@inproceedings{
bonet2024mirror,
title={Mirror and Preconditioned Gradient Descent in Wasserstein Space},
author={Cl{\'e}ment Bonet and Th{\'e}o Uscidda and Adam David and Pierre-Cyril Aubin-Frankowski and Anna Korba},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=N12B6wvA55}
}
As the problem of minimizing functionals on the Wasserstein space encompasses many applications in machine learning, different optimization algorithms on $\mathbb{R}^d$ have received their counterpart analog on the Wasserstein space. We focus here on lifting two explicit algorithms: mirror descent and preconditioned gradient descent. These algorithms have been introduced to better capture the geometry of the function to minimize and are provably convergent under appropriate (namely relative) smoothness and convexity conditions. Adapting these notions to the Wasserstein space, we prove guarantees of convergence of some Wasserstein-gradient-based discrete-time schemes for new pairings of objective functionals and regularizers. The difficulty here is to carefully select along which curves the functionals should be smooth and convex. We illustrate the advantages of adapting the geometry induced by the regularizer on ill conditioned optimization tasks, and showcase the improvement of choosing different discrepancies and geometries in a computational biology task of aligning single-cells.
Mirror and Preconditioned Gradient Descent in Wasserstein Space
Clément Bonet Théo Uscidda Adam David Pierre-Cyril Aubin-Frankowski Anna Korba