AC-Sampler: Accelerate and Correct Diffusion Sampling with Metropolis-Hastings Algorithm
Authors: Minsang Park
Gyuwon Sim
Hyungho Na
Jiseok Kwak
Sumin Lee
Richard Lee Kim
Donghyeok Shin
Byeonghu Na
Yeongmin Kim
Il-chul Moon
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion model
Metropolis-Hastings Algorithm
Langevin Dynamics
Primary area: generative models
TL;DR: Accelerate and Correct Diffusion Sampling with Metropolis-Hastings Algorithm

Abstract: Diffusion-based generative models have recently achieved state-of-the-art performance in high-fidelity image synthesis. These models learn a sequence of denoising transition kernels that gradually transform a simple prior distribution into a complex data distribution. However, requiring many transitions not only slows down sampling but also accumulates approximation errors.
We introduce the Accelerator-Corrector Sampler (AC-Sampler), which accelerates and corrects diffusion sampling without fine-tuning. It generates samples directly from intermediate timesteps using the Metropolis–Hastings (MH) algorithm while correcting them to target the true data distribution. We derive a tractable density ratio for arbitrary timesteps with a discriminator, enabling computation of MH acceptance probabilities. Theoretically, our method yields samples better aligned with the true data distribution than the original model distribution. Empirically, AC-Sampler achieves FID 2.38 with only 15.8 NFEs, compared to the base sampler’s FID 3.23 with 17 NFEs on unconditional CIFAR-10. On CelebA-HQ 256×256, it attains FID 6.6 with 98.3 NFEs. AC-Sampler can be combined with existing acceleration and correction techniques, demonstrating its flexibility and broad applicability. Our code is available at \href{https://github.com/aailab-kaist/AC-Sampler}{https://github.com/aailab-kaist/AC-Sampler.}

Published as a conference paper at ICLR 2026
AC-SAMPLER: ACCELERATE AND CORRECT DIFFU- SION AMPLING WITH METROPOLIS-HASTIN ALGO- RITHM
Minsang Park'; Gyuwon Sim', Hyungho Na², Jiscok Kwak', Sumin Lee', Richard Lee Kim', Dongyhyeok Shin', Byeonghu Na', Yeongmin Kim', Il-Chul Moon 1:3+ KAIST, UNIST, 3 'summary.ai
ABSTRACT
Diffusion-based generative models have recently achieved state-of-the-art per- formance in high-fidelity image synthesis. These models learn a sequence of denoising transition kernels that gradually transform a simple prior distribution into a complex data distribution. However, requiring many transitions not only slows down sampling but also accumulates approximation errors. We introduce the Seclerator-Correcet Sampler (AC-Sampler) which accelerates and corrects diffuusion sampling without fine-tuning. It generates samples directly from interme- diate timesteps using the Metropolis-Hasting (MH) algorithm while correcting them to target the true data distribution. We derive a tractable density ratio for arbitrary timesteps with discriminator, enabling computation of MH acceptance probabilities. Theoretically, our method yields samples better aligned with the true data distribution than the original model distribution. Empirically, AC-Sampler achieves FID 2.38 with only 15.8 NFEs, compared to the base sampler's 's FID 3.23 with 17 NFES on unconditional CIFAR-I0. On CelebA-HQ 256x256, attains FID 6.6 with 98.3 NFEs. AC-Sampler can be combined with existing acceleration and correction techniques, demonstrating its flexibility and broad applicability. Our code is available at https://github..cmm cwtpo/oooccalaaaaaaaaaaaaa wttpa/aaaaaaaa//-SSSaaaaaa kaist/ACCSSmpller
INTRODUCTION
Diffusion- based generative models (Ho et al. 2020 Sohl-Dickstein et al. 2015 Song et al. 202 have become one of the most popular approaches in recent years due to their strong ability to generate
diverse types of data such as high-fidelle fichefiiyiity images (Dhariwal Nichol, 2021 Rombach et al.) 2022)
and videos (Ho et al. 2022 Voleti et al. 2022). Building these models, strong pre- pre-trained variants have emerged (Rombach et al. [022] Karras et al., 2022 In spite of many variations, these models share fundamental structure: they start by sampling from a simple prior distribution and iteratively transform the samples through a series of learned transition kernels to approximate the complex data distribution. This iterative generation causes two problems. First, the sampling process is slow due to the large number of kernel transition calculations required (Song et al., 2021b Zhang & Chen 2023). Second, errors can accumulate during the sampling process if the kernel transition does not accurately reflect the true reverse diffusion process (Xu et al. 2023b).
