Score-Based Metropolis-Hastings

• Score-Based Metropolis-Hastings defines a framework where only the gradient of the log-target is used to reconstruct acceptance probabilities, enabling sampling when the full density is unknown.
• The methodology leverages line-integral estimators, stochastic Bernoulli factory techniques, and trained neural acceptance networks to reconcile the detailed balance condition using solely first-order score information.
• Empirical results and theoretical guarantees indicate that SB-MH methods reduce discretization bias and maintain unbiasedness and reversibility, making them valuable for complex high-dimensional and heavy-tailed sampling tasks.

Score-Based Metropolis-Hastings (SB-MH) algorithms generalize the classical Metropolis-Hastings (MH) framework to scenarios where only the score function—i.e., the gradient of the log-target density—is available, without requiring access to the underlying energy or density function. This paradigm is central to recent advances in score-based diffusion models, fractional and minibatch MCMC, and various settings where the likelihood is intractable but score estimators can be reliably learned. SB-MH methods provide mechanisms for incorporating acceptance-rejection corrections within Markov chain Monte Carlo (MCMC) procedures, thus eliminating or dramatically reducing finite-step and discretization biases, while retaining rigorous target invariance. The core methodology relies on reconciling the detailed balance condition using only score information, leveraging either analytical identities, stochastic (Bernoulli factory/Barker) procedures, or differentiable surrogate objectives for acceptance probability construction.

1. Theoretical Foundations and Motivation

Classical Metropolis-Hastings sampling requires evaluating acceptance probabilities of the form

$$a(x, x') = \min\left\{1,\ \frac{p(x')\,q(x|x')}{p(x)\,q(x'|x)}\right\}$$

for a given target $p(x)$ and proposal $q(x'|x)$. In modern generative modeling, and in particular for score-based diffusion models, explicit evaluation of $p(x)$ is impossible: only the score $s(x) \approx \nabla_x \log p(x)$ is available from a pre-trained network. This motivates algorithms that reconstruct acceptance probabilities from score information alone, so that MH corrections and their theoretical properties—unbiasedness, reversibility, and convergence to $p(x)$—are maintained (Aloui et al., 2024, Sjöberg et al., 2023, Lam et al., 10 May 2026).

Key to the SB-MH framework are the following principles:

• The log-density ratio, $\log p(x') - \log p(x)$, can be expressed as an integral of the score along a continuous path from $x$ to $x'$:

$$\log \frac{p(x')}{p(x)} = \int_0^1 s(x + u(x'-x)) \cdot (x'-x)\, du$$

• The detailed balance condition (reversibility) can be enforced using only first-order score information by matching the gradients of acceptance probability functions with the difference in scores and proposal gradients (Aloui et al., 2024, Aloui et al., 31 Jan 2026).

This viewpoint enables a wide variety of SB-MH approaches, each adapted to the structure of the proposal and target at hand.

2. Methodologies for Score-Based Acceptance Construction

2.1 Integral-Based and Line-Integral Approaches

For discretized diffusions (e.g., Unadjusted Langevin Algorithm steps), SB-MH correctors compute acceptance probabilities via a path integral of the score, typically along the straight line between the current point $p(x)$0 and the proposal $p(x)$1. This yields the exact MH acceptance as

$p(x)$2

Fast, unbiased estimators for the exponential of such integrals are constructed via two-coin Bernoulli factory algorithms, providing exact Barker acceptance for the chain without evaluating $p(x)$3 (Lam et al., 10 May 2026). Approximate deterministic acceptance ratios can alternatively be obtained via high-order quadrature methods (e.g., Simpson's rule), achieving discretization bias $p(x)$4 while requiring only additional midpoint score evaluations (Lam et al., 10 May 2026).

