A Particle-Based Algorithm for Distributional Optimization on \textit{Constrained Domains} via Variational Transport and Mirror Descent

We consider the optimization problem of minimizing an objective functional, which admits a variational form and is defined over probability distributions on the \textit{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 and particle-based algorithm, named Mirrored Variational Transport (\textbf{mirrorVT}). For each iteration, \textbf{mirrorVT} maps particles to a unconstrained dual space 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 space. Through simulated experiments, we demonstrate the effectiveness of \textbf{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.