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2021 High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion.
Pierre Monmarché
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Electron. J. Statist. 15(2): 4117-4166 (2021). DOI: 10.1214/21-EJS1888

Abstract

The efficiency of a Markov sampler based on the underdamped Langevin diffusion is studied for high dimensional targets with convex and smooth potentials. We consider a classical second-order integrator which requires only one gradient computation per iteration. Contrary to previous works on similar samplers, a dimension-free contraction of Wasserstein distances and convergence rate for the total variance distance are proven for the discrete time chain itself. Non-asymptotic Wasserstein and total variation efficiency bounds and concentration inequalities are obtained for both the Metropolis adjusted and unadjusted chains. In particular, for the unadjusted chain, in terms of the dimension d and the desired accuracy ε, the Wasserstein efficiency bounds are of order dε in the general case, dε if the Hessian of the potential is Lipschitz, and d14ε in the case of a separable target, in accordance with known results for other kinetic Langevin or HMC schemes.

Funding Statement

P. Monmarché acknowledges financial support from the French ANR grant EFI (Entropy, flows, inequalities, ANR-17-CE40-0030).

Acknowledgments

P. Monmarché thanks Tony Lelièvre, Benedict Leimkuhler, Gabriel Stoltz and Alaind Durmus for stimulating discussions on the topic of Langevin integrators.

Citation

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Pierre Monmarché. "High-dimensional MCMC with a standard splitting scheme for the underdamped Langevin diffusion.." Electron. J. Statist. 15 (2) 4117 - 4166, 2021. https://doi.org/10.1214/21-EJS1888

Information

Received: 1 August 2020; Published: 2021
First available in Project Euclid: 9 September 2021

Digital Object Identifier: 10.1214/21-EJS1888

Subjects:
Primary: 65C05

Keywords: Hamiltonian Monte Carlo , Langevin diffusion , Markov chain Monte Carlo , Wasserstein curvature

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Vol.15 • No. 2 • 2021
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