This work considers the problem of sampling from a probability distribution known up to a normalization constant while satisfying a set of statistical constraints specified by the expected values of general nonlinear functions. This problem finds applications in, e.g., Bayesian inference, where it can constrain moments to evaluate counterfactual scenarios or enforce desiderata such as prediction fairness. Methods developed to handle support constraints, such as those based on mirror maps, barriers, and penalties, are not suited for this task. This work therefore relies on gradient descent-ascent dynamics in Wasserstein space to put forward a discrete-time primal-dual Langevin Monte Carlo algorithm (PD-LMC) that simultaneously constrains the target distribution and samples from it. We analyze the convergence of PD-LMC under standard assumptions on the target distribution and constraints, namely (strong) convexity and log-Sobolev inequalities. To do so, we bring classical optimization arguments for saddle-point algorithms to the geometry of Wasserstein space. We illustrate the relevance and effectiveness of PD-LMC in several applications.

本文解决了在满足一组统计约束条件下，从已知归一化常数的概率分布中进行采样的问题，特别是在贝叶斯推断中的应用。文章提出了一种新的原始-对偶朗之万蒙特卡洛算法（PD-LMC），该算法利用Wasserstein空间中的渐变下降-上升动态，能够同时约束目标分布并进行采样。研究表明，PD-LMC在多个应用中具有良好的效果和相关性。

具有约束采样的原始-对偶朗之万蒙特卡洛