Understanding MCMC Dynamics as Flows on the Wasserstein Space

By Chang Liu, Jingwei Zhuo, Jun Zhu. 2019

23 citations recorded by the source corpus.

It is known that the Langevin dynamics used in MCMC is the gradient flow of the KL divergence on the Wasserstein space, which helps convergence analysis and inspires recent particle-based variational inference methods (ParVIs). But no more MCMC dynamics is understood in this way. In this work, by developing novel concepts, we propose a theoretical framework that recognizes a general MCMC dynamics as the fiber-gradient Hamiltonian flow on the Wasserstein space of a fiber-Riemannian Poisson manifold. The "conservation + convergence" structure of the flow gives a clear picture on the behavior of general MCMC dynamics. The framework also enables ParVI simulation of MCMC dynamics, which enriches the ParVI family with more efficient dynamics, and also adapts ParVI advantages to MCMCs. We develop two ParVI methods for a particular MCMC dynamics and demonstrate the benefits in experiments.

Subfields: Radiology, Nuclear Medicine and Imaging, Statistical and Nonlinear Physics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Radiology, Nuclear Medicine and Imaging

1. Zhenyi Wang

In Statistical and Nonlinear Physics

1. Andrew Gordon Wilson
2. Changyou Chen
3. Chunyuan Li
4. Zhenyi Wang
5. Eric P. Xing
6. Hannes Nickisch
7. Zhiting Hu

In Statistics and Probability

1. Chunyuan Li
2. Changyou Chen
3. Andrew Gordon Wilson
4. Ping Yu
5. Jianyi Zhang
6. Le Fang
7. Li Shen
8. Ruqi Zhang

Citations in the corpus

• Improving Task-free Continual Learning by Distributionally Robust Memory Evolution (2022, methodology): “We then introduce three specific memory evolution methods to solve the Dynamic DRO, including Langevin Dynamics (Welling & Teh, 2011), Stein Variational Gradient Descent (Liu & Wang, 2016), and Hamiltonian flow (Ma et al., 2015; Liu et al., 2019).”
• Bayesian Meta Sampling for Fast Uncertainty Adaptation (2020, methodology): “Recently, a particle-optimization sampling framework that unifies SG-MCMC and SVGD has also been proposed (Chen et al., 2018b), followed by some recent developments (Liu et al., 2019b;a).”
• Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning (2019, background): “We would like to point out that the convergence theorems developed in the above several sections can be potentially applied to study the convergence of the Wasserstein gradient flows (Santambrogio, 2017), which can be regarded as a continuous-time MCMC (Chen et al., 2018; Liu et al., 2019).”