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Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations
Assyr Abdulle, Grigorios A. Pavliotis, Gilles Vilmart
TL;DR
Sampling high-dimensional probability measures motivates modified Langevin dynamics with improved sampling properties. The paper adds a measure-preserving Stratonovich perturbation to overdamped Langevin dynamics; it remains reversible while improving convergence and asymptotic variance.
Problem
Sampling high-dimensional probability measures is an important problem, and nonreversible Langevin perturbations can introduce oscillatory transients.
Source: §1 · pp. 2–3Method
The paper perturbs overdamped Langevin dynamics with a divergence-free Stratonovich noise that preserves the target measure and yields reversible dynamics.
Source: §1 · p. 3Results
The perturbation increases the spectral gap and reduces asymptotic variance relative to standard overdamped Langevin dynamics, improving mean-squared error.
Source: §1 · p. 4Takeaways & Limitations
The resulting sampler retains the convergence and variance advantages of nonreversible perturbations without their oscillatory transient behavior.
Source: §1 · pp. 3–4The amount of asymptotic-variance improvement depends on the observable's projection onto the perturbation's null space.
Source: §2 · p. 5
Abstract
from arXiv · showhide
In this paper we propose a new approach for sampling from probability measures in, possibly, high dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in~[C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Ann. Appl. Probab. 1993] and studied in, e.g. [T. Lelievre, F. Nier, and G. A. Pavliotis J. Stat. Phys., 2013] and [A. B. Duncan, T. Leli\`evre, and G. A. Pavliotis J. Stat. Phys., 2016], while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution.
1. Introduction
Sampling high-dimensional probability measures motivates modified Langevin dynamics with better sampling properties. The paper introduces a Stratonovich perturbation that preserves the target measure while retaining reversibility and improving convergence and asymptotic variance.
1. Introduction
- Earlier nonreversible perturbations preserve the target measure and improve sampling, but their nonselfadjoint generator can introduce oscillatory transients.An appropriate splitting numerical scheme is noted as a possible way to address that transient behavior.Source: §1 · p. 3
- The perturbation is reversible and preserves the target invariant measure, avoiding the oscillatory transient associated with the nonselfadjoint generator of nonreversible perturbations.The perturbing vector field satisfies the divergence-free condition, and the added scalar Brownian motion is independent of the original noise.Source: §1 · p. 3
- The reversibility and invariant-measure results extend to general ergodic SDEs, including non-gradient drifts and degenerate diffusions such as underdamped Langevin dynamics.When the target distribution's functional form is unknown, computing a suitable divergence-free vector field can be difficult.Source: §1 · p. 3
- Multidimensional Brownian perturbations and preconditioned or Riemannian-manifold Langevin extensions are identified as directions for future work.The multidimensional perturbation uses divergence-free diffusion functions.Source: §1 · p. 3
- Compared with standard overdamped Langevin dynamics, the Stratonovich perturbation gives a larger spectral gap and reduced asymptotic variance, improving mean-squared error.The variance improvement depends on the observable's projection onto the perturbation's null space; if that projection is zero, the inequality is strict.Source: §1 · p. 4
- For a two-dimensional anisotropic Gaussian, the reversible perturbation achieves λS=O(1) with δ~λ^-1/2 and θ=1/2; its optimal choice is shared across convergence and variance criteria.The standard Langevin rate is λL=λ, while the nonreversible rate reaches O(1) with θ=1.Source: §1 · p. 4
2. Proof of the main results
The proofs use the divergence-free perturbation to establish reversibility and invariant-measure preservation, then compare spectral gaps and asymptotic variances. The variance improvement depends on how the observable projects onto the perturbation’s null space.
2. Proof of the main results
- Skew-symmetry of A and symmetry of A² support the symmetry of the Stratonovich-perturbed generator.The proof writes the perturbed generator using the original generator and A².Source: §2 · pp. 4–5
- The perturbed dynamics has π as its unique invariant measure.The proof uses L*π = A*π = 0.Source: §2 · p. 4
- The spectral-gap comparison is framed through Rayleigh quotients and the quadratic forms of the original and perturbed generators.The proof gives the quadratic forms for both generators.Source: §2 · p. 5
- The asymptotic-variance comparison uses Poisson-equation solutions and reduces the proof to nonnegativity of an inner product involving A.The proof introduces solutions for both generators and identifies the relevant inner product.Source: §2 · p. 5
- The variance improvement depends on the observable’s projection onto the null space of A²; a zero projection makes the inequality strict.A² is only negative semidefinite, so its null space is larger than the constants-only null space of the overdamped Langevin generator.Source: §2 · p. 5
3. Numerical experiments
The experiments compare the Stratonovich perturbation with standard and nonreversible Langevin dynamics on a two-dimensional warped Gaussian target. Both perturbations increase convergence speed and reduce asymptotic variance relative to standard dynamics, while the Stratonovich perturbation avoids oscillatory transients.
3. Numerical experiments
- The experiment uses a two-dimensional warped Gaussian target with warping parameter b = 0.05.The authors choose this lower-dimensional example for simplicity while aiming to illustrate the theoretical findings.Source: §3 · p. 5
- The comparison includes standard Langevin, a nonreversible perturbation with θ = 1, and a Stratonovich perturbation with θ = 1/2.The perturbations are tested at several sizes.Source: §3 · p. 5
- The estimator averages the observable over trajectories starting at (0, 0), with the SDE approximated by Euler–Maruyama using Δt = 10^-5.The plotted quantity is the error between the trajectory-average estimate and π(φ).Source: §3 · p. 5
- Both perturbations increase convergence speed and reduce asymptotic variance versus standard Langevin; the Stratonovich perturbation shows no oscillatory behavior.The figure tracks error over time; oscillations in the equilibrium phase are attributed to Monte Carlo error.Source: §3 · pp. 5–6, Figure 1 · p. 6