Speed and accuracy are usually considered separate research topics, as improving both simultancously is often challenging. Some approaches for acceleration diffusion sampling (Kim & Ye 2023 Zheng jet al. 2023) focus on reducing the NFE while maintaining image quality, but they lack theoretical analysis for converging to the true data distribution, leaving model errors unaddressed. On the
contrary, previous correction methods (Kim et al. 2023 Na et al. 2024 Xu et al. 2023b) makes additional computational burden, which makes the methods unscalable in real-world services.
To address both challenges, we propose Accelerator-Correttue Sampler (AC-Sampler). It ac- celerates and corrects the diffusion sampling process without any fine-tuning of the pre-trained model. AC-Sampler directly proposes samples at intermediate timesteps, which enables acceleration.
"Corresooniing authors: Minsang Park <pagemu @ krk Il-Chul Moon <iemoon kaist.ac.kr

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Published as a conference paper at ICLR 2026
Using Metropolis- MetropoliisHasttnns correction, these proposals guaranteed to theoretically follow the true marginal distribution. Since the pre-trained model approximates the score function, we can construct an effective proposal distribution using Langevin dynamics (Grenander & Millerr [994). For computing the acceptance probability, we only train a time-dependent discriminator (Kim et al. Na et al. 2024), which can be learned much lower cost than the diffusion model itself.
2023
Real-world services (xAI, [025 Midjourney, Inc., 2025 Microsoft 2025) and diverse downstream tasks (Hiranaka et al. 2025: Trabucco al. 2024 Azizi et al. (023) typically require generating
multiple samples quickly and accurately. Our method meets this demand by producing multiple samples that are both faster and more accurate, effectively overcoming the limitations of previous methods while offering the flexibility to be integrated with various sampling methods.
Our contributions are as follows:
We propose the AC-Sampler, which accelerates diffusion sampling by generating samples from intermediate timesteps rather than the initial prior distribution. It also corrects the accumulated error in the sampling process with Metropolis- -aatting (MH) algorithms.
We provide a theoretical analysis showing that training the discriminator and following the MH chain in our method leads a tighter bound on the data distribution compared to that of a pre-trained diffusion model. Furthermore. we provide theoretic analysis of the expected reduction in the number of function evaluations (NFE).
We validate our theoretical claim through experiments on benchmark datasets and toy settings, and demonstrate that our method applies effectively to diverse pre-trained models in both unconditional and conditional settings.
Our contribution is orthogonal to advances in training-free samplers, so the two gains are complementary and can be realized simultaneously. Also, the utilize discriminator is simple and does not require ad-hoc structures on the pre-trained diffusion model.
2
RELATED WORK
Acceleration Methods To reduce the computational burden associated with additional training, acceleration methods have been developed to speed up the sampling process without modifying the original diffusion model. DDIM (Song et al. 2021a) reformulates the reverse diffusion deterministic ODE, achieving significant speedups with fewer steps while preserving the pre-trained network. Building on this foundation, various works have further improved ODE solvers through
high-order numerical methods and exponential integration, leading to significant gains in sampling
efficiency (Liu al. 2022 Lu al. 2022a b Dockhorn al. 2022a Karras et al. 2022 Zhang &
Chen 2023 Zhao et al. 2023 Zheng et al. 2023 Tong et al. 2025).
In parallel, other lines of work have explored fundamentally different perspectives on diffusion model acceleration. For example, PDS (Ma et al., (202)) treats diffusion sampling as an Markoy Chain Monte Carlo (MCMC) process, incorporating frequency-domate preconditioning to improve high-frequency details. DLG (Kim & Ye 2023) formulates the sampling process over the product space of data and time, enabling joint Langevin-based Gibbs sampling. This sampling process identifies the intermediate perturbed data with low noise for the initialization of the reverse process, which shortens the subsequent diffusion trajectory. However, as shown in Appendix A. DLG lacks theoretical convergence guarantees, leaving room for improvement.
Correction Methods Several studies have been conducted to improve the sampling quality of pre-trained diffusion models. DG (Kim al. 2023) proposes a correction method using time- dependent discriminator when score estimation is inaccurate, thereby improving the accuracy of the transition kernel. Restart (Xu et al.) (023b) theoretically demonstrates that repeating forward and backward steps within fixed time interval [fiin, fmmxx] in pre-trained model can reduce sampling error. DiftRS (Na et al.) (024)) aims to sample from the true distribution by applying a rejection sampling scheme with time-dependent discriminator. ES (Ning et al. 2024) proposes a training- free correction schedule to compensate for the scale gap in score norms between the training and sampling phases. However, there is theoretical guarantee that simply matching the norms leads to distributional equivalence. While these methods focus on sampling correction, they do not reduce the base NFE and may even increase it, leading to slower sampling.
