Non-geodesically-convex optimization in the Wasserstein space
<|reference_start|>Non-geodesically-convex optimization in the Wasserstein space: We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is \emph{nonconvex} along generalized geodesics. When the regularization term is the negative entropy, the optimization problem becomes a sampling problem where it minimizes the Kullback-Leibler divergence between a probability measure (optimization variable) and a target probability measure whose logarithmic probability density is a nonconvex function. We derive multiple convergence insights for a novel {\em 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.<|reference_end|>
@article{luu2024non-geodesically-convex,
    title={Non-geodesically-convex optimization in the Wasserstein space},
    author={Hoang Phuc Hau Luu, Hanlin Yu, Bernardo Williams, Petrus Mikkola,
  Marcelo Hartmann, Kai Puolam"aki, Arto Klami},
    journal={arXiv preprint arXiv:2406.00502},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.00502},
    primaryClass={math.OC cs.LG}
}