High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2016

High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm


We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density wrt the Lebesgue measure on $\mathbb{R}^d$, known up to a normalisation factor $x \mapsto \mathrm{e}^{−U (x)} / \int_{\mathbb{R}^d} \mathrm{e}^{−U (y)}\mathrm{d}y$. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that $U$ is continuously differentiable, $\nabla U$ is globally Lipschitz and $U$ is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order $2$ and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of the obtained bounds is studied to demonstrate the applicability of this method. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are either Lipchitz continuous or measurable and bounded. An illustration to a Bayesian inference for binary regression is presented.
Fichier principal
Vignette du fichier
main.pdf (598.05 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-01304430 , version 1 (19-04-2016)
hal-01304430 , version 2 (09-12-2016)


  • HAL Id : hal-01304430 , version 2


Alain Durmus, Éric Moulines. High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm. 2016. ⟨hal-01304430v2⟩
907 View
1223 Download


Gmail Facebook X LinkedIn More