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.

This problem configuration encompasses a broad range of applications in ML scenarios, e.g.

We propose a method for analyzing the distributed random coordinate descent algorithm for solving separable resource allocation problems in the context of an open multiagent system, where agents can be replaced during the process. In particular, we characterize the evolution of the distance to the minimizer in expectation by following a time-varying optimization approach which builds on two components. First, we establish the linear convergence of the algorithm in closed systems, in terms of the estimate towards the minimizer, for general graphs and appropriate step-size. Second, we estimate the change of the optimal solution after a replacement, in order to evaluate its effect on the distance between the current estimate and the minimizer. From these two elements, we derive stability conditions in open systems and establish the linear convergence of the algorithm towards a steady-state expected error. Our results enable to characterize the trade-off between speed of convergence and robustness to agent replacements, under the assumptions that local functions are smooth, strongly convex, and have their minimizers located in a given ball. The approach proposed in this paper can moreover be extended to other algorithms guaranteeing linear convergence in closed system.

Sampling from a log-concave distribution function on $\mathbb{R}^d$ (with $d\gg 1$) is a popular problem that has wide applications. In this paper we study the application of random coordinate descent method (RCD) on the Langevin Monte Carlo (LMC) sampling method, and we find two sides of the theory: 1. The direct application of RCD on LMC does reduce the number of finite differencing approximations per iteration, but it induces a large variance error term. More iterations are then needed, and ultimately the method gains no computational advantage; 2. When variance reduction techniques (such as SAGA and SVRG) are incorporated in RCD-LMC, the variance error term is reduced. The new methods, compared to the vanilla LMC, reduce the total computational cost by $d$ folds, and achieve the optimal cost rate. We perform our investigations in both overdamped and underdamped settings.