arxiv: 2605.09654 · v1 · submitted 2026-05-10 · 📊 stat.ML · cs.LG· stat.CO

Recognition: 2 theorem links

· Lean Theorem

Metropolis-Adjusted Diffusion Models

Arnaud Doucet, Christopher Williams, Jun Yang, Kevin H. Lam, Tyler Farghly, Yee Whye Teh

Pith reviewed 2026-05-12 04:09 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.CO

keywords diffusion modelsscore-based generative modelsLangevin dynamicsMetropolis-Hastings algorithmBernoulli factorybias correctionsampling methodsimage synthesis

0 comments

The pith

Diffusion models can use Metropolis-Hastings adjustments on their Langevin correctors by computing acceptance probabilities from the score function alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors aim to eliminate discretization bias in diffusion model sampling by replacing unadjusted Langevin corrector steps with Metropolis-adjusted versions. They achieve this without access to the target density by devising ways to calculate the required acceptance probability using only the score function. Their exact approach employs a two-coin Bernoulli factory algorithm, and they add a low-cost Simpson's rule approximation that reaches high-order accuracy. When tested, these changes produce higher-quality samples on both simple distributions and complex image datasets.

Core claim

We present the first exact procedure for Metropolis-adjusted Langevin correctors in score-based diffusion models, relying on a two-coin Bernoulli factory to evaluate acceptance probabilities from the score function. An accompanying approximation via Simpson's rule delivers fifth-order accuracy in the step size with negligible overhead. Experiments confirm that these adjusted correctors deliver consistent improvements in sample quality, reflected in reduced Fréchet Inception Distance on image generation tasks.

What carries the argument

A two-coin Bernoulli factory algorithm that constructs the exact Metropolis-Hastings acceptance decision from the score function evaluated at current and proposed points.

Load-bearing premise

It is possible to obtain the exact Metropolis-Hastings acceptance probability needed for bias correction solely from the score function without new sources of error.

What would settle it

Applying the exact adjustment method to sample from a simple, fully known Gaussian distribution and verifying that the output distribution matches the target exactly would confirm or refute the claim of unbiased sampling.

Figures

Figures reproduced from arXiv: 2605.09654 by Arnaud Doucet, Christopher Williams, Jun Yang, Kevin H. Lam, Tyler Farghly, Yee Whye Teh.

**Figure 2.** Figure 2: Properties of integral estimators. Under this assumption, we obtain |I| ≤ C(x, xe) almost surely. To estimate the exponential of the integral, we use a Poisson-truncated power-series estimator. Let N ∼ Poisson(2C(x, xe)), let I1, . . . , IN be independent copies of I and define the estimator, W = Y N j=1 1 2 + Ij 2C(x, xe) . (13) The intuition is that the random product expands the exponential series f… view at source ↗

**Figure 3.** Figure 3: DDM sampling using an ancestral sampler predictor, and MADM correction. Mean [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Optimal acceptance rate for Algorithm 2 Let g(w) = (1 + exp(−w))−1 represent Barker’s function. Because g is continuous and bounded, g(w) ∈ (0, 1), the function f(z, w) = z 2 g(w) is continuous. By the Continuous Mapping Theorem: Vd := d 1/3 (˜x1 − x1) 2αd(x, x˜) D −→ ℓ 2 z 2 1 g(W). (26) To pass the limit inside the expectation, we require the sequence Vd to be uniformly integrable. Indeed, since 0 ≤ αd ≤… view at source ↗

read the original abstract

Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a biased sampler, as it discretises a continuous diffusion process. In this work, we consider adjusted Langevin correctors that employ Metropolis--Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order $5/2$ in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fr\'echet Inception Distance (FID) on the latter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The claimed exact adjustment via Bernoulli factory cannot recover the target density ratio from two score values, so it does not deliver detailed balance.

read the letter

The paper's central claim is that a two-coin Bernoulli factory gives the first exact Metropolis-Hastings adjustment for Langevin correctors inside diffusion samplers, using only the score function. That does not work. The acceptance step requires the ratio p(x')/p(x), which is the line integral of the score between the points. Two endpoint scores do not fix that integral, so any acceptance probability built from them alone cannot satisfy detailed balance with the true target.

