Laplace-type results characterize the limit of sequence of measures $(π_\varepsilon)_{\varepsilon >0}$ with density w.r.t the Lebesgue measure $(\mathrm{d} π_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$ when the temperature $\varepsilon>0$ converges to $0$. If a limiting distribution $π_0$ 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 $π_\varepsilon$ and $π_0$ 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.

翻译：Laplace型结果刻画了具有Lebesgue测度密度$(\mathrm{d} π_\varepsilon / \mathrm{d} \mathrm{Leb})(x) \propto \exp[-U(x)/\varepsilon]$的测度序列$(π_\varepsilon)_{\varepsilon >0}$在温度$\varepsilon>0$趋近于$0$时的极限。若极限分布$π_0$存在，则其集中于势函数$U$的极小值点。经典结果要求$U$的Hessian矩阵可逆以建立此类渐近性。本文研究范数型势函数$U$的特例，在广义Jacobian可逆条件下，建立了$π_\varepsilon$与$π_0$关于1阶Wasserstein距离的定量界。证明的关键要素是使用几何测度论工具，如余面积公式。我们将所得结果应用于最大熵模型（微正则/宏正则分布）的研究以及随机梯度Langevin动力学（SGLD）算法在低温非凸极小化中迭代序列收敛性的分析。