# Moreau-Yoshida Variational Transport: A General Framework For Solving Regularized Distributional Optimization Problems

arXiv: https://arxiv.org/abs/2307.16358 · HTML: https://mlsanity.com/paper/2307.16358/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 29 references that are in the map.
- First posted on arXiv 31 Jul 2023.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [GOT: An Optimal Transport framework for Graph comparison](https://mlsanity.com/paper/1906.02085/) (2019)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Learning to Draw Samples with Amortized Stein Variational Gradient Descent](https://mlsanity.com/paper/1707.06626/) (2017)
- [Learning to Draw Samples: With Application to Amortized MLE for Generative Adversarial Learning](https://mlsanity.com/paper/1611.01722/) (2016)
- [On a linear fused Gromov-Wasserstein distance for graph structured data](https://mlsanity.com/paper/2203.04711/) (2022)
- [Learning subtree pattern importance for Weisfeiler-Lehmanbased graph kernels](https://mlsanity.com/paper/2106.04739/) (2021)
- [A Particle-Based Algorithm for Distributional Optimization on textitConstrained Domains via Variational Transport and Mirror Descent](https://mlsanity.com/paper/2208.00587/) (2022)

## Built on by (most cited first, 3 of 3)

- [Wasserstein Gradient Flow over Variational Parameter Space for Variational Inference](https://mlsanity.com/paper/2310.16705/) (2023)
- [Multiple Wasserstein Gradient Descent Algorithm for Multi-Objective Distributional Optimization](https://mlsanity.com/paper/2505.18765/) (2025)
- [Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization](https://mlsanity.com/paper/2601.19220/) (2026)

## Abstract (arXiv)

We consider a general optimization problem of minimizing a composite objective functional defined over a class of probability distributions. The objective is composed of two functionals: one is assumed to possess the variational representation and the other is expressed in terms of the expectation operator of a possibly nonsmooth convex regularizer function. Such a regularized distributional optimization problem widely appears in machine learning and statistics, such as proximal Monte-Carlo sampling, Bayesian inference and generative modeling, for regularized estimation and generation. We propose a novel method, dubbed as Moreau-Yoshida Variational Transport (MYVT), for solving the regularized distributional optimization problem. First, as the name suggests, our method employs the Moreau-Yoshida envelope for a smooth approximation of the nonsmooth function in the objective. Second, we reformulate the approximate problem as a concave-convex saddle point problem by leveraging the variational representation, and then develope an efficient primal-dual algorithm to approximate the saddle point. Furthermore, we provide theoretical analyses and report experimental results to demonstrate the effectiveness of the proposed method.