Normalizing constants of log-concave densities

We derive explicit bounds for the computation of normalizing constants Z for log-concave densities π = exp(-U)/Z with respect to the Lebesgue measure on ℝd. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm (High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm, A. Durmus and E. Moulines). Polynomial bounds in the dimension d are obtained with an exponent that depends on the assumptions made on U. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established.