Multilevel-Langevin pathwise average for Gibbs sampling

Abstract

In this talk, I will introduce a new multilevel-Langevin method dedicated to the Gibbs sampling. More precisely, this procedure both inspired by [Szpruch et al., 2016] et [Pagès-Panloup, 2018] is based on a multilevel combination of pathwise averages of discretized schemes of overdamped-Langevin diffusions. In a general setting, we first show that for any positive ε, an appropriate choice of the parameters (including steps/number of layers/ length of the path,...) leads to an ε-approximation with a cost proportional to ε−2, i.e. proportional to a Monte-Carlo method without bias. In a second part, we investigate, in the uniformly convex setting, the dependence in the dimension and optimize the choice of the parameters of the multilevel strategy in terms of d. We show that it is possible to obtain a complexity in (dε^{-2})\log^3(dε−2), which is very close to the unbeatable complexity dε^{-2}. This talk is based on a joint work with M. Egea.