  Sampling from various kinds of distributions is an issue of paramount
importance in statistics since it is often the key ingredient for constructing
estimators, test procedures or confidence intervals. In many situations, the
exact sampling from a given distribution is impossible or computationally
expensive and, therefore, one needs to resort to approximate sampling
strategies. However, there is no well-developed theory providing meaningful
nonasymptotic guarantees for the approximate sampling procedures, especially in
the high-dimensional problems. This paper makes some progress in this direction
by considering the problem of sampling from a distribution having a smooth and
log-concave density defined on \(\RR^p\), for some integer \(p>0\). We
establish nonasymptotic bounds for the error of approximating the target
distribution by the one obtained by the Langevin Monte Carlo method and its
variants. We illustrate the effectiveness of the established guarantees with
various experiments. Underlying our analysis are insights from the theory of
continuous-time diffusion processes, which may be of interest beyond the
framework of log-concave densities considered in the present work.
