Unadjusted Langevin Algorithm for Non-convex Weakly Smooth Potentials
Dao Nguyen , Xin Dang , Yixin Chen
Communications in Mathematics and Statistics ›› 2025, Vol. 13 ›› Issue (4) : 979 -1036.
Unadjusted Langevin Algorithm for Non-convex Weakly Smooth Potentials
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
Langevin Monte Carlo / Kullback–Leibler divergence / Log-Sobolev inequality / Convexification / Mixture weakly smooth / 60J22 / 62F15 / 62M45 / 65C05
School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature
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