We study a class of optimization problems in the Wasserstein space (the space of probability measures) where the objective function is \emph{nonconvex} along generalized geodesics. When the regularization term is the negative entropy, the optimization problem becomes a sampling problem where it minimizes the Kullback-Leibler divergence between a probability measure (optimization variable) and a target probability measure whose logarithmic probability density is a nonconvex function. We derive multiple convergence insights for a novel {\em semi Forward-Backward Euler scheme} under several nonconvex (and possibly nonsmooth) regimes. Notably, the semi Forward-Backward Euler is just a slight modification of the Forward-Backward Euler whose convergence is -- to our knowledge -- still unknown in our very general non-geodesically-convex setting.

研究了定义在Wasserstein空间（概率测度空间）中的一类优化问题，其中目标函数沿广义测地线是非凸函数。当正则化项为负熵时，优化问题转化为一个采样问题，在概率测度（优化变量）和目标概率测度之间最小化Kullback-Leibler散度，其中概率密度的对数是一个非凸函数。在几个非凸（可能非光滑）情况下，通过一种新的半正向-反向Euler方案，得出了多个收敛性洞察。值得注意的是，这种半正向-反向Euler方案只是正向-反向Euler方案的轻微修改，在我们非常普遍的非测地线凸设置中，正向-反向Euler方案的收敛性 -- 这是我们所知道的 -- 仍然未知。

在Wasserstein空间中的非测地凸优化