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

TL;DR

This paper introduces mirrorVT, a particle-based algorithm inspired by mirror descent and variational transport, for optimizing probability distributions over constrained domains, with proven convergence and demonstrated effectiveness.

Contribution

The paper develops mirrorVT, a novel particle-based algorithm that extends variational transport to constrained domains using mirror maps, with theoretical analysis and empirical validation.

Findings

Effective minimization of functionals over probability distributions on constrained domains.

Theoretical proof of convergence to the global minimum.

Successful experiments on simplex and Euclidean ball domains.

Abstract

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.