  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 of the 2-Wasserstein distance between a Gibbs
distribution and the law of general Langevin-type algorithms based on the
Liptser--Shiryaev theory and Poincar\'{e} 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.
