Normalizing constants of log-concave densities

TL;DR

This paper develops explicit bounds for computing normalizing constants of log-concave densities using Gaussian annealing and Langevin algorithms, with results applicable in high dimensions and supported by numerical experiments.

Contribution

It introduces a novel method combining Gaussian annealing with Langevin bounds to estimate normalization constants in high-dimensional log-concave densities.

Findings

Polynomial bounds in dimension for normalization constant estimation

A theoretically grounded annealing sequence for variance

Numerical experiments validating the bounds

Abstract

We derive explicit bounds for the computation of normalizing constants $Z$ for log-concave densities $\pi = \exp(-U)/Z$ 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.