Couplings and quantitative contraction rates for Langevin dynamics

Abstract

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 ). * Financial support from DAAD and French government through the PROCOPE program, and from the German Science foundation through the Hausdorff Center for Mathematics is gratefully acknowledged.MSC 2010 subject classifications: Primary 60J60, 60H10, 35Q84, 35B40

Sampling can be faster than optimization

“…It naturally allows us to combine convergence rates for sampling strongly log-concave posteriors and those for sampling smooth posteriors in a bounded region. Indeed, our upper bounds on convergence rates generalize existing results for strongly log-concave posteriors [17,19,18,15,13,21,36,37] and also strengthen recent work using the Wasserstein metric to the KL divergence [22,9,14,35].…”

Ergodicity and Lyapunov Functions for Langevin Dynamics with Singular Potentials

“…Because many ergodic sampling algorithms are built using discretizations/and or slight modifications of Langevin dynamics, this problem seems relevant from this perspective. There are instances where one can do such estimates; see, for example, the paper [8] and references therein in specific cases of U , but this problem is saved for future research. Another natural problem is to see if the perturbative methods developed here work for different type of dynamics, like adaptive thermostats such as variations of the Nosé-Hoover equation or modifications of the kinetic energy functional.…”

On sampling from a log-concave density using kinetic Langevin diffusions

“…On the one hand, (Cheng et al, 2018) provide results only for a fixed value of parameters (γ, u) = (2, 1/M ). On the other hand, if we instantiate results of (Eberle et al, 2017) to the case of convex functions f , convergence to the invariant density is proved under the condition γ 2 ≥ 30M u. This is to be compared to the conditions of Theorem 1 that establishes exponential convergence as soon as γ 2 > M u.…”

“…As discussed above, the quality of the resulting sampler will depend on two key properties of the process: rate of mixing and smoothness of sample paths. The rate of mixing of kinetic diffusions has been recently studied by Eberle et al (2017) under conditions that are more general than strong convexity of f . In strongly convex case, a more tractable result has been obtained by .…”