  In this paper, we provide new insights on the Unadjusted Langevin Algorithm.
We show that this method can be formulated as a first order optimization
algorithm of an objective functional defined on the Wasserstein space of order
$2$. Using this interpretation and techniques borrowed from convex
optimization, we give a non-asymptotic analysis of this method to sample from
logconcave smooth target distribution on $\mathbb{R}^d$. Based on this
interpretation, we propose two new methods for sampling from a non-smooth
target distribution, which we analyze as well. Besides, these new algorithms
are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD)
algorithm, which is a popular extension of the Unadjusted Langevin Algorithm.
Similar to SGLD, they only rely on approximations of the gradient of the target
log density and can be used for large-scale Bayesian inference.
