Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures. (22nd December 2021)

• Record Type: Journal Article
• Authors: Barbier, Jean; Panchenko, Dmitry; Sáenz, Manuel
• Is Part Of: Information and inference. Volume 11:Number 3(2022)
• Page Start: 1079
• Page End: 1108
• Publication Date: 2021-12-22
• Subjects: Bayesian inference -- disordered systems -- multioverlap concentration -- replica symmetry; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaab027
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: We consider a generic class of log-concave, possibly random, (Gibbs) measures. We prove the concentration of an infinite family of order parameters called multioverlaps. Because they completely parametrize the quenched Gibbs measure of the system, this implies a simple representation of the asymptotic Gibbs measures, as well as the decoupling of the variables in a strong sense. These results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimization, maximum a-posteriori inference or M-estimation. We believe that they may be applicable in establishing some type of 'replica symmetric formulas' for the free energy, inference or generalization error in such settings.