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3 PRELIMINARY
3.1
NETROPOLIS-HASTIIE ALGORITHM AND LANGEVIN PROPOSAL
The
algorithm (MH algorithm) Metroopoiis et al., 1953 Hastings 1970) is MCMC method used to sample from target distribution when direct sampling is not possible. It constructs Markov chain whose stationary distribution is the target distribution by satisfying the detailed balance condition. Based on this condition, the algorithm defines an acceptance probability, which is used to determined whether to accept proposals drawn from simple proposal distribution.
Given target distribution g(-) and a proposal distribution Ppropesal (- .), the acceptance probability for a proposed sample x and current sample x
a = min (1. 11,
((x) Pproposal [ x)
(((x) Pproposal
X x
which guarantees that q()) is the stationary distribution of the Markov chain.
To improve mixing and convergence of MH algorithm, gradient-based proposals have been stud- ied (Parisi 1988 Neal et al. 201T). i.c. the Langevin proposal. Specifically, the proposal is derived from the Euler-Maruyan discretization of the overdamped Langevin dynamics (Roberts & Tweedie 1996). Formally, with the target distribution q(),, the Langevin proposal is defined as
Pproposal (-1x) = N (x+ 7 V log q(x), nl).
2
where yj is the step size and Vx log ((x) is the score function of q(x). This method is commonly referred to as the Metropolis- Adjjssed Langevin Algorithm (MALA) (Grenander & Miller 1994).
3.2 DIFFUSION MODELS
Diffusion models (Ho al. 2020 Song al. 2021b) probabilistic generative models that approximate data distributions by adding noise and reversing this process. They consist of a forward process that corrupts data into noise and reverse process that removes noise generate samples.
Let go (xo) denote the true data distribution and p (xo) denote the distribution of generated samples from the model. particular, the forward process is fixed Markov chain where Gaussian noise is added using pre-defined variance schedule. This creates a sequence of random variables X1:2:
Qx: xo) = 11 (xx
where each q (x/[xx-1) Gaussian transition with increasing noise levels. This process transforms the data distribution into tractable prior distribution (xTx) T. The reverse process is modeled as a Gaussian distribution that denoises a Pr (xx) iteratively:
(Xor))) := Pr (x))
p.. Pf (x)))
(X,
where pi-11e is the transition kernel that generate data from prior distribution pr. The model is trained by maximizing a variational bound on the log- likelihood of the data. practice, this can be achieved via denoising score matching loss (Vincent 2011 Ho et al., 2020), given by:
LDSM (0) (0t=UU(0.1)xxx000ee (xa)
| t) Xx, log Le (xx
(3)
With this loss, 8 (xx, optimizes to follow the true score Vx, log qe (x₂). The transition kernel is parameterized as a Gaussian distribution whose mean is a function of the score function:
Pi (x xx) NX(x-1; (x 8 (xr, t)),o())
where IfI denotes dependeendnnt variance. After training, samples are generated by iteratively applying the transition kernel from T to O.
We assume that we have access to pre-trained diffusion model, denoted by 0. Let qi (x;) and pf (x;) denote the marginal distributions timestep I, defined by forward diffusion processes starting from
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Prior
Data
T
EDM
MH Transition
DG
DIfTRS
Demoising Step
(Ours)
AC-Sampler
Burn-In
2.0
12
15
18
21
24
27
30
NFE
12.5
DPM
10.0
UniPC
DPM v3
7.5
DPM v3
AC [Ours)
5.0
10
NFE
Acceleration Gain No prior sampling needed
Correction Gain
Distribution correction with Metropolii-fistinngs
Figure FID-NFE graph on uncond. CIFAR-10: (Top) Correction methods
Figure 1: Overall figure of AC-Sampler
(Bottom) Acceleration methods.
Yo (xo) and p (xo). respectively. Pre-trained diffusion model provides 8 (x+, () xx, log ze (x+).
The mean of the transition kernel depends on both X; and the score function s (x: t). Therefore, treat the score function and the transition kernel p/-1je (x as equivalent parametrizations, since both are derived from the same model and same input x,,
Due to this structure, diffusion models are inherently limited by slow sampling speed from the large number of transition steps, and by the accumulation approximation errors the transition kernels.
