu et al. "Non-Geodesically-Convex Optimization in the Wasserstein Space." Neural Information Processing Systems, 2024. doi:10.52202/079017-0535

Markdown

[Luu et al. "Non-Geodesically-Convex Optimization in the Wasserstein Space." Neural Information Processing Systems, 2024.](https://mlanthology.org/neurips/2024/luu2024neurips-nongeodesicallyconvex/) doi:10.52202/079017-0535

BibTeX

@inproceedings{luu2024neurips-nongeodesicallyconvex,
  title     = {{Non-Geodesically-Convex Optimization in the Wasserstein Space}},
  author    = {Luu, Hoang Phuc Hau and Yu, Hanlin and Williams, Bernardo and Mikkola, Petrus and Hartmann, Marcelo and Puolamäki, Kai and Klami, Arto},
  booktitle = {Neural Information Processing Systems},
  year      = {2024},
  doi       = {10.52202/079017-0535},
  url       = {https://mlanthology.org/neurips/2024/luu2024neurips-nongeodesicallyconvex/}
}