Couplings and quantitative contraction rates for Langevin dynamics

By Andreas Eberle, Arnaud Guillin, Raphael Zimmer. Annals of Probability, 2017

241 citations recorded by the source corpus.

We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance a, we obtain a lower bound for the contraction rate of order Ω(a−1) provided the friction coefficient is of order Θ(a−1)

Subfields: Spectroscopy, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4a1768d9f5f90c669af875524704d6f262fb9ebf

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Citations in the corpus

Log-concave sampling: Metropolis-Hastings algorithms are fast! (2018, methodology): “In this section, we describe general MCMC algorithms and review the rates of convergence of existing random walks for log-concave distributions.”
Underdamped Langevin MCMC: A non-asymptotic analysis (2017, methodology): “The continuous-time process has been studied extensively [12, 13, 14, 15, 16, 17, 18, 19, 20].”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, background): “The first steps in this direction were made in [20], where a corresponding coupling approach for kinetic Langevin diffusions was introduced.”
Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices (2019, background): “There are recent works that analyze convergence of sampling using various techniques such as reflection coupling [28], kernel methods [29], and higher-order integrators [40], albeit still under some strong conditions such as distant dissipativity, which is similar to strong logconcavity outside a bo”
KALE: When Energy-Based Learning Meets Adversarial Training (2020, methodology): “This allows to directly apply (Eberle et al., 2017)(Corollary 2.6.) which implies that: where c is a positive constant and C 1 only depends on π ∞ and the initial distribution π 0 .”
Generalized Energy Based Models (2020, unknown): “This allows to directly apply (Eberle et al., 2017)(Corollary 2.”