Abstract
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 $\mathbb{R}^{d}$. 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.
Citation
Nicolas Brosse. Alain Durmus. Éric Moulines. "Normalizing constants of log-concave densities." Electron. J. Statist. 12 (1) 851 - 889, 2018. https://doi.org/10.1214/18-EJS1411
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