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Sampling from various kinds of distributions is an issue of paramountimportance in statistics since it is often the key ingredient for constructingestimators, test procedures or confidence intervals. In many situations, theexact sampling from a given distribution is impossible or computationallyexpensive and, therefore, one needs to resort to approximate samplingstrategies. However, there is no well-developed theory providing meaningfulnonasymptotic guarantees for the approximate sampling procedures, especially inthe high-dimensional problems. This paper makes some progress in this directionby considering the problem of sampling from a distribution having a smooth andlog-concave density defined on \(\RR^p\), for some integer \(p>0\). Weestablish nonasymptotic bounds for the error of approximating the targetdistribution by the one obtained by the Langevin Monte Carlo method and itsvariants. We illustrate the effectiveness of the established guarantees withvarious experiments. Underlying our analysis are insights from the theory ofcontinuous-time diffusion processes, which may be of interest beyond theframework of log-concave densities considered in the present work.

Theoretical guarantees for approximate sampling from smooth and log-concave densities