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Regular version of the site

Article

Normalizing constants of log-concave densities

Electronic journal of statistics. 2018. Vol. 12. No. 1. P. 851-889.
Moulines E., Brosse N., Durmus A.

We derive explicit bounds for the computation of normalizing constants Z for log-concave densities \pi=e^{−U}/Z w.r.t. the Lebesgue measure on Rd. 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.