# A Particle-Based Algorithm for Distributional Optimization on textitConstrained Domains via Variational Transport and Mirror Descent

arXiv: https://arxiv.org/abs/2208.00587 · HTML: https://mlsanity.com/paper/2208.00587/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 17 references that are in the map.
- First posted on arXiv 1 Aug 2022.

## Builds on (most cited first, 9 of 9)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)

## Built on by (most cited first, 2 of 2)

- [Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems](https://mlsanity.com/paper/2307.16358/) (2023)
- [Nonlinear Distributionally Robust Optimization](https://mlsanity.com/paper/2306.03202/) (2023)

## Abstract (arXiv)

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.