The tamed unadjusted Langevin algorithm

Abstract

In this article, we consider the problem of sampling from a probability measure π having a density on R d proportional to x → e −U (x) . The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V -total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

“…A second difficult feature of such a Hamiltonian is the singularity of the interacting function W , which typically results in numerical instability. A standard stabilization procedure is to «tame» the dynamics [38,11], which is the strategy adopted in [55]. However, this smoothing of the force induces a supplementary bias in the invariant measure, as shown in [11] for regular Hamiltonians.…”

Higher order Langevin Monte Carlo algorithm

“…Proof of Theorem 2. The proof follows along the same lines as the proof of Theorem 4 in [2], but for the completeness, the details are given below. By Proposition 1, for all n ∈ N and x ∈ R d , we have…”

“…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]. However, recent research has developed new types of explicit numerical schemes for SDEs with superlinear coefficients, and it has been shown in [15], [17], [16], [18], [13], [19], that the tamed Euler (Milstein) scheme converges to the true solution of the SDE (1) in L p on any given finite time horizon with optimal rate.…”

“…However, recent research has developed new types of explicit numerical schemes for SDEs with superlinear coefficients, and it has been shown in [15], [17], [16], [18], [13], [19], that the tamed Euler (Milstein) scheme converges to the true solution of the SDE (1) in L p on any given finite time horizon with optimal rate. This progress led to the creation of the tamed unadjusted Langevin algorithm (TULA) in [2], where the aforementioned convergence results are extended to an infinite time horizon and, moreover, one obtains rate of convergence results in total variation and in Wasserstein distance. In this article, a new higher order LMC algorithm is constructed, which is based on the order 1.5 scheme (3) of the SDE (1).…”

“…One observes that the scheme (2) coincides with (3) in distribution, and the latter is obtained using an Itô-Taylor expansion (see [22]). By extending the techniques used in [2] and [20], it can be shown that the scheme (2) has a unique invariant measure π γ , where γ ∈ (0, 1) is the stepsize of the scheme, and one obtains the improved convergence results between π γ and the target distribution π. More precisely, assume the potential U is three times differentiable, and its third derivative is locally Hölder continuous with exponent β ∈ (0, 1].…”