Random Coordinate Descent on the Wasserstein Space of Probability Measures

Researchers Yewei Xu and Qin Li have published a new paper proposing a randomized coordinate descent framework specifically designed for optimization on the Wasserstein manifold. The work introduces two novel algorithms: Random Wasserstein Coordinate Descent (RWCD) and Random Wasserstein Coordinate Proximal-Gradient (RWCP) for composite objectives. These methods address a key bottleneck in modern machine learning and mean-field modeling, where traditional optimization over probability measures using full Wasserstein gradients suffers from high computational overhead, particularly in high-dimensional or ill-conditioned settings.

The core innovation lies in adapting the classic coordinate descent approach—which updates only a subset of coordinates at each iteration—to the geometric structure of the Wasserstein space. By exploiting coordinate-wise structures, the methods can adapt to anisotropic objective landscapes where full-gradient approaches typically struggle. The researchers provide rigorous convergence analysis across various landscape geometries, establishing guarantees under non-convex, Polyak-Łojasiewicz, and geodesically convex conditions, mirroring classic convergence properties found in Euclidean space.

Numerical experiments on ill-conditioned energies demonstrate that this framework offers significant speedups over conventional full-gradient methods. 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. This represents an important advancement for applications ranging from generative modeling and optimal transport to mean-field games and distributional reinforcement learning.