Non-asymptotic analysis of Langevin-type Monte Carlo algorithms

Abstract We study Langevin-type algorithms for sampling from Gibbs distributions such that the potentials are dissipative and their weak gradients have finite moduli of continuity not necessarily convergent to zero. Our main result is a non-asymptotic upper bound on the 2-Wasserstein distance between a Gibbs distribution and the law of general Langevin-type algorithms based on a Liptser–Shiryaev-type condition for change of measures and Poincaré inequalities. We apply this bound to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy if the potentials are dissipative and their gradients are uniformly continuous. We also propose Langevin-type algorithms with spherical smoothing for distributions whose potentials are not convex or continuously differentiable and show their polynomial complexities.