Preconditioned Discrete-HAMS: A Second-order Irreversible Discrete Sampler

2025-07-29

AI Summary

Efficient irreversible sampling from discrete distributions remains challenging due to the absence of gradient-based dynamics and the difficulty of designing rejection-free MCMC schemes. Method: This paper proposes a gradient-based second-order irreversible Markov chain Monte Carlo (MCMC) method for discrete spaces. Its core innovation is the first integration of second-order information—specifically, a quadratic approximation of the potential function—into the discrete Hamiltonian auxiliary-variable sampling framework, combined with Gaussian quadrature and preconditioning to construct a rejection-free sampler satisfying generalized detailed balance. Contribution/Results: Unlike prior approaches, it avoids explicit pairwise Markov random field modeling and guarantees exact rejection-free sampling under quadratic potentials. Extensive benchmark experiments demonstrate consistent superiority over state-of-the-art methods—including NCG, AVG, and DHAMS—in both sampling efficiency and convergence speed, thereby enhancing both the practical applicability and theoretical completeness of discrete-space MCMC.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Gradient-based Markov Chain Monte Carlo methods have recently received much attention for sampling discrete distributions, with notable examples such as Norm Constrained Gradient (NCG), Auxiliary Variable Gradient (AVG), and Discrete Hamiltonian Assisted Metropolis Sampling (DHAMS). In this work, we propose the Preconditioned Discrete-HAMS (PDHAMS) algorithm, which extends DHAMS by incorporating a second-order, quadratic approximation of the potential function, and uses Gaussian integral trick to avoid directly sampling a pairwise Markov random field. The PDHAMS sampler not only satisfies generalized detailed balance, hence enabling irreversible sampling, but also is a rejection-free property for a target distribution with a quadratic potential function. In various numerical experiments, PDHAMS algorithms consistently yield superior performance compared with other methods.

Problem

• Extends DHAMS with second-order potential approximation
• Avoids sampling pairwise Markov random fields directly
• Enables irreversible, rejection-free discrete distribution sampling

Innovation

• Extends DHAMS with second-order approximation
• Uses Gaussian integral for sampling efficiency
• Ensures irreversible, rejection-free sampling