  We consider the problem of sampling from a high-dimensional target
distribution $\pi_\beta$ on $\mathbb{R}^d$ with density proportional to
$\theta\mapsto e^{-\beta U(\theta)}$ using explicit numerical schemes based on
discretising the Langevin stochastic differential equation (SDE). In recent
literature, taming has been proposed and studied as a method for ensuring
stability of Langevin-based numerical schemes in the case of super-linearly
growing drift coefficients for the Langevin SDE. In particular, the Tamed
Unadjusted Langevin Algorithm (TULA) was proposed in [Bro+19] to sample from
such target distributions with the gradient of the potential $U$ being
super-linearly growing. However, theoretical guarantees in Wasserstein
distances for Langevin-based algorithms have traditionally been derived
assuming strong convexity of the potential $U$. In this paper, we propose a
novel taming factor and derive, under a setting with possibly non-convex
potential $U$ and super-linearly growing gradient of $U$, non-asymptotic
theoretical bounds in Wasserstein-1 and Wasserstein-2 distances between the law
of our algorithm, which we name the modified Tamed Unadjusted Langevin
Algorithm (mTULA), and the target distribution $\pi_\beta$. We obtain
respective rates of convergence $\mathcal{O}(\lambda)$ and
$\mathcal{O}(\lambda^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for
the discretisation error of mTULA in step size $\lambda$. High-dimensional
numerical simulations which support our theoretical findings are presented to
showcase the applicability of our algorithm.
