Constrained Sampling with Primal-Dual Langevin Monte Carlo (2411.00568v2)

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.

• The paper introduces the PD-LMC algorithm that couples Langevin dynamics with primal-dual optimization to enforce statistical constraints in sampling.
• It rigorously proves convergence under conditions like strong convexity and log-Sobolev inequalities within Wasserstein space.
• Experiments validate the method’s effectiveness in applications such as fairness in predictive modeling and counterfactual Bayesian inference.

Constrained Sampling with Primal-Dual Langevin Monte Carlo: An Overview

The paper introduces a novel approach to the problem of constrained sampling from probability distributions where the distribution is known up to a normalization constant and subject to statistical constraints. These constraints are specified via expected values of nonlinear functions, which are common in Bayesian inference scenarios for tasks such as incorporating fairness constraints or evaluating counterfactual scenarios. This approach is crucial because conventional MCMC methods lack natural mechanisms to enforce such statistical constraints.

Core Contributions

The authors present a new discrete-time algorithm, Primal-Dual Langevin Monte Carlo (PD-LMC), which involves coupling Langevin dynamics adapted for Wasserstein space with saddle-point optimization methods. This dual-ascent methodology employs primal-dual algorithms that maintain constraint satisfaction while conducting sampling. Notably, PD-LMC overcomes the support constraints by modifying the sampling process to not only adhere to the desired target distribution's properties but also integrate constraints seamlessly.

Key contributions of this paper include:

1. Algorithm Development:
   • Introduction of the PD-LMC that intertwines constrained optimization techniques within the Langevin Monte Carlo framework.
   • Implementation of simultaneous sampling and constraint adherence using gradient descent-ascent dynamics within Wasserstein space, thus addressing challenges of conventional methods in handling statistical constraints.
2. Theoretical Analysis:
   • The paper rigorously demonstrates the convergence of the proposed algorithm under standard assumptions, particularly focusing on strong convexity and log-Sobolev inequalities of the target distribution.
   • Expansion of classical optimization arguments to the geometry of Wasserstein space to support the convergence analysis of the PD-LMC algorithm.
3. Practical Implications and Experiments:
   • Validation of the PD-LMC algorithm's effectiveness through multiple applications, demonstrating its practical potential in scenarios such as ensuring fairness in predictive models and exploring counterfactual scenarios within Bayesian frameworks.

Mathematical Insights

The proposed PD-LMC method serves as a sampling counterpart of gradient descent-ascent methods. The transformation of the constrained sampling problem into a dual problem offers substantial computational advantages. The characterization of solutions using Lagrange multipliers further ties optimization techniques with sampling strategies by exploiting dual problem properties in probabilistic space.

Performance & Implications

Numerically, the paper claims strong empirical results that underscore PD-LMC’s efficacy and flexibility in tackling complex constrained sampling problems. The methodology could advance AI applications by providing more robust sampling techniques that are preparatory for applications in fairness, robust model assessments, and potentially new realms where constraints need enforcement within probabilistic models.

Future Directions

Extending the analysis to almost sure convergence results and incorporating accelerated methods or proximal algorithms could enhance the PD-LMC. This development would be particularly crucial in addressing high-dimensional scenarios or more intricate statistical constraints. Further exploring these prospects might unlock novel applications or theoretical insights, thus broadening constrained sampling's utility in AI research.

The paper offers a competent blend of algorithmic innovation and theoretical backbone, paving the way for enhanced structured sampling methods aligned with statistical constraints. Given its highly pertinent contributions, it signifies a substantial step forward in probabilistic modeling and AI research domains focusing on principled constraint handling.