Variance Reduction Using Nonreversible Langevin Samplers

Abstract

A standard approach to computing expectations with respect to a given target measure is to introduce an overdamped Langevin equation which is reversible with respect to the target distribution, and to approximate the expectation by a time-averaging estimator. As has been noted in recent papers [30, 37, 61, 72], introducing an appropriately chosen nonreversible component to the dynamics is beneficial, both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution. In this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility. Our theoretical findings are supported by numerical simulations.

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…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 . This study was undertaken in [14] for processes of the form (7).…”

“…Remark 12 In [14], the authors consider the task of finding optimal perturbations J for the nonreversible overdamped Langevin dynamics given in (15). In the Gaussian case this optimization problem turns out be equivalent to the one considered in this section.…”

“…In the Gaussian case this optimization problem turns out be equivalent to the one considered in this section. Indeed, equation (39) of [14] can be rephrased as f ∈ ker(J q · ∇) ⊥ . Therefore, Algorithm 1 and its generalization Algorithm 2 (described in Sect.…”

“…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]. In all these works it was shown that, in theory [i.e.…”

“…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:…”