  Discretization of continuous-time diffusion processes is a widely recognized
method for sampling. However, the canonical Euler Maruyama discretization of
the Langevin diffusion process, referred as Unadjusted Langevin Algorithm
(ULA), studied mostly in the context of smooth (gradient Lipschitz) and
strongly log-concave densities, is a considerable hindrance for its deployment
in many sciences, including statistics and machine learning. In this paper, we
establish several theoretical contributions to the literature on such sampling
methods for non-convex distributions. Particularly, we introduce a new mixture
weakly smooth condition, under which we prove that ULA will converge with
additional log-Sobolev inequality. We also show that ULA for smoothing
potential will converge in $L_{2}$-Wasserstein distance. Moreover, using
convexification of nonconvex domain \citep{ma2019sampling} in combination with
regularization, we establish the convergence in Kullback-Leibler (KL)
divergence with the number of iterations to reach $\epsilon$-neighborhood of a
target distribution in only polynomial dependence on the dimension. We relax
the conditions of \citep{vempala2019rapid} and prove convergence guarantees
under isoperimetry, and non-strongly convex at infinity.
