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Discretization of continuous-time diffusion processes is a widely recognizedmethod for sampling. However, the canonical Euler Maruyama discretization ofthe Langevin diffusion process, referred as Unadjusted Langevin Algorithm(ULA), studied mostly in the context of smooth (gradient Lipschitz) andstrongly log-concave densities, is a considerable hindrance for its deploymentin many sciences, including statistics and machine learning. In this paper, weestablish several theoretical contributions to the literature on such samplingmethods for non-convex distributions. Particularly, we introduce a new mixtureweakly smooth condition, under which we prove that ULA will converge withadditional log-Sobolev inequality. We also show that ULA for smoothingpotential will converge in $L_{2}$-Wasserstein distance. Moreover, usingconvexification of nonconvex domain \citep{ma2019sampling} in combination withregularization, we establish the convergence in Kullback-Leibler (KL)divergence with the number of iterations to reach $\epsilon$-neighborhood of atarget distribution in only polynomial dependence on the dimension. We relaxthe conditions of \citep{vempala2019rapid} and prove convergence guaranteesunder isoperimetry, and non-strongly convex at infinity.

Unadjusted Langevin algorithm for non-convex weakly smooth potentials