Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting

Abstract

We consider the problem of sampling from a high-dimensional target distribution $π_β$ on $\mathbb{R}^d$ with density proportional to $θ\mapsto e^{-βU(θ)}$ 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 $π_β$. We obtain respective rates of convergence $\mathcal{O}(λ)$ and $\mathcal{O}(λ^{1/2})$ in Wasserstein-1 and Wasserstein-2 distances for the discretisation error of mTULA in step size $λ$. High-dimensional numerical simulations which support our theoretical findings are presented to showcase the applicability of our algorithm.