METHOD
To address the two key challenges previously discussed, we propose AC-Sampler, novel diffusion sampling framework. The overall sampling procedure is follows. First, we perform denoising from the prior distribution up to target timestep т, which serves as the initial sample of the MCMC chain. Starting from this initial sample, repeatedly generate new candidates using score- based proposal distribution. At each step, MH correction is applied. These steps are performed as described in Algorithm. D After sufficient burn-in period, the resulting samples corrected true marginal distribution, 4r Finally, each accepted sample is further denoised to obtain the final outputs. The overall sampling process is illustrated in Fig.
This approach has two advantages. First, samples are created directly at T without denoising from T. This enables faster sampling (denoted Acceleration Gain Fig. D Second, because of MH correction, the resulting samples follow true marginal distribution gr- (denoted as Correction Gain in Fig. [T. Our method accelerates and corrects the sampling process without requiring any fine-tuning of the underlying diffusion model. For this process, we require the design of the proposal distribution (Sec. 4.T and the computation of the acceptance probability (Sec. 4.2
4.1 PROPOSAL DISTRIBUTION
To sample from the intermediate timestep use MALA. With target distribution (xx.), construct the proposal distribution using pre-trained score model (xx, t) as:
0
(-1x;) ( _ (x + " t1,
Pproposal,
where is Langevin step size. We adaptively set the value of to maintain constant signal-to-noise ratio (SNR) during sampling (Song Ermon 2019 Song et al. 2021b). An important advantage of framework is that both the denoising step and the proposal distribution rely the same score value. As a result, a single network evaluation sufficient for both operations, enabling efficient integration of denoising and exploration within AC-Sampler.
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Algorithm 1 MALAOneStep
Input:
Target timestep Previous sample x,, Score output s (x (XTTT) 7), (x-i7)
Likelihood ratio I : 1-dd(x,,,) Score network s Discriminator d
Output: Next sample X,
1: repeat
2: Propose &, from proposal distribution
p Prroposaa, - (: x+) (Eq.)
3: Get score s + (x,,, , T), and likelihood ratio L + 1 1- 1x (T)
d &
4: Calculate acceptance probability a t á(x,, X, 8, L. L (Eq.
5: Sample ~ (0, 1)
6: until u < 0
7:
return X,, S, L
4.2 ACCEPTANCE PROBABILITY
ge (xx Peopoall (x: (xx]
The acceptance probability for target distribution (r(-) ar = min 1,
where
(x(x)) pppppeatt (xx1
x, is a sample from the proposal distribution P xx). To make the acceptance probability tractable, we first decompose qu (Xt)/ (x:)) as stated in the following theorem.
(
Theorem 4.1. Let Xt and X, be two arbitrary samples at diffussin timestep Then, for any fixed Xt-1, the density ratio of the true marginal distribution qt is given by:
(x().)
Qqp--((8, x(-1) T2 (Re.)) Pi-1j((xe-11 xx)
(6)
(x(x))
geje-1 (x,, x(-1) (x pe pi-1j(xx-11 (8t)
where [x,,,,,)) == ge (x.) / (xx) denotes the likelihood ratio between the data and model marginal distributions timestep t.
denote the mean of the reverse transition Proof is provided in Appendix A.2 Let Me (x sx,, (xx,t) t))
kernel p. Ile- It is possible to choose 8;-1 == (X,, s (x,, t)) + t s5, (X,, t))) since Theorem holds for arbitrary x With this choice, the transition kernel terms in Eq. 6 are
pi 11 -
which cancel in the density ratio because both are Gaussian with the same variance
1 1 [x.)
and - is equidistant from their means. With chosen X acceptance probability follows:
a min 1, 4:)e1 (Xt Sc-1) ( 8(-1) Le(xe, f)
Le 1) Peroposal, (Xe (Xt)
pe Pproposal, (x(xx)
To access (x,, f), we use time-dependent discriminator d following the approach of DG (Kim ct al. 2023). The discriminator trained to distinguish between qe and p at all timesteps. achieve this, weighted binary cross-entropy loss is used for training the discriminator:
LaCE($) = x(1) Rxx [- log d (xt, ()] + Ex, ~ 1-oog(1 - dd (x.. O)] dt,
-A(0))
(8)
The optimal discriminator satisfies do' (xe,t) (x(x))+pp (x,) so the density ratio
(xx)
(((x))
becomes
pf (x,)
I- (x, Having access (xx,
109 (x)..)
