Understanding MCMC Dynamics as Flows on the Wasserstein Space
arXiv:1902.00282
Abstract
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.
References refined
References in corpus (18)
- A Conceptual Introduction to Hamiltonian Monte Carlo
- Stochastic Gradient Hamiltonian Monte Carlo
- On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators
- Stein Variational Gradient Descent as Gradient Flow
- Measuring Sample Quality with Stein's Method
- On the Geometric Ergodicity of Hamiltonian Monte Carlo
- Stein Variational Policy Gradient
- A Unified Particle-Optimization Framework for Scalable Bayesian Sampling
- Lagrangian Dynamical Monte Carlo
- Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem
- Langevin Monte Carlo and JKO splitting
- Stochastic Gradient MCMC with Repulsive Forces
- Analysis of Langevin Monte Carlo via convex optimization
- VAE Learning via Stein Variational Gradient Descent
- Towards Unifying Hamiltonian Monte Carlo and Slice Sampling
- Relativistic Monte Carlo
- Stochastic Gradient Monomial Gamma Sampler
- Asymptotic gradient flow structures of a nonlinear Fokker-Planck equation