Abstract
A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi}$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally H\"{o}lder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ \beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.
Highlights
In Bayesian statistics and machine learning, one challenge, which has attracted substantial attention in recent years due to its high importance in data-driven applications, is the creation of algorithms which can efficiently sample from a high-dimensional target probability distribution π
The corresponding numerical scheme of the Langevin SDE obtained by using the Euler-Maruyama (Milstein) method yields the unadjusted Langevin algorithm (ULA), known as the Langevin Monte Carlo (LMC), which has been well studied in the literature
As for the case of superlinear ∇U, the difficulty arises from the fact that the algorithms constructed based on explicit numerical schemes, for example ULA, is unstable, and its Metropolis adjusted version, MALA, loses some of its appealing properties as discussed in [25] and demonstrated numerically in [1]
Summary
In Bayesian statistics and machine learning, one challenge, which has attracted substantial attention in recent years due to its high importance in data-driven applications, is the creation of algorithms which can efficiently sample from a high-dimensional target probability distribution π. Recent research has developed new types of explicit numerical schemes for SDEs with superlinear coefficients, and it has been shown in [3], [13], [16], [26], [27], [30], 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 [1], where the aforementioned convergence results are extended to an infinite time horizon and, one obtains rate of convergence results in total variation and in Wasserstein distance. For the HOLA algorithm (2), by extending the techniques used in [1] and [28], it can be shown that the scheme (2) has a unique invariant measure πγ, and one can obtain convergence results between πγ and the target distribution π in some proper distance.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
