Motivated by approximation Bayesian computation using mean-field variational approximation and the computation of equilibrium in multi-species systems with cross-interaction, this paper investigates the composite geodesically convex optimization problem over multiple distributions. The objective functional under consideration is composed of a convex potential energy on a product of Wasserstein spaces and a sum of convex self-interaction and internal energies associated with each distribution. To efficiently solve this problem, we introduce the Wasserstein Proximal Coordinate Gradient (WPCG) algorithms with parallel, sequential and random update schemes. Under a quadratic growth (QC) condition that is weaker than the usual strong convexity requirement on the objective functional, we show that WPCG converges exponentially fast to the unique global optimum. In the absence of the QG condition, WPCG is still demonstrated to converge to the global optimal solution, albeit at a slower polynomial rate. Numerical results for both motivating examples are consistent with our theoretical findings.

翻译：受均值场变分逼近的近似贝叶斯计算及具有交叉相互作用的多物种系统平衡计算的驱动，本文研究了多分布上的复合测地线凸优化问题。所考虑的目标泛函由Wasserstein空间乘积上的凸势能以及各分布相关的凸自相互作用能与内能之和构成。为高效求解该问题，我们提出了具有并行、顺序和随机更新模式的Wasserstein邻近坐标梯度（WPCG）算法。在弱于目标泛函通常强凸性要求的二次增长（QG）条件下，我们证明了WPCG以指数速率快速收敛至唯一全局最优解。当QG条件不成立时，WPCG仍被证明以较慢的多项式速率收敛至全局最优解。两个动机实例的数值结果与我们的理论发现一致。