Electronic Communications in Probability

Variance reduction for irreversible Langevin samplers and diffusion on graphs

Luc Rey-Bellet and Konstantinos Spiliopoulos

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In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is,from several points of view,  advantageous when compared to sampling from a time-reversible one. Adding an appropriate irreversible drift to the overdamped Langevin equation results  in a larger large deviations rate function for the empirical measure of the process, a smaller variance for the long time average of  observables of the process,  as well as a larger spectral gap. In this work, we concentrate on irreversible Langevin samplers with a drift of increasing intensity. The asymptotic variance  is monotonically decreasing with respect to the growth of the drift and we characterize its limiting behavior. For a Gibbs measure whose potential has one or more critical points, adding a large irreversible drift results in a decomposition of the process in a slow and fast component  with fast motion along the level sets of the potential and slow motion in the orthogonal direction. This result helps understanding the variance reduction, which can be explained  at the process level by the induced fast motion of the process along the level sets of the potential. Correspondingly the limit of the asymptotic variance is the  asymptotic variance of the limiting slow motion which is a diffusion process on a graph.

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Electron. Commun. Probab., Volume 20 (2015), paper no. 15, 16 pp.

Accepted: 14 February 2015
First available in Project Euclid: 7 June 2016

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Rey-Bellet, Luc; Spiliopoulos, Konstantinos. Variance reduction for irreversible Langevin samplers and diffusion on graphs. Electron. Commun. Probab. 20 (2015), paper no. 15, 16 pp. doi:10.1214/ECP.v20-3855. https://projecteuclid.org/euclid.ecp/1465320942

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