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

Laplace-type results characterize the limit of sequence of measures (π_varepsilon)varepsilon >0 with density w.r.t the Lebesgue measure (d π_varepsilon / d Leb)(x) propto exp[-U(x)/varepsilon] when the temperature varepsilon>0 converges to 0. If a limiting distribution π0 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 π_varepsilon and π0 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.