• Bernoulli
  • Volume 25, Number 4A (2019), 2854-2882.

High-dimensional Bayesian inference via the unadjusted Langevin algorithm

Alain Durmus and Éric Moulines

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We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density w.r.t. the Lebesgue measure on $\mathbb{R}^{d}$, known up to a normalization constant $x\mapsto\pi(x)=\mathrm{e}^{-U(x)}/\int_{\mathbb{R}^{d}}\mathrm{e}^{-U(y)}\,\mathrm{d}y$. 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 these bounds is explicit. 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 measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.

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Bernoulli, Volume 25, Number 4A (2019), 2854-2882.

Received: July 2017
Revised: July 2018
First available in Project Euclid: 13 September 2019

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Langevin diffusion Markov chain Monte Carlo Metropolis adjusted Langevin algorithm rate of convergence total variation distance


Durmus, Alain; Moulines, Éric. High-dimensional Bayesian inference via the unadjusted Langevin algorithm. Bernoulli 25 (2019), no. 4A, 2854--2882. doi:10.3150/18-BEJ1073.

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Supplemental materials

  • Supplement to “High-dimensional Bayesian inference via the unadjusted Langevin algorithm”. Most proofs and derivations are postponed and carried out in a supplementary paper.