  In this paper, we revisit the recently established theoretical guarantees for
the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth
and (strongly) log-concave density. We improve the existing results when the
convergence is measured in the Wasserstein distance and provide further
insights on the very tight relations between, on the one hand, the Langevin
Monte Carlo for sampling and, on the other hand, the gradient descent for
optimization. Finally, we also establish guarantees for the convergence of a
version of the Langevin Monte Carlo algorithm that is based on noisy
evaluations of the gradient.