2.2 Detailed-Balance Matching and Neural Acceptance Estimation

When the log-density difference cannot be tractably or stably estimated, acceptance networks can be trained to satisfy the gradient form of detailed balance. Specifically, one minimizes the loss

$p(x)$5

possibly augmented with entropy regularization to avoid degenerate solutions. The converged acceptance network $p(x)$6 then specifies the acceptance probability for proposing $p(x)$7 from $p(x)$8, using only score and proposal gradients (Aloui et al., 2024, Aloui et al., 31 Jan 2026).

This approach is applicable even when the proposal is complex or heavy-tailed (e.g., fractional Langevin with $p(x)$9-stable noise), via closed-form proxy gradients derived from the proposal's score structure (Aloui et al., 31 Jan 2026).

2.3 Stochastic Gradient and Minibatch MH Proposals

For massive datasets, SB-MH can be combined with minibatch-based proposals—such as reversible stochastic gradient Langevin dynamics (RSGLD)—to create mini-batch MH transitions that balance unbiased stochastic score estimation with a rigorously computable approximate acceptance probability. This induces controlled tempering and preserves the mode structure of the original target, provided suitable conditions on batch size and scaling constants are met (Wu et al., 2019).

3. Algorithmic Implementations and Pseudocode Structure

Implementation of SB-MH methods requires assembling the following core components:

• Score Network: $q(x'|x)$0 trained via denoising, score-matching, or equivalent schemes.
• Proposal Distribution: ULA, MALA, RSGLD, pCN, or tailored discretizations, frequently parameterized by the score.
• Acceptance Mechanism:
  • Exact (Bernoulli factory / two-coin): For small proposals, uses score-averaged stochastic estimators (Lam et al., 10 May 2026).
  • Quadrature-based (Simpson, trapezoidal): Approximates the line integral deterministically (Lam et al., 10 May 2026, Sjöberg et al., 2023).
  • Neural Acceptance Network: Trained via detailed balance or gradient matching objectives (Aloui et al., 2024, Aloui et al., 31 Jan 2026).

A generic SB-MH sampler operates as follows (schematically), taking as input the current state $q(x'|x)$1, proposing $q(x'|x)$2, evaluating an acceptance probability $q(x'|x)$3 via one of the above mechanisms, and accepting with this probability.

In advanced settings (e.g., model composition, annealed MCMC), the same framework is applied at every time-step of a reverse-diffusion sampler or on a composition of multiple score fields (Sjöberg et al., 2023, Lam et al., 10 May 2026).

4. Bias Reduction, Unbiasedness, and Theoretical Guarantees

A central advantage of SB-MH methods is the substantial reduction or elimination of sampler bias:

• Unadjusted Langevin correctors under finite discretizations introduce $q(x'|x)$4 marginal bias; SB-MH correctors, through exact acceptance or sufficiently accurate quadrature, restore invariance to $q(x'|x)$5 or reduce bias to $q(x'|x)$6 (Lam et al., 10 May 2026).
• Exact two-coin Barker-Langevin correctors are shown to terminate almost surely, with mean computational cost $q(x'|x)$7 per step under standard score regularity conditions.
• In high dimension $q(x'|x)$8, optimal scaling of the acceptance mechanism balances acceptance rate and computational cost (e.g., $q(x'|x)$9 yields target acceptance $p(x)$0 with sublinear cost scaling) (Lam et al., 10 May 2026).
• Theoretical properties such as reversibility, stationarity, and ergodicity of the generated Markov chain hold under population-level satisfaction of detailed balance conditions, and are maintained by both stochastic (Bernoulli factory) and differentiable neural approaches (Aloui et al., 2024, Aloui et al., 31 Jan 2026).
• For fractional or heavy-tailed proposals, score-based acceptance functions trained via Score Balance Matching (SBM) optimize a loss function directly equivalent to gradient-form detailed balance, which is theoretically proven to guarantee stationarity of $p(x)$1 under minor assumptions (Aloui et al., 31 Jan 2026).