Rd a (x(,.)) =: I. (x,, (), the acceptance probability can be
(xx)
1-d0(xx.1)
calculated as below:
1. (xt x(-1)
P Prrsposall (x, (xx) p proposal (X, (x)
6(x,, s. S.
İ) min
(9)
q (X; xx-1)
Forward term
Likelihood ratio
Proposal
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where s, s, L, L denotes (x,, t), (Xx, t), (x,, (xx, t), L t), respectively. The acceptance prob- ability consists of three terms. The forward and proposal terms are tractable Gaussian distributions, and the likelihood ratio is computed using discriminator. Together, these components make the ac- ceptance probability fully tractable. Note that with any tractable proposal distribution, the acceptance probability also remains tractable.
4.3 THEORETICAL ANALYSIS
We prove that our method can theoretically achieve sampling acceleration and correction.
Proposition 4.2. Let the reverse diffusion process consist of a total number of timesteps T, and let the larget fimestep in AC-Sampler be т. Suppose that 1 denotes the length of the Markov chain at timestep T. Let NFER denote the average reduction in NFE per sample achieved by AC-Sampler.
the acceptance probability a satisfies a > T-++I+ then, as co, the expected NFE reduction remains strictly positive, while its variance converges zero:
lim E[[FER]] > O, lim Var(NFER) = 0. (030
(10)
1-000
In practice. it is intractable to set / to infinity. However, we experimentally verified that significant reduction in the NFE be achieved even with finite This demonstrates that with score-guided proposal distribution, the aforementioned condition > T- is not hard constraint. Further discussion of the acceptance probability is provided in Appendix
To show that method not only accelerates sampling but also corrects errors, theoretically demonstrate that the data distribution induced by our sampler is closer to the true data distribution than that of the baseline model.
Theorem 4.3. Let pÿ- Po 0.0" denote the model distribution and refined distribution by AC-Sampler with optimal discriminator 6", respectively. Then, the KL divergence between the true data distribu- tion and the refined distribution Po 0.4 is bounded by: (2xr(90(x)))ll'
(xo)) Drr(99(x0)]lp00 0.0'
(11)
Theorem 4.4. Let T, be the transition kernel of MALA at timestep Also, 0.4".01 (0) denotes marginal
distribution at timestep T after the 1-th MALA transition from p². and Po
0.4" denotes the data
0.40 with denoising transition kernel Pl-11 C denotes a space distribution generated from Pr
pr 200
(x+) (xx))
of function which satisfies (fz [f(p < <o. (x E L", "log where a, B. E co] satisfy 1 + distribution go and the refined distribution Po
E EC2, Tr E LT,
(-(x-)
+ 1, then the KL divergence between the true data J
0. ((++)) is bounded by:
DxL(90(xo)11p 0.0" .1I+1) (xo)) s DKL(g0(xx))1)))) 0.0'.0' (xo))
(12)
The proofs of each theoretical result and detailed analysis are provided in Appendix A.4 and A.5 respectively. When our model has sufficiently converged, Theorem 443 suggests that it can generate samples that are closer the true data distribution than those produced by the baseline model. Theorem [.44] further shows that applying more MALA steps progressively moves the samples closer the true data distribution. Note that = 0 denotes the base diffusion model, without MALA.
Mitigating Empirical Distribution Distortion Although the theoretical results above are derived under the standard MH formulation, where rejected proposals are retained as the next state, this assumption is suboptimal in practical generative sampling. In continuous sample space, observing the exact same point twice is a probability event; such repetition would mathematically require the distribution to contain Dirac delta mass, which is incompatible with the smooth target density 4.. Thus. repeated states arise only as an artifact of rejection, not as a reflection of the underlying distribution dr- Storing these duplicated states leads to noticeably distorted empirical distribution

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Table 1: Performance on unconditional CIFAR- 10 generation. Values that are better compared the baseline are highlighted in bold.
Table 2: Performance on unconditional CIFAR- 10 generation with (Top) correction and (Bottom) acceleration methods.
Model
FID
NFEL
Method
HIDL
NFE]
FIDL
NFEL
FIDL
NFEL
Unconditional Generation
EDM (Heun)
2,05
2.23
3:23
DDifRS (Na
2024
2.17
3.26
7:79
VDM Kingmma al. 2021
7.41
1000
+DG Ki
2023
1,3
2.12
3.62
DDPM (Ho et 2020
3.17
1000
iDDPM (Nfctio & Dhaiwal 2021
2.90
1000
(Ours)
36.19
2.10
22.78
2.38
DDIM (Song al.
