High-dimensional Bayesian inference via the unadjusted Langevin algorithm

Abstract

We consider in this paper the problem of sampling a high-dimensional probability distribution π having a density w.r.t. the Lebesgue measure on R d , known up to a normalization constant x → π(x) = e −U(x) / R d e −U(y) dy. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that U is continuously differentiable, ∇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. 1

Higher order Langevin Monte Carlo algorithm

“…To the best of the authors' knowledge, these are the first such results which provide a higher rate of convergence in Wasserstein distance compared to the existing literature. As for the total variation distance, [11] proves that the rate of convergence is 1 for the case of a strongly convex U , whereas our result yields the same convergence rate without assuming convexity.…”

“…The corresponding numerical scheme of the Langevin SDE obtained by using the Euler-Maruyama (Milstein) method yields the unadjusted Langevin algorithm (ULA), known also as the Langevin Monte Carlo (LMC), which has been well studied in the literature. For a globally Lipschitz ∇U , the non-asymptotic bounds in total variation and Wasserstein distance between the n-th iteration of the ULA algorithm and π have been provided in [11], [12] and [10]. As for the case of superlinear ∇U , the difficulty arises from the fact that ULA is unstable (see [23]), and its Metropolis adjusted version, MALA, loses some of its appealing properties as discussed in [7] and demonstrated numerically in [2].…”

“…An important feature of the newly proposed, higher order LMC is its computational efficiency. It can be seen in key paradigms of the area, such as sampling from a high-dimensional Gaussian distribution, since one obtains with limited additional computational effort, that the distribution of the algorithm converges faster to the target distribution π compared to existing algorithms (see [9], [11] and references therein). In other words, it takes fewer steps for the error to be less than a given ε, where the error is measured using the notion of a 2-Wasserstein distance between the n-th iteration of the algorithm (3) and π.…”

“…whereC > 0 is some constant that is proportional to d and x * is the unique minimizer of U . Thus, the algorithm (3) requires much fewer steps to reach a certain precision level compared to results in [9] and [11]. However, one notes that for the case of a Lipschitz gradient with nonzero L 2 and L in assumptions H5 and H6 (see below), the dependency of the Wasserstein bound on the dimension increases from d to d 2 .…”

Variance reduction for Markov chains with application to MCMC

“…Under mild technical conditions, the Langevin diffusion admits π as its unique invariant distribution. We consider the sampling method based on the Euler-Maruyama discretization of (13). This scheme referred to as unadjusted Langevin algorithm (ULA), defines the discrete-time Markov chain (X k ) k≥0 given by…”