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 introduce a new tamed unadjusted Euler scheme termed adTULA and a new tamed randomized midpoint scheme, termed adTRLMC. Building on the shifted- composition approach of [1], we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE in KL divergence for adTULA, and in total variation for adTRLMC valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimall iteration complexity for adTULA 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 complexity for adTRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.
ZOOM: https://uoc-gr.zoom.us/j/88177475079?pwd=ik9vJRdyknP8xneS5NumlqgQdnplaz.1
