[Paper Review] Constructing sampling schemes via coupling: Markov semigroups and optimal transport

N. Nuesken, Grigorios A. Pavliotis|arXiv (Cornell University)|Jun 28, 2018

Markov Chains and Monte Carlo Methods46 references3 citations

TL;DR

This paper develops a unified framework for constructing efficient Markov chain Monte Carlo (MCMC) samplers using coupling techniques, linking them to Markov semigroups and optimal transport. It shows that minimizing asymptotic variance via optimal couplings reduces to multimarginal optimal transport problems, leading to novel antithetic variate methods and a modified Poincaré inequality for convergence analysis.

ABSTRACT

In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.

Motivation & Objective

• To develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers for both diffusion and piecewise deterministic processes.
• To show that optimizing performance criteria like asymptotic variance reduces to problems in optimal transport theory.
• To derive a modified Poincaré inequality for analyzing convergence to equilibrium of coupled processes.
• To propose novel sampling techniques inspired by antithetic variates for variance reduction in Bayesian posterior estimation.
• To establish structural properties of couplings, including a singularity theorem with geometric and probabilistic interpretations.

Proposed method

• Formulates coupled Markov processes on product spaces with fixed marginals, using infinitesimal generators to characterize coupling dynamics.
• Reframes variance minimization as a multimarginal optimal transport problem, enabling analytical and numerical solutions.
• Applies the theory of Markov semigroups to analyze ergodicity and convergence of coupled processes to invariant measures.
• Derives a modified Poincaré inequality to quantify convergence rates of coupled processes to equilibrium.
• Uses the Poisson equation for the coupling generator to express asymptotic variance in terms of solutions to transport-related equations.
• Employs L’Hôpital’s rule and integrability conditions to analyze boundary behavior of solutions, ensuring well-posedness of the coupling framework.

Experimental results

Research questions

• RQ1 How can coupling techniques be systematically used to reduce the asymptotic variance in MCMC sampling?
• RQ2 In what way do optimal transport principles govern the construction of efficient couplings for Markov processes?
• RQ3 What structural properties, such as a singularity theorem, emerge in the space of couplings for diffusion and piecewise deterministic processes?
• RQ4 How can the convergence to equilibrium of coupled processes be quantified, and what new inequalities arise?
• RQ5 Can novel antithetic variate-like methods be derived from coupling frameworks to improve sampling efficiency in Bayesian inference?

Key findings

• The asymptotic variance of coupled samplers is minimized when the coupling corresponds to a solution of a multimarginal optimal transport problem.
• A singularity theorem is established, revealing geometric and probabilistic constraints on the structure of efficient couplings.
• The framework enables the construction of novel antithetic variate techniques for Bayesian posterior variance estimation, reducing variance without increasing computational cost.
• A modified Poincaré inequality is derived, providing a quantitative bound on the convergence rate of coupled processes to equilibrium.
• Numerical experiments support the theoretical findings, showing that approximate solutions to the optimal transport problems yield effective couplings with reduced variance.
• Boundary behavior of solutions to the Poisson equation is rigorously analyzed, ensuring integrability and uniqueness under mild conditions on the potential function.