4.16
100
Method
FIDJ
NFE]
FID.
NFFE
[ID]
NFF.
Score Song ot al 202 lb
2.20
2000
Soft Truncation Kim et a. 2022b
2.33
2000
DPM
11.85
2.91
STF et al. 2022
1,90
35
UniPC
2025
11.10
6
3,86
2.85
10
DPM-03
Zhong es
2025
8.73
3.62
2.65
10
CLD:SGM (Dockhorn al. 20226
2.25
312
INDM Kim K0/44
2.28
2000
DPM-v3 (Ours)
7.12
5.61
3.09
LSGM Validi 20M4
2.10
138
PFGM++ [Xu el 21767
1.93
35
Table 3: FID and NFE unconditional CelebA-
PSLD
Panddey & Mandi 2025
2.10
246
Flow Matching Lapman ef 2023
6.35
142
HQ 256 generation.
Rectified Flow
Liu et al. 2023
2.58
127
ES (Nin al
2024
1.95
35
Method
FID
NFE
FIDJ
NFE1
FID.
NEE]
ScorsSDE (KAR)) 1
121.27
40
122.74
98
125.15
198
HDM (Heun) Karra
2022
2.01
35
+DLG
20.19
21:21
29.12
107.21
EDM (leen))
AC (Ours)
1.97
26.19
+AC (Ours)
15.13
15:94
22.55
44.06
15.69
87.26
DDO (Heun) ZZeng
2025
1.42
35
ScorEDE (KAR2)
83.21
40
57.28
98
29.74
198
DDO (Heun) AC (Ours)
1.41
29.41
+DLG (Kim Yo. 2025
17.92
21.21
12.12
107.21
4AC (Ours)
8.45
20,05
9.55
40.05
6.60
98.27
Acceleration Method
DPM + Lu et al. 2022a
24.54
5
Table 4: Performance on conditional ImageNet (Top) 64x64, (Bottom) 256x256 generation.
UniPC Zhao el al. 2023
23.52
5
DPM-v3 Zheng al. 2023
12.41
5
DPM-x3 + AC (Ours)
9.88
4.78
Method
FID
NFE]
FID
NEEJ
FIDI
NFF.
Correction Method
EDM (SDE)
2.30
61
1.78
127
1.43
511
Restart Xu et al. 2023b
1.95
43
+AC (Ours)
2.25
58.75
1.77
121.98
1.42
483.86
DifRS
2.02
30.73
DG (Kim 2045
1.93
27
Method
FID]
NFE]
Precision?
Recall
DIT IDPM
Peebles
Xie
2.35
250
0,829
DG AC (Ours)
1.84
26.19
+AC (Ours)
2.31
234.38
0.817
0.592
because the number of samples that can be generated is limited in realistic setting. To address this, design: proposals are repeatedly drawn until one is
in Algorithm[] [ we adopt a propose-until-acee accepted, and only the accepted sample is recorded. By contrast, this eliminates such artifacts and uses the entire sampling budget to explore new regions of the space. While this deviates from the strict MH setting assumed in theory, it yields substantially better sample approximation in practice. Further discussion is provided in Section 5.3 and Appendix
EXPERIMENTS
Experimental setting We employ our methods on various pre-trained networks trained on CIFAR-10 (Krizhevsky) 2009), mnageNer64x64 and 256x256 (Deng et al., 2009). CelebA-HQ
256x256 Karras et al., 2017). In text-conditioned generation, we use Stable Diffusion v v1.5 (Rom bach et al. 2022). We report the Fréchet Inception Distance (FID) (Heusel et al. 2017), ) Preci-
sion/Recall |Kynkäänniemi et al. 2019). For text-conditioned generation, we further evaluate CLIP
Score (CLIP) (Hessel et al.) (021)) ImageReward (IR) (Xu et al. 2023a), GenEval Score GGhosh
et al. 2023) for evaluate text-image alignment. We use Heun, SDE sampler Karras et al.
(002))
KARI [deterministtc) KAR2(stochastic) sampler (Kim & Ye (023), and DDPM (Ho al.
