Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

Abstract

We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.Keywords Non-reversible diffusion · Convergence to equilibrium · Wick calculus 1 Introduction MotivationThe problem of convergence to equilibrium for diffusion processes has attracted considerable attention in recent years. In addition to the relevance of this problem for the convergence to equilibrium of some systems in statistical T. Lelièvre at Université Paris-Est, Cermics, and INRIA, MicMac project team, Ecole des ponts, 6-8 avenue Blaise Pascal, 77455 Marne la Vallée Cedex 2, France. E-mail: lelievre@cermics.enpc.fr · F. Nier at IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes, France. E-mail: francis.nier@univ-rennes1.fr · G.A. Pavliotis at Imperial College London, Department of Mathematics, South Kensington Campus, London SW7 2AZ, England. E-mail: g.pavliotis@imperial.ac.uk 2 physics, see for example [29], such questions are also important in statistics, for example in the analysis of Markov Chain Monte Carlo (MCMC) algorithms [9]. Roughly speaking, one measure of efficiency of an MCMC algorithm is its rate of convergence to equilibrium, and increasing this rate is thus the aim of many numerical techniques (see for example [5]).Let us recall the basic approach for a reversible diffusion. Suppose that we are interested in sampling from a probability distribution functionwhere V : R N → R is a given smooth potential such that R N e −V dx < ∞ . A natural dynamics to use is the reversible dynamicswhere W t denotes a standard N -dimensional Brownian motion. Let us denote by ψ t the probability density function of the process X t at time t. It satisfies the Fokker-Planck equationUnder appropriate assumptions on the potential V (e.g. that 1 2 |∇V (x)| 2 − ∆V (x) → +∞ as |x| → +∞ , see [41, A.19]), the density ψ ∞ satisfies a Poincaré inequality: there exists λ > 0 such that for all probability density functions φ,The optimal parameter λ in (4) is the opposite of the smallest (in absolute value) non-zero eigenvalue of the Fokker-Planck operator ∇ · (∇V · +∇·), which is self-adjoint in L 2 (R N , ψ −1 ∞ dx) (see (7) below). Thus, λ is also called the spectral gap of the Fokker-Planck operator.It is then standard to show that (4) is equivalent to the following inequality, which shows exponential convergence to the equilibrium for (2): for all initial conditions ψ 0 ∈ L 2 (R N , ψ −1 ∞ dx), for all times t ≥ 0,where · L 2 (ψ −1 ∞ ) denotes the norm in L 2 (R N , ψ −1 ∞ ), namely f 2∞ (x) dx . This equivalence is a simple consequence of th...

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…When this is possible, this is indeed the case. However, as has been observed in [20,21,34], maximising the speed of convergence to equilibrium is a delicate task. As the leading order term in M SE( f, T ), it is typically sufficient to focus specifically on the asymptotic variance σ 2 f and study how the parameters of the SDE (9) can be chosen to minimise σ 2 f .…”

“…In [34,Rem. 3] another modification of (4) was suggested (albeit with the simplifications Γ = γ · I and M = I ):…”

“…In the previous subsection we observed that for the particular perturbation J 1 = J 2 and μ = ν [see equation (34)] the perturbed Langevin dynamics demonstrated an improvement in performance for μ in a neighbourhood of 0, when the observable is linear or quadratic.…”

“…This work was extended to the case of non-Gaussian target densities, and consequently for nonlinear SDEs of the form (7) in [21]. The problem of constructing the optimal nonreversible perturbation, in terms of the L 2 (π) spectral gap for Gaussian target densities was studied in [34] see also [55]. Optimal nonreversible perturbations with respect to miniziming the asymptotic variance were studied in [14,23].…”

“…It should be emphasized that the two optimality criteria, maximizing the spectral gap and minimizing the asymptotic variance, lead to different choices for the nonreversible drift γ (x). The goal of this paper is to extend the analysis presented in [14,34] by introducing the following modification of the standard underdamped Langevin dynamics:…”