Constrained Sampling with Primal-Dual Langevin Monte Carlo

2024-11-01

arXiv.org

AI Summary

This work addresses constrained sampling from unnormalized probability distributions subject to nonlinear statistical constraints—e.g., prescribed expectations—arising in fair Bayesian inference and counterfactual analysis. Existing methods lack theoretical convergence guarantees. To bridge this gap, we introduce the first provably convergent primal-dual Langevin Monte Carlo (PD-LMC) framework for such constraints, grounded in gradient descent–ascent dynamics on the Wasserstein space. Our method integrates Wasserstein duality, discrete-time Langevin dynamics, and leverages strong convexity together with the logarithmic Sobolev inequality to establish rigorous convergence. We prove that PD-LMC converges to the constrained optimal distribution at an explicit rate under standard assumptions. Experiments demonstrate its efficacy in satisfying complex statistical constraints and show significant improvements over baseline methods in fair inference tasks.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Constrained Sampling
• Bayesian Inference
• Probabilistic Models

Innovation

• PD-LMC Algorithm
• Bayesian Inference
• Restricted Sampling