2020
DDIM (Song et al. 2021a) sampler. We highlight the best-performing results compare to baseline
model in bold. Detailed hyperparameters and settings are provided in Appendix C and H
5.1 UNCONDITIONAL GENERATION
CIFAR-10
The upper part of Table T shows results for unconditional generation on CIFAR-10. Our method is compatible with both EDM and DDO checkpoints. Both methods were re-tested without
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Stable Diffusion
AC-sampler (Ours)
Method
FID
NFE
CLIP
IR
0.3202
+AC (Ours)
23.16
45.24
0.3210
0.4531
GenEval Score
SD-v1.5 (DDIM)
+AC (Ours)
Position
4.00
6.25
Counting
35.62
35.62
couple of moter bikes parked on top of e green surfece
Two object
34.85
38.13
Colors
76.33
81.38
Single object
96.56
97.81
Color attribute
5.73
8.00
0.4219
0.4453
NFE
50.00
47.10
A wearing collar shit with blue
Table 5: FID, CLIP Score (Top) and
Figure 3: Generated image with SD-v1.5 (left) and AC- GenEval Score (Bottom) for SD-v1.5 and sampler (right). The red box indicates base model error. AC-sampler.
applying seed fixing in the EDM setting. For the EDM checkpoint, our sampler improves the FID from 2.01 with 35 NFE to 1.97 with only 26.19 NFE. For the stronger DDO checkpoint, we achieve 1.41 FID while reducing NFE from 35 29.41. Although the gain in FID is marginal for highly capable pre-trained models, our method consistently reduces the NFE, demonstrating its efficiency.
CelebA-HQ 256x256 We employ the pre-trained time classifier released by DLG to reproduce their reported performance. We extend the MALA algorithm to the joint space of (x+; 7), as detailed in Appendix
[ to support effective sampling on high-dimensional benchmark datasets a restricted NFE setting. As shown in Table 3 the AC-Sampler demonstrates clear improvement in FID compared to other methods with lower NFE.
Method Compatibility The lower part of Table Trpprrs results when applying AC-Sampler top of existing acceleration and correction methods. When combined with DPM-v3, which originally yields 12.41 FID NFEs, method improves performance to 9.88 FID with only 4.78 NFEs. Similarly, DG achieves 1.93 FID at 27 NFEs, while AC-Sampler applied to DG further reduces this 1.84 FID 26.19 NFEs. These experiments highlight that our method is orthogonal to existing acceleration and correction methods, and can flexibly enhance them. Table 2] and Figure present the FID-NFE trade-offs of our method compared existing acceleration and correction techniques. We observe that our method achieves better trade-offs in most NFE regimes.
5.2 CONDITIONAL GENERATION
Class- Conditional Table presents the results on conditional ImageNet 64x64 and x56 generation. For ImageNet 64,64, we employed conditional score network, while for ImageNet 256 256, we generate images with classifier-fre. free guid- ance (Ho & Salimans 2022) scale w = 1.5. We find effective improvements in NFE and slight improvements in FID. Although cass-condiitoned settings inherently limit the length of the MCMC chain, our results demonstrate that the proposed method can still be applied effectively under this constraint. We provide further discussion of the class diversity and related experiments in Appendix F22
Figure AC-Sampler on CelebA-HQ 256x256 (Top, unconditional) and ImageNet 256x256 (Bottom, condi- tion on "Weasel"). Panel presents the final results the MALA chains, ordered from left to right.
CoxttConditional Our method also applicable to high-dimensional text-to-image generation. We conducted experiments the Stable Diffusion v1.5 (Rombach al. 2022) with 5,000 randomly selected COCO validation prompts (Lin et al. 2014). Following the design real-world text-to- image systems (e.g., DALL-E-2 (OpenAl 2025), Midjourney (Midjourney, Inc. (025). Bing Image Creator (Microsoft) 2025),. we generated four images per prompt. This setting aligns naturally with method, which is designed to accelerate and improve the generation process when multiple
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25
DDAM
Table 6: Performance of AC-Sampler apply EDM. Time denotes the wall-clock sec- onds required to generate 100 images.
Mit correction
Mode Cover
23
2.0
Method
FID1
NFEL
Precisionf
Recallf
Timel
300
EDM (Heun)
2.01
35
0.704
0.627
6.46
5.26
EDM (Heun)
2.24
23
0.703
0.625
4.30
Figure 5: Effect of an MH correction on AC- Sampler.
Figure 6: Mode cover with different T in 25- Gaussian toy experiment.
