A Generative Sampler for distributions with possible discrete parameter based on Reversibility

2026-03-10

AI Summary

This work addresses the failure of conventional gradient-based sampling methods in systems with discrete or mixed variables by proposing a universal generative sampling framework that does not rely on target gradients. The method uniquely enforces time reversibility—specifically, detailed balance—as a direct statistical constraint during sampler training. It constructs Markov trajectories using Metropolis–Hastings transition kernels and minimizes the maximum mean discrepancy (MMD) between forward and reverse joint distributions, requiring only evaluations of the energy function. Avoiding continuous relaxations or score functions, the framework uniformly handles continuous, discrete, and hybrid variable spaces. It accurately reproduces thermodynamic quantities and effectively captures mode-switching behavior in multimodal Gaussian mixtures, high-dimensional Ising models, and discrete–continuous coupled systems, demonstrating both broad applicability and high precision.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Learning to sample from complex unnormalized distributions is a fundamental challenge in computational physics and machine learning. While score-based and variational methods have achieved success in continuous domains, extending them to discrete or mixed-variable systems remains difficult due to ill-defined gradients or high variance in estimators. We propose a unified, target-gradient-free generative sampling framework applicable across diverse state spaces. Building on the fact that detailed balance implies the time-reversibility of the equilibrium stochastic process, we enforce this symmetry as a statistical constraint. Specifically, using a prescribed physical transition kernel (such as Metropolis-Hastings), we minimize the Maximum Mean Discrepancy (MMD) between the joint distributions of forward and backward Markov trajectories. Crucially, this training procedure relies solely on energy evaluations via acceptance ratios, circumventing the need for target score functions or continuous relaxations. We demonstrate the versatility of our method on three distinct benchmarks: (1) a continuous multi-modal Gaussian mixture, (2) the discrete high-dimensional Ising model, and (3) a challenging hybrid system coupling discrete indices with continuous dynamics. Experiments show that our framework accurately reproduces thermodynamic observables and captures mode-switching behavior across all regimes, offering a physically grounded and universally applicable alternative for equilibrium sampling.

Problem

• discrete sampling
• unnormalized distributions
• mixed-variable systems
• equilibrium sampling
• generative sampling

Innovation

• reversibility
• gradient-free sampling
• Maximum Mean Discrepancy
• discrete-continuous systems
• detailed balance