论文标题
基于粒子的算法，用于通过变异传输和镜像下降对\ textit {约束域}进行分布优化
A Particle-Based Algorithm for Distributional Optimization on \textit{Constrained Domains} via Variational Transport and Mirror Descent
论文作者
论文摘要
我们考虑最小化目标功能的优化问题，该问题可以接受变分形式，并根据约束域上的概率分布定义，这对理论分析和算法设计都构成了挑战。受镜下降算法的启发，我们提出了一种基于迭代粒子的算法，称为镜像变异传输（MirrorVT），该算法是从变异传输框架[7]延伸的，用于处理约束域。特别是，对于每次迭代，镜像映射到镜像图引起的无约束的双域，然后通过推动粒子来定义的分布的歧管上大约执行wasserstein梯度下降。在迭代结束时，将粒子映射回原始的约束域。通过模拟实验，我们证明了MirrorVT在单纯形和欧几里得球受约束域上的概率分布中最小化功能的有效性。我们还分析了其理论特性，并将其融合到目标功能的全局最小值。
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.
