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arXiv · 1707.00460

Normalizing constants of log-concave densities

Nicolas Brosse·Alain Durmus·Éric Moulines
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Abstract

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 $\mathbb{R}^d$. 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.

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Nicolas Brosse, Alain Durmus, Éric Moulines. 2018-02-28. Normalizing constants of log-concave densities. https://arxiv.org/abs/1707.00460

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