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

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