Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

Duncan, Anthony; Nüsken, Nikolas; Pavliotis, Grigorios A.

doi:10.1007/s10955-017-1906-8

Cited by 35 publications

(39 citation statements)

References 49 publications

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“…l Proof of Theorem 1. 3 We write the generator of the Stratonovich-perturbed dynamics as L S "´B 7 BÁ 7 A with B " ∇, A " g¨∇, A 7 "´A. The quadratic form associated to L S is x´L S φ, φy π " }Bφ} 2 π`} Aφ} 2 π for all φ P H 1 pπq the weighted Sobolev space that is defined in the standard manner.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations

Abdulle

Pavliotis

Vilmart

2019

Comptes Rendus. Mathématique

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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

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations

Abdulle

Pavliotis

Vilmart

2019

Comptes Rendus. Mathématique

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“…Another work [8,18] only analyzed skew acceleration for LD. A previous work [17] combined a non-reversible drift term with ULD. Unlike our method, this work's purpose was to reduce the asymptotic variance of the expectation of a test function and is mainly focusing on sampling from Gaussian distribution.…”

Section: Relation To Non-reversible Methodsmentioning

confidence: 99%

“…For example, skew acceleration has been analyzed focusing on sampling from Gaussian distributions [13][14][15][16][17], although assuming Gaussian distributions in Bayesian models is restrictive in practice. A recent study [8] theoretically showed that skew acceleration accelerates the dynamics around the local minima and saddle points for non-convex functions.…”

Section: Introductionmentioning

confidence: 99%

Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices

Futami

Iwata

Sato

2021

Entropy

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Langevin dynamics (LD) has been extensively studied theoretically and practically as a basic sampling technique. Recently, the incorporation of non-reversible dynamics into LD is attracting attention because it accelerates the mixing speed of LD. Popular choices for non-reversible dynamics include underdamped Langevin dynamics (ULD), which uses second-order dynamics and perturbations with skew-symmetric matrices. Although ULD has been widely used in practice, the application of skew acceleration is limited although it is expected to show superior performance theoretically. Current work lacks a theoretical understanding of issues that are important to practitioners, including the selection criteria for skew-symmetric matrices, quantitative evaluations of acceleration, and the large memory cost of storing skew matrices. In this study, we theoretically and numerically clarify these problems by analyzing acceleration focusing on how the skew-symmetric matrix perturbs the Hessian matrix of potential functions. We also present a practical algorithm that accelerates the standard LD and ULD, which uses novel memory-efficient skew-symmetric matrices under parallel-chain Monte Carlo settings.

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“…. , n, J i ∈ R d×d skew being skew-symmetric matrices and Q i : R d → R d×d sym being positive definite matrix-valued functions, as discussed in [28,Section 2]. Note that in this case the processes (X i t ) t≥0 are not copies of each other, since Q i and J i may not be the same for different particles.…”

Section: The General Casementioning

confidence: 99%

“…Remark 29. Generalising the above to the case where the particles have different frictions γ i and masses M i or some or all of them move according to perturbed versions of underdamped Langevin dynamics as considered in [28] is straightforward (see also Remark 24).…”

Section: Underdamped Langevin Dynamicsmentioning

confidence: 99%

Constructing Sampling Schemes via Coupling: Markov Semigroups and Optimal Transport

Nüsken¹,

Pavliotis²

2019

SIAM/ASA J. Uncertainty Quantification

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In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.

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Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

Cited by 35 publications

References 49 publications

Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations

Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations

Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices

Constructing Sampling Schemes via Coupling: Markov Semigroups and Optimal Transport

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