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:我们考虑的是基于高维目标分布 $\ pi ⁇ beta 的抽样问题。 美元= mathbb{ R ⁇ d$, 密度与 $\theta\ mapsto e\\\\\\\\beta U (\theta)} 使用基于朗埃文随机差异方程式( SDE ) 的清晰数字方案。 在最近的文献中, 提议并研究调制方法, 以确保朗埃文基数字方案在朗埃文SDE超级线性增长流系数的情况下的稳定性。 特别是, [Bro+19] 提议采用密度与美元成比例成正比的不调整的朗埃文Algooral- liforal=lation。 我们的诺埃文基调调调调调调调调调调调调调和调和调和调和调和的美元- 美元- 美元=lal=lal=lal 。