On quantitative Laplace-type convergence results for some exponential probability measures, with two applications

TL;DR

This paper provides quantitative bounds on the convergence of exponential measures to their limits as temperature approaches zero, using geometric measure theory and Wasserstein distances.

Contribution

It establishes new convergence bounds for measures with norm-like potentials under invertibility conditions, extending classical results without requiring Hessian invertibility.

Findings

Quantitative Wasserstein bounds for measures with norm-like potentials.

Application to maximum entropy models and low-temperature SGLD convergence.

Use of geometric measure theory tools like the coarea formula.

Abstract

Laplace-type results characterize the limit of sequence of measures $(\pi_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} \pi_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $\pi_0$ exists, it concentrates on the minimizers of the potential $U$. Classical results require the invertibility of the Hessian of $U$ in order to establish such asymptotics. In this work, we study the particular case of norm-like potentials $U$ and establish quantitative bounds between $\pi_\varepsilon$ and $\pi_0$ w.r.t. the Wasserstein distance of order $1$ under an invertibility condition of a generalized Jacobian. One key element of our proof is the use of geometric measure theory tools such as the coarea formula. We apply our results to the study of maximum entropy models (microcanonical/macrocanonical distributions) and to the convergence of the iterates of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm at low temperatures for non-convex minimization.