No explicit nonasymptotic convergence rate or finite-sample bounds are given for the data-driven acceptance network; empirical results suggest robust finite-time accuracy especially in the challenging heavy-tailed or multimodal regimes.

5. Empirical Performance and Applications

SB-MH correctors demonstrate consistent improvements over standard unadjusted schemes across domains:

Model/Dataset	Unadjusted FID	SB-MH FID	Other Metrics
CIFAR-10 (Heun)	1.96	1.95
CIFAR-10 (Euler)	7.62	7.52
FFHQ (Heun)	2.47	2.43
AFHQv2 (Heun)	2.04	2.03
ImageNet-64 (Heun)	2.32	2.15

• On synthetic 2D targets (e.g., spiral, pinwheel), ULA often yields supports with significant outliers; SB-MH (MAD-MH, line-integral, or acceptance network-based) sharply concentrates the sample support onto the data manifold (Sjöberg et al., 2023, Lam et al., 10 May 2026).
• For heavy-tailed mixtures and combinatorial optimizations, Metropolis-adjusted fractional Langevin (MAFLA) with neural acceptance rules outperforms unadjusted dynamics in both Wasserstein distance and quantile error across a range of stability indices, step sizes, and dimensions (Aloui et al., 31 Jan 2026).
• In high-dimensional neural network posteriors, mini-batch SB-MH (MHBT+RSGLD) achieves lower classification error and greater robustness to step size than conventional SGD or SGLD, attributed to rigorously correct acceptance mechanisms counteracting stochastic proposal errors (Wu et al., 2019).
• On geometric toy distributions (e.g., Swiss Roll, Moons, Pinwheel), Score MALA and Score RW reduce Wasserstein-1 and MMD distances by orders of magnitude over ULA (Aloui et al., 2024).

6. Extensions: Model Composition, Annealed MCMC, and Beyond

SB-MH methods extend naturally to settings requiring the composition of multiple models or adaptation to continuously changing targets:

• For model composition—targeting a product of marginal distributions at each step—the score field is the sum of component scores, and the line-integral MH framework readily applies, supporting unbiased sampling from composed or guided distributions (Sjöberg et al., 2023).
• In annealed MCMC for diffusion models, SB-MH corrections at every reverse-diffusion iteration dramatically reduce accumulated bias, allowing granular control over sample quality at all noise levels (Lam et al., 10 May 2026).
• Generalization to heavy-tailed, stable, or non-Gaussian proposals is enabled by proxy proposal score evaluations based solely on sample differences and score approximations, with acceptance probabilities learned or computed as described above (Aloui et al., 31 Jan 2026).
• For scenarios where neither proposal nor target densities are available, such as combinatorial relaxations, closed-form score computation and acceptance network training permit SB-MH samplers to deliver improved sampling accuracy with minimal assumptions (Aloui et al., 31 Jan 2026).

7. Significance, Challenges, and Ongoing Directions

SB-MH algorithms bridge the gap between score-based generative modeling and the powerful, unbiased sampling guarantees of classical MCMC. By offering exact or controlled-approximation acceptance probabilities without requiring energy function access, these methods open rigorous MCMC methodology to a wide spectrum of score-parametric models. Their demonstrated performance in sampling efficiency, robustness to proposal specification, and empirical bias reduction establish them as foundational tools in modern generative modeling and Bayesian computation (Lam et al., 10 May 2026, Aloui et al., 31 Jan 2026, Aloui et al., 2024, Sjöberg et al., 2023, Wu et al., 2019).

Open challenges include:

• Formal finite-sample theoretical error bounds for neural acceptance approximation.
• Efficient high-dimensional scaling, particularly for very large score networks and small step sizes.
• Adaptive and composite-proposal SB-MH schemes for multimodal or structured targets.
• Further empirical and theoretical assessment of entropy regularization's role in acceptance network convergence.

The foundational structure of SB-MH—realizing detailed balance from score information—positions it as a unifying framework across a variety of evolving MCMC-corrected generative models.