# A Distributed Algorithm for Measure-valued Optimization with Additive Objective

arXiv: https://arxiv.org/abs/2202.08930 · HTML: https://mlsanity.com/paper/2202.08930/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 17 Feb 2022.

## Builds on (most cited first, 18 of 18)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Variational Policy Gradient Method for Reinforcement Learning with General Utilities](https://mlsanity.com/paper/2007.02151/) (2020)
- [Mean Field Analysis of Neural Networks: A Central Limit Theorem](https://mlsanity.com/paper/1808.09372/) (2018)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks](https://mlsanity.com/paper/2106.00774/) (2021)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Policy Optimization as Wasserstein Gradient Flows](https://mlsanity.com/paper/1808.03030/) (2018)
- [A Smoothed Dual Approach for Variational Wasserstein Problems](https://mlsanity.com/paper/1503.02533/) (2015)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Gradient Flows in Filtering and Fisher-Rao Geometry](https://mlsanity.com/paper/1710.00064/) (2017)
- [Probability Functional Descent: A Unifying Perspective on GANs, Variational Inference, and Reinforcement Learning](https://mlsanity.com/paper/1901.10691/) (2019)
- [Gradient Flow Algorithms for Density Propagation in Stochastic Systems](https://mlsanity.com/paper/1908.00533/) (2019)
- [Generalized Sinkhorn iterations for regularizing inverse problems using optimal mass transport](https://mlsanity.com/paper/1612.02273/) (2016)
- [Proximal Recursion for Solving the Fokker-Planck Equation](https://mlsanity.com/paper/1809.10844/) (2018)
- [Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems](https://mlsanity.com/paper/1704.00102/) (2017)
- [Proximal Recursion for the Wonham Filter](https://mlsanity.com/paper/1909.05827/) (2019)

## Abstract (arXiv)

We propose a distributed nonparametric algorithm for solving measure-valued optimization problems with additive objectives. Such problems arise in several contexts in stochastic learning and control including Langevin sampling from an unnormalized prior, mean field neural network learning and Wasserstein gradient flows. The proposed algorithm comprises a two-layer alternating direction method of multipliers (ADMM). The outer-layer ADMM generalizes the Euclidean consensus ADMM to the Wasserstein consensus ADMM, and to its entropy-regularized version Sinkhorn consensus ADMM. The inner-layer ADMM turns out to be a specific instance of the standard Euclidean ADMM. The overall algorithm realizes operator splitting for gradient flows in the manifold of probability measures.