A Particle-Based Algorithm for Distributional Optimization on \textit{Constrained Domains} via Variational Transport and Mirror Descent
This work addresses optimization challenges for probability distributions on constrained domains, which is incremental as it builds on prior variational transport methods.
The paper tackles the problem of optimizing objective functionals over probability distributions on constrained domains by proposing a particle-based algorithm called Mirrored Variational Transport (mirrorVT), which extends an existing framework to handle constraints via mirror maps and demonstrates effectiveness in simulated experiments on simplex- and Euclidean ball-constrained domains with theoretical convergence guarantees.
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.