Wasserstein accelerated proximal gradient algorithms
Abstract
We propose accelerated proximal gradient methods for composite optimization problems on the quadratic Wasserstein space of probability measures, including Kullback--Leibler divergence minimization. The objective function consists of a smooth geodesically convex functional and a possibly nonsmooth regularizer that is convex along generalized geodesics. The convexity and smoothness conditions on the smooth component are imposed only on the feasible set. As a preliminary result, we establish the equivalence between geodesic convexity of the smooth functional on the feasible set and convexity of its Hilbert-space lift on the corresponding feasible set. Assuming exact proximal steps, we prove an O(k−2) convergence rate for the objective gap when the smooth component is geodesically convex and accelerated linear convergence when it is geodesically strongly convex. We also describe particle approximations of these algorithms.
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BibTeX
@misc{ushiyama2026wasserstein,
title = {{Wasserstein accelerated proximal gradient algorithms}},
author = {Kansei Ushiyama and Yi Feng and Kaito Fujii and Yuma Fujimoto and Noboru Isobe},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03524},
eprint = {2611.03524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03524},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wasserstein accelerated proximal gradient algorithms
AU - Kansei Ushiyama
AU - Yi Feng
AU - Kaito Fujii
AU - Yuma Fujimoto
AU - Noboru Isobe
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03524
UR - https://arxiv.org/abs/2611.03524
AB - We propose accelerated proximal gradient methods for composite optimization problems on the quadratic Wasserstein space of probability measures, including Kullback--Leibler divergence minimization. The objective function consists of a smooth geodesically convex functional and a possibly nonsmooth regularizer that is convex along generalized geodesics. The convexity and smoothness conditions on the smooth component are imposed only on the feasible set. As a preliminary result, we establish the equivalence between geodesic convexity of the smooth functional on the feasible set and convexity of its Hilbert-space lift on the corresponding feasible set. Assuming exact proximal steps, we prove an O(k^−2) convergence rate for the objective gap when the smooth component is geodesically convex and accelerated linear convergence when it is geodesically strongly convex. We also describe particle approximations of these algorithms.
ER -