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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.01606 |
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| _version_ | 1866917380042522624 |
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| author | Xu, Yewei Li, Qin |
| author_facet | Xu, Yewei Li, Qin |
| contents | Optimization over the space of probability measures endowed with the Wasserstein-2 geometry is central to modern machine learning and mean-field modeling. However, traditional methods relying on full Wasserstein gradients often suffer from high computational overhead in high-dimensional or ill-conditioned settings. We propose a randomized coordinate descent framework specifically designed for the Wasserstein manifold, introducing both Random Wasserstein Coordinate Descent (RWCD) and Random Wasserstein Coordinate Proximal{-Gradient} (RWCP) for composite objectives. By exploiting coordinate-wise structures, our methods adapt to anisotropic objective landscapes where full-gradient approaches typically struggle. We provide a rigorous convergence analysis across various landscape geometries, establishing guarantees under non-convex, Polyak-Łojasiewicz, and geodesically convex conditions. Our theoretical results mirror the classic convergence properties found in Euclidean space, revealing a compelling symmetry between coordinate descent on vectors and on probability measures. The developed techniques are inherently adaptive to the Wasserstein geometry and offer a robust analytical template that can be extended to other optimization solvers within the space of measures. Numerical experiments on ill-conditioned energies demonstrate that our framework offers significant speedups over conventional full-gradient methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01606 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random Coordinate Descent on the Wasserstein Space of Probability Measures Xu, Yewei Li, Qin Machine Learning Optimization and Control Optimization over the space of probability measures endowed with the Wasserstein-2 geometry is central to modern machine learning and mean-field modeling. However, traditional methods relying on full Wasserstein gradients often suffer from high computational overhead in high-dimensional or ill-conditioned settings. We propose a randomized coordinate descent framework specifically designed for the Wasserstein manifold, introducing both Random Wasserstein Coordinate Descent (RWCD) and Random Wasserstein Coordinate Proximal{-Gradient} (RWCP) for composite objectives. By exploiting coordinate-wise structures, our methods adapt to anisotropic objective landscapes where full-gradient approaches typically struggle. We provide a rigorous convergence analysis across various landscape geometries, establishing guarantees under non-convex, Polyak-Łojasiewicz, and geodesically convex conditions. Our theoretical results mirror the classic convergence properties found in Euclidean space, revealing a compelling symmetry between coordinate descent on vectors and on probability measures. The developed techniques are inherently adaptive to the Wasserstein geometry and offer a robust analytical template that can be extended to other optimization solvers within the space of measures. Numerical experiments on ill-conditioned energies demonstrate that our framework offers significant speedups over conventional full-gradient methods. |
| title | Random Coordinate Descent on the Wasserstein Space of Probability Measures |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2604.01606 |