Hypocoercivity of piecewise deterministic Markov process-Monte Carlo

Andrieu, Christophe; Durmus, Alain; Nüsken, Nikolas; Roussel, Julien

doi:10.1214/20-aap1653

Cited by 29 publications

(67 citation statements)

References 47 publications

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“…and the Zig-Zag process targets µ as in (2). For all k ′ < k, there exist β ∈ (0, 1) and δ, B > 0 such that for all (x, θ) ∈ R × {−1, +1},…”

Section: Resultsmentioning

confidence: 99%

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A Note on the Polynomial Ergodicity of the One-Dimensional Zig-Zag process

Vasdekis¹,

Roberts²

2021

Preprint

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“…and the Zig-Zag process targets µ as in (2). For all k ′ < k, there exist β ∈ (0, 1) and δ, B > 0 such that for all (x, θ) ∈ R × {−1, +1},…”

Section: Resultsmentioning

confidence: 99%

“…On the other hand, it was shown in Theorem 1 of [9] that the process will converge to the invariant measure under very mild assumptions, including the heavy tailed case. Furthermore, [1] used hypocoercivity techniques (see [2]) to prove polynomial rates of convergence for the Zig-Zag process on heavy tailed targets in arbitrary dimension.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Polynomial Ergodicity of the One-Dimensional Zig-Zag process

Vasdekis¹,

Roberts²

2021

Preprint

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“…We adopt the Zig-Zag sampler (Bierkens et al (2019)) which is a sampler based on the theory of piecewise deterministic Markov processes (see Fearnhead et al (2018), Bouchard-Côté et al (2015), Andrieu and Livingstone (2019), Andrieu et al (2018)). The main reasons motivating this choice are:…”

Section: Contribution Of the Papermentioning

confidence: 99%

“…(e) The process is non-reversible: as shown, for example, in Diaconis et al (2000), nonreversibility generally enhances the speed of convergence to the invariant measure and mixing properties of the sampler. For an advanced analysis on convergences results for this class of non-reversible processes, we refer to the articles Andrieu and Livingstone (2019) and Andrieu et al (2018).…”

Section: Contribution Of the Papermentioning

confidence: 99%

A piecewise deterministic Monte Carlo method for diffusion bridges

Bierkens¹,

Grazzi²,

Meulen³

et al. 2020

Preprint

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We introduce the use of the Zig-Zag sampler to the problem of sampling of conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy-Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber-Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of a local version of the Zig-Zag sampler that allows to exploit the sparse dependency structure of the coefficients of the Faber-Schauder expansion to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples. Contrary to some other Markov Chain Monte Carlo methods, our approach works well in case of strong nonlinearity in the drift and multimodal distributions of sample paths.

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“…The dimensional complexity of BPS and ZZ was studied in (Bierkens et al, 2018;Deligiannidis et al, 2018;Andrieu et al, 2018). For the case of an isotropic target distribution, the rate of reflections per unit of time is constant for BPS and proportional to d for ZZ with unit speeds in all directions.…”

Section: Scaling With Dimensionmentioning

confidence: 99%

The Boomerang Sampler

Bierkens,

Grazzi,

Kamatani

et al. 2020

Preprint

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This paper introduces the boomerang sampler as a novel class of continuous-time non-reversible Markov chain Monte Carlo algorithms. The methodology begins by representing the target density as a density, e −U , with respect to a prescribed (usually) Gaussian measure and constructs a continuous trajectory consisting of a piecewise elliptical path. The method moves from one elliptical orbit to another according to a rate function which can be written in terms of U . We demonstrate that the method is easy to implement and demonstrate empirically that it can out-perform existing benchmark piecewise deterministic Markov processes such as the bouncy particle sampler and the Zig-Zag. In the Bayesian statistics context, these competitor algorithms are of substantial interest in the large data context due to the fact that they can adopt data subsampling techniques which are exact (ie induce no error in the stationary distribution). We demonstrate theoretically and empirically that we can also construct a control-variate subsampling boomerang sampler which is also exact, and which possesses remarkable scaling properties in the large data limit. We furthermore illustrate a factorised version on the simulation of diffusion bridges.1. We relax a condition which restricts the covariance function of the auxiliary velocity process to be isotropic. This generalisation is crucial to ensure good convergence properties of the algorithm.2. Furthermore we extend the Boomerang Sampler to allow for exact subsampling (as introduced above),

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Hypocoercivity of piecewise deterministic Markov process-Monte Carlo

Cited by 29 publications

References 47 publications

A Note on the Polynomial Ergodicity of the One-Dimensional Zig-Zag process

A Note on the Polynomial Ergodicity of the One-Dimensional Zig-Zag process

A piecewise deterministic Monte Carlo method for diffusion bridges

The Boomerang Sampler

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