Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
Résumé
Sampling distribution over high-dimensional state-space is a problem which has recently attracted a lot of research efforts; applications include Bayesian non-parametrics, Bayesian inverse problems and aggregation of estimators.
All these problems boil down to sample a target distribution $\pi$ having a density \wrt\ the Lebesgue measure on $\mathbb{R}^d$, known up to a normalisation factor $x \mapsto \mathrm{e}^{-U(x)}/\int_{\mathbb{R}^d} \mathrm{e}^{-U(y)} \mathrm{d} y$ where $U$ is continuously differentiable and smooth. In this paper, we study a sampling technique based on the Euler discretization of the Langevin stochastic differential equation. Contrary to the Metropolis Adjusted Langevin Algorithm (MALA), we do not apply a Metropolis-Hastings correction. We obtain for both constant and decreasing step sizes in the Euler discretization, non-asymptotic bounds for the convergence to stationarity in both total variation and Wasserstein distances. A particular attention is paid on the dependence on the dimension of the state space, to demonstrate the applicability of this method in the high dimensional setting, at least when $U$ is convex. These bounds are based on recently obtained estimates of the convergence of the Langevin diffusion to stationarity using Poincar{\'e} and log-Sobolev inequalities. These bounds improve and extend the results of (Dalalyan, 2014). We also investigate the convergence of an appropriately weighted empirical measure and we report sharp bounds for the mean square error and exponential deviation inequality for Lipschitz functions. A limited Monte Carlo experiment is carried out to support our findings.
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