Referee Report

1 major / 0 minor

Summary. The paper proposes Metropolis-adjusted Langevin correctors within score-based diffusion models to reduce discretization bias from unadjusted Langevin steps. It claims to introduce the first exact adjustment method via a two-coin Bernoulli factory algorithm that computes acceptance probabilities using only score evaluations at current and proposed points, along with a low-cost Simpson's rule approximation achieving O(h^{5/2}) accuracy in the step size. Empirical results on synthetic data and image datasets report consistent FID improvements.

Significance. Bias reduction in diffusion sampling is a central practical concern. An exact, score-only Metropolis adjustment would be a meaningful theoretical advance if achievable, and the claimed higher-order approximation offers efficiency advantages. The reported FID gains on images indicate potential practical value, but these rest on the validity of the unbiasedness claims.

major comments (1)

[Abstract] Abstract and the description of the exact method: The claim of an 'exact' two-coin Bernoulli factory procedure for the Metropolis-Hastings acceptance probability is not supported. Standard MH requires the target ratio p(x')/p(x) = exp(∫_γ s · dx) for any path γ connecting x and x'. This line integral is not determined by the endpoint values s(x) and s(x') alone. No algorithm using only these two score evaluations (even via Bernoulli factory) can recover the exact ratio in general, so the resulting kernel cannot satisfy detailed balance w.r.t. the true target without additional assumptions on s. This directly undermines the central claim of an unbiased exact corrector.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments, which have helped us identify areas where the manuscript's claims require clarification and revision. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and the description of the exact method: The claim of an 'exact' two-coin Bernoulli factory procedure for the Metropolis-Hastings acceptance probability is not supported. Standard MH requires the target ratio p(x')/p(x) = exp(∫_γ s · dx) for any path γ connecting x and x'. This line integral is not determined by the endpoint values s(x) and s(x') alone. No algorithm using only these two score evaluations (even via Bernoulli factory) can recover the exact ratio in general, so the resulting kernel cannot satisfy detailed balance w.r.t. the true target without additional assumptions on s. This directly undermines the central claim of an unbiased exact corrector.

Authors: We appreciate the referee's precise identification of this issue. The referee is correct that the Metropolis-Hastings ratio requires the line integral of the score along a path, which cannot be recovered from endpoint score values alone in general. Our proposed two-coin Bernoulli factory method was intended to enable sampling of the acceptance decision using only the available score evaluations at the current and proposed points. However, we acknowledge that this does not yield exact detailed balance with respect to the true target density without further assumptions (e.g., on the conservativeness of the score or the integration procedure). We will revise the abstract, the method description, and related claims to remove the unqualified 'exact' terminology, clarify the conditions under which the procedure is unbiased, and discuss the limitations relative to standard MH. We will also strengthen the connection to the Simpson's rule approximation in light of this point. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper introduces novel algorithmic constructions for Metropolis-adjusted Langevin correctors within diffusion models, specifically a two-coin Bernoulli factory method claimed to be exact and a Simpson's rule approximation of order 5/2. These are derived from standard Monte Carlo techniques (Bernoulli factories for probability simulation and numerical quadrature) applied to the score function, without any reduction of the central claims to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain is self-contained and presents independent algorithmic proposals rather than renaming known results or smuggling ansatzes via citation. No enumerated circular patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; none can be identified.

pith-pipeline@v0.9.0 · 5509 in / 1014 out tokens · 27353 ms · 2026-05-12T04:09:24.752916+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We introduce the first exact method for adjusting Langevin corrections... based on a two-coin Bernoulli factory algorithm... Simpson’s rule that achieves accuracy of order 5/2
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear
log(r_pt(x,ex)) = ∫_0^1 <s(x+u(ex-x),t), ex-x> du

Reference graph

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