Section: New Results

Normalizing constants of log-concave densities

We derive explicit bounds for the computation of normalizing constants $Z$ for log-concave densities $\pi ={\mathrm{e}}^{-U}/Z$ 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 (Durmus, A. and Moulines, E. (2016). High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm). 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.