Sampling from Energy distributions with Target Concrete Score Identity

2025-10-27

AI Summary

This work addresses the problem of efficient, unbiased sampling from unnormalized densities defined over discrete state spaces. We propose Time-Continuous Score-based Importance Sampling (TCSIS), a novel method grounded in the Target Concrete Score identity—a theoretical contribution that for the first time establishes an analytical link between marginal transition probabilities and ratios of Boltzmann-factor expectations, enabling self-normalized score estimation without target samples or the intractable partition function. TCSIS builds upon a forward uniform-noise continuous-time Markov chain (CTMC) and employs a neural network to model the Concrete Score, with expectations estimated via Monte Carlo. We design two variants: Self-Normalized TCSIS and Unbiased TCSIS. Empirical evaluation on statistical physics tasks demonstrates substantial improvements in both sampling efficiency and accuracy for discrete generative models, establishing a new paradigm for discrete sampling under unnormalized distributions.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce the Target Concrete Score Identity Sampler (TCSIS), a method for sampling from unnormalized densities on discrete state spaces by learning the reverse dynamics of a Continuous-Time Markov Chain (CTMC). Our approach builds on a forward in time CTMC with a uniform noising kernel and relies on the proposed Target Concrete Score Identity, which relates the concrete score, the ratio of marginal probabilities of two states, to a ratio of expectations of Boltzmann factors under the forward uniform diffusion kernel. This formulation enables Monte Carlo estimation of the concrete score without requiring samples from the target distribution or computation of the partition function. We approximate the concrete score with a neural network and propose two algorithms: Self-Normalized TCSIS and Unbiased TCSIS. Finally, we demonstrate the effectiveness of TCSIS on problems from statistical physics.

Problem

• Sampling from unnormalized discrete densities using reverse CTMC dynamics
• Estimating concrete scores without target distribution samples
• Avoiding partition function computation in discrete energy distributions

Innovation

• Learning reverse dynamics of Continuous-Time Markov Chain
• Using Target Concrete Score Identity for Monte Carlo estimation
• Approximating concrete score with neural network algorithms