Normalizing constants of log-concave densities

We derive explicit bounds for the computation of normalizing constants Z for log-concave densities π = e^−U/Z w.r.t. the Lebesgue measure on R^d. Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm [15]. 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.