In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler--Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE -- in KL divergence for kTULA, and in total variation for tRLMC -- valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.

翻译：本文研究具有局部Lipschitz、超线性增长漂移的随机微分方程的数值离散化，及其对满足对数Sobolev不等式的非对数凹分布采样的影响。在此框架下，经典Euler-Maruyama格式（作为非调整Langevin算法ULA的基础）已知是不稳定的。我们分析了KL加速的驯服非调整Langevin算法（kTULA），并引入了一种新的驯服随机化中点格式，称为tRLMC。基于\cite{chewi2024local}的移位组合方法，我们发展了两个新的局部误差框架，针对一般局部Lipschitz漂移，分别在kTULA的KL散度和tRLMC的总变差范数下，得到关于底层SDE的有限时间非渐近误差估计。将这些框架专门应用于对数Sobolev不等式下的采样问题，我们获得了kTULA在KL散度下的近最优$\widetilde{O}(\varepsilon^{-1/2})$迭代复杂度，以及对应的总变差和Wasserstein距离保证。此外，我们首次建立了超线性漂移增长下驯服随机化Langevin格式在总变差下的非渐近保证，同时给出了相应的Wasserstein距离界，tRLMC的复杂度均为$\widetilde{O}(\varepsilon^{-1})$。作为推论，两种格式均对非凸超额风险优化问题给出了非渐近界。