Normalizing constants of log-concave densities

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 (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.
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Submitted on : Tuesday, March 6, 2018 - 9:16:38 AM
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Nicolas Brosse, Alain Durmus, Éric Moulines. Normalizing constants of log-concave densities. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.851-889. ⟨10.1214/18-EJS1411⟩. ⟨hal-01648666v2⟩

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