提出一种新采样算法，能同时满足复杂约束并生成符合分布的数据。
Constrained Sampling with Primal-Dual Langevin Monte Carlo
- 基于对偶梯度下降-上升机制，在Wasserstein空间中设计离散算法
- 在强凸和log-Sobolev条件下保证收敛，理论严谨
- 适合需要约束条件的贝叶斯推断、公平性建模等场景
本文研究在仅知归一化常数前提下从概率分布中采样的问题，同时满足由一般非线性函数期望值定义的统计约束。该问题在贝叶斯推断中有应用，例如通过约束矩来评估反事实情景或实现预测公平性。现有处理支持集约束的方法（如镜映映射、障碍项和惩罚项）不适用于此类问题。为此，本文基于Wasserstein空间中的梯度下降-上升动力学，提出一种离散时间的对偶-原始Langevin蒙特卡洛算法（PD-LMC），可同时约束目标分布并从中采样。在目标分布与约束满足标准假设（强凸性和log-Sobolev不等式）的条件下，分析了PD-LMC的收敛性。方法将经典鞍点优化的论证引入Wasserstein几何框架。通过多个应用场景展示了PD-LMC的有效性。
原文摘要 · Abstract (English)
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.
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