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Score-Based Metropolis-Hastings Algorithms

Ahmed Aloui, Ali Hasan, Juncheng Dong, Zihao Wu, Vahid Tarokh

TL;DR

This work addresses the challenge of combining Metropolis-Hastings with score-based generative models, which traditionally lack an energy function and thus an accessible MH adjustment. It introduces Score Balance Matching (SBM), a learning objective that estimates the MH acceptance function $a(x',x)$ from an estimated score and data, ensuring detailed balance and enabling MH corrections for score-based samplers such as RW, MALA, and pCN. The authors show that SBM, especially with entropy regularization, yields more reliable and efficient sampling across multimodal and heavy-tailed distributions, and demonstrate improvements in diffusion-model-like tasks such as MNIST. By enabling MH adjustments within score-based frameworks, the paper broadens the toolkit for generative modeling and improves sampling quality and stability in challenging regimes.

Abstract

In this paper, we introduce a new approach for integrating score-based models with the Metropolis-Hastings algorithm. While traditional score-based diffusion models excel in accurately learning the score function from data points, they lack an energy function, making the Metropolis-Hastings adjustment step inaccessible. Consequently, the unadjusted Langevin algorithm is often used for sampling using estimated score functions. The lack of an energy function then prevents the application of the Metropolis-adjusted Langevin algorithm and other Metropolis-Hastings methods, limiting the wealth of other algorithms developed that use acceptance functions. We address this limitation by introducing a new loss function based on the \emph{detailed balance condition}, allowing the estimation of the Metropolis-Hastings acceptance probabilities given a learned score function. We demonstrate the effectiveness of the proposed method for various scenarios, including sampling from heavy-tail distributions.

Score-Based Metropolis-Hastings Algorithms

TL;DR

This work addresses the challenge of combining Metropolis-Hastings with score-based generative models, which traditionally lack an energy function and thus an accessible MH adjustment. It introduces Score Balance Matching (SBM), a learning objective that estimates the MH acceptance function from an estimated score and data, ensuring detailed balance and enabling MH corrections for score-based samplers such as RW, MALA, and pCN. The authors show that SBM, especially with entropy regularization, yields more reliable and efficient sampling across multimodal and heavy-tailed distributions, and demonstrate improvements in diffusion-model-like tasks such as MNIST. By enabling MH adjustments within score-based frameworks, the paper broadens the toolkit for generative modeling and improves sampling quality and stability in challenging regimes.

Abstract

In this paper, we introduce a new approach for integrating score-based models with the Metropolis-Hastings algorithm. While traditional score-based diffusion models excel in accurately learning the score function from data points, they lack an energy function, making the Metropolis-Hastings adjustment step inaccessible. Consequently, the unadjusted Langevin algorithm is often used for sampling using estimated score functions. The lack of an energy function then prevents the application of the Metropolis-adjusted Langevin algorithm and other Metropolis-Hastings methods, limiting the wealth of other algorithms developed that use acceptance functions. We address this limitation by introducing a new loss function based on the \emph{detailed balance condition}, allowing the estimation of the Metropolis-Hastings acceptance probabilities given a learned score function. We demonstrate the effectiveness of the proposed method for various scenarios, including sampling from heavy-tail distributions.
Paper Structure (38 sections, 3 theorems, 51 equations, 16 figures, 3 tables, 2 algorithms)

This paper contains 38 sections, 3 theorems, 51 equations, 16 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Score-Based Models.
  4. Metropolis-Hastings.
  5. Score-Based Metropolis-Hastings
  6. Score Matching
  7. Metropolis-Hastings
  8. Detailed Balance.
  9. Score Balance Matching
  10. Motivation
  11. Algorithm
  12. Entropy Regularization.
  13. Implementation.
  14. Empirical Results
  15. Stability with respect to Step Size: Score MALA vs. ULA.
  16. ...and 23 more sections

Key Result

Proposition 1

We have that, for every function $a:\mathcal{X} \times \mathcal{X} \rightarrow [0,1]$

Figures (16)

  • Figure 1: Comparison of sampling methods for a mixture of two Gaussians: (a) Original distribution, (b) ULA with the true scores, (c) RW with the true distribution, and (d) MALA with true scores and distribution. The plots demonstrate the impact of slow mixing between modes, highlighting the challenges in low-density regions. We report the empirical weights of the mixture given by each sampling method.
  • Figure 2: Visualizing the acceptance probabilities for different values of $x'$ for a given initial state $x=(0.0,0.0)$. We plot the results for training an acceptance network with three different regularization values $\lambda\in\{0,0.1,1.0\}$
  • Figure 3: Comparison of different methods on the Pinwheel dataset.
  • Figure 4: Comparison of different methods on the Moons dataset.
  • Figure 5: Comparison of different methods on the S-curve dataset.
  • ...and 11 more figures

Theorems & Definitions (9)

  • Definition 1: Fisher Divergence
  • Definition 2: Sliced Score Matching
  • Definition 3: Denoising Score Matching
  • Proposition 1
  • Proposition 2: Finite-Sample Generalization Bound
  • Proposition 3
  • proof : Proof of Proposition 1
  • proof : Proof of Proposition 2
  • proof : Proof of Proposition 3