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.

翻译：我们考虑从高维目标分布 $\pi_\beta$ 中采样的问题，该分布定义在 $\mathbb{R}^d$ 上，其密度正比于 $\theta\mapsto e^{-\beta U(\theta)}$，采样方法基于离散化朗之万随机微分方程（SDE）的显式数值格式。在近期文献中，驯服方法被提出并研究，用以确保当朗之万SDE的漂移系数超线性增长时，基于朗之万的数值格式的稳定性。特别地，[Bro+19]中提出了驯服非调整朗之万算法（TULA），用于从势函数梯度超线性增长的此类目标分布中采样。然而，传统上基于朗之万的算法的Wasserstein距离理论保证通常是在假设势函数 $U$ 强凸的条件下推导的。本文提出了一种新的驯服因子，并在势函数 $U$ 可能非凸且其梯度超线性增长的设定下，推导了我们的算法（命名为修正驯服非调整朗之万算法，mTULA）的分布与目标分布 $\pi_\beta$ 之间在Wasserstein-1和Wasserstein-2距离上的非渐近理论界。我们得到了mTULA在步长 $\lambda$ 下的离散化误差在Wasserstein-1和Wasserstein-2距离上分别为 $\mathcal{O}(\lambda)$ 和 $\mathcal{O}(\lambda^{1/2})$ 的收敛速率。文中给出了支持我们理论发现的高维数值模拟，以展示我们算法的适用性。