Section: New Results

The Tamed Unadjusted Langevin Algorithm

We consider the problem of sampling from a probability measure $\pi$ having a density on ${ℝ}^d$ known up to a normalizing constant, $x\to {\mathrm{e}}^{-U\left(x\right)}/Z$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable in a precise sense, 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 $\pi$, as well as weak error bounds. Numerical experiments support our findings.