Nonasymptotic convergence analysis for the tamed unadjusted stochastic Langevin algorithm

In this work, we consider sampling from a target distribution $\pi_{\beta}$ characterized by the density function $ \pi_{\beta}( \theta) = e^{-\beta U(\theta) }/ \int_{\mathbb R^d} e^{-\beta U(\theta)} \, \mathrm{d} \theta$ with $\beta>0$. It is well-known that the Euler-Maruyama discretization of overdamped Langevin stochastic differential equations (SDEs) exhibits instability when the potentials exhibit superlinear growth. Building upon the approach proposed in \cite{brosse2019tamed} for mitigating the impact of superlinear drift coefficients in SDEs, we propose a novel Langevin dynamics-based algorithm, termed the Tamed Unadjusted Stochastic Langevin Algorithm (TUSLA), to address the aforementioned sampling problem and establish rigorous performance guarantees. Specifically, we establish a sharp non-asymptotic convergence guarantee in Kullback–Leibler (KL) divergence with the optimal rate of order one, by combining tools from the logarithmic Sobolev inequality (LSI) and the Fokker–Planck equation. As a direct consequence, we further obtain an $O(\lambda^{1/2})$ convergence rate in both Wasserstein-2 and total variation distances, thereby strengthening and generalizing the best-known results in the current literature. Our theoretical findings are supported by comprehensive high-dimensional experiments.