We consider the optimization problem of minimizing an objective functional,
which admits a variational form and is defined over probability distributions
on the constrained domain, which poses challenges to both theoretical analysis
and algorithmic design. Inspired by the mirror descent algorithm for
constrained optimization, we propose an iterative particle-based algorithm,
named Mirrored Variational Transport (mirrorVT), extended from the Variational
Transport framework [7] for dealing with the constrained domain. In particular,
for each iteration, mirrorVT maps particles to an unconstrained dual domain
induced by a mirror map and then approximately perform Wasserstein gradient
descent on the manifold of distributions defined over the dual space by pushing
particles. At the end of iteration, particles are mapped back to the original
constrained domain. Through simulated experiments, we demonstrate the
effectiveness of mirrorVT for minimizing the functionals over probability
distributions on the simplex- and Euclidean ball-constrained domains. We also
analyze its theoretical properties and characterize its convergence to the
global minimum of the objective functional.

本文提出了一种基于粒子的迭代算法 —— 反射变分传输（mirrorVT），应用于概率分布的约束优化问题，特别是在单纯形和欧几里得球约束域上，该算法通过推送粒子对定义在对偶空间上的分布流形执行 Wasserstein 梯度下降，实现了概率分布函数的优化，并且在理论分析和模拟实验中验证了其收敛性和有效性。