Mirror variational transport: a particle-based algorithm for distributional optimization on constrained domains

Dai Hai Nguyen, Tetsuya Sakurai

Abstract: We consider the optimization problem of minimizing an objective functional, which admits a variational form and is defined over probability distributions on a constrained domain, which poses challenges to both theoretical analysis and algorithmic design. We propose Mirror Variational Transport (mirrorVT), which uses a set of samples, or particles, to represent the approximating distribution and deterministically updates the particles to optimize the functional. To deal with the constrained domain, in each iteration, mirrorVT maps the particles to an unconstrained dual domain, induced by a mirror map, and then approximately performs Wasserstein Gradient Descent on the manifold of distributions defined over the dual space to update each particle by a specified direction. At the end of each iteration, particles are mapped back to the original constrained domain. Through experiments on synthetic and real world data sets, we demonstrate the effectiveness of mirrorVT for the distributional optimization on the constrained domain. We also analyze its theoretical properties and characterize its convergence to the global minimum of the objective functional.