+AC
2.21
20.67
0.707
0.632
4.15
EDM (Houn)
3.32
17
0,683
0.622
3.20
+AC
2.41
15.78
0.699
0.623
3.19
images are generated for a single prompt. Also, we use classifier-free guidance (Ho & Salimans 2021) scale = 7.5. Figure 3 presents samples generated by our method. For quantitative evaluation, we randomly selected one of the four generated images for each prompt. Our method achieves both faster sampling and improved FID compared to the base sampler as shown in Table
We also conduct experiment on GenEval(Ghoshl et al. 2023) benchmark. GenEval evaluates how well the generated images follow the prompt instructions. Since GenEval requires 4 different images per prompt for evaluation, we make four samples per prompt for computing metric. Ours consistently performs well as better than the baseline across all metries shown in Table (Botttom..
5.3 ABLATION STUDIES
Distribution Alignment To validate the necessity of the MH correction, we conducted an ablation study where the MH accept-reject step was omitted (i.e., all proposals were accepted). As illustrated in Fig. []] the FID deteriorates significantly without the MH correction. This result empirically demonstrates that the MH step is crucial for effectively aligning the generated samples with the target distribution. Furthermore, it validates that the acceptance probabilities derived from the discriminator provide reliable density ratio estimates consistent with the underlying theory.
Jump Markov Chain Conventional MH (MHc) retains rejected proposals as part of the chain, which ensures detailed balance but is inefficient for generative sampling under limited sample bud- get. As shown in Table 7 the MHc formulation involving rejected samples results in both lower recall and worse FID. In contrast, our propose-until-aceee
Table 7: Comparison of conven- tional MH (MHc) and Alg.
Method
RDD,
NFF,
Recallt
EDM (Rase)
2.05
27
0.627
design in Algorithm|] | encourages transitions to new regions of the sample space, thereby improving both diver- sity and fidelity. This approach can be formally interpreted as a
+AC with MHl
3.22
+AC with Alg.
1,97
26.19
0.625
Jump Markov chain (Rosenthal et al. 2021). and further details are provided in Appendix
Wall-clock time Since our method employs an ad- ditional discriminator at inference time, the wall-
Mormaltzed inference Time [Baseline-10) 1.0)
CIFAR1O
COCO
2
- 1.57
FID:
7,81
FID: 24 31 23.16
clock time could potentially be slower even with the
1.0
1.0
5.7%
same NFE. To evaluate this, we measured the average time (in seconds) required to generate 100 samples using both the base sampler and our method on a sin- gle RTX 3090. As shown in Tanle Figure 7 our approach not only improved sample quality but also achieved faster wall-clock time across various sam- pling strategies. Unlike conventional approaches that reduce NFE by enlarging the time step and thereby increasing discretization error, our method generates
0.9
6.46s
2.421
11.2%
0.9
10.83m
10.21m
2.15:
0.8
5.20:
0.8
0.7
0.7
0.4
0.4
Houn
AC
DPNv3
AC
DOIM
+
AC
Figure 7: Wall-clock Time Comparison. intermediate samples without coarsening the time grid. As a result, the improvement in sample quality is particularly pronounced in the low-NFE regime.
Sample Diversity Because our method relies on MH, successive samples can be correlated. Thus, assessing whether it still produces diverse samples is crucial. In Table 6 the recall metric, which is an indicator of sample diversity, is comparable to the base sampler. These results indicate that, method can preserve sample diversity. Together with the improved FID and reduced NFE, these

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Published as a conference paper at ICLR 2026
results demonstrate both the effectiveness and efficiency of our approach. The MALA chains benchmark datasets are shown in Fig. 4 where we observe that our method successfully generates diverse and high-quality images. Further analysis is provided in Appendix
Toy experiment We applied our method to a 25- Gaussian toy task. We generated 100 samples and counted mode as covered if any sample fell within 2a of mean. Across 10 trials, DDPM covered 23.5 modes on average, whereas our method consistently covered all as in Fig- 6 The solid lines indicate the mean, and the shaded areas are the standard de- viation. This confirms that our method effectively performs distribution correction and ensures diverse mode coverage. When low-7 regimes, more samples required for sufficient cover. Fig. 8 shows the results with 100 generated samples. Red circles de- note uncovered modes. Ours reduces NFE while fully covering all modes of the 25-Gaussian mixture.
#
1
4
4
4
4
(a) DDPM (NFE=1000)
(b) +AC (NFE=504.5)
Figure 8: Toy experiment results. Red circles denote uncovered mode.
Hyperparameter We primarily tune two parameters: the target timestep and the signal-to-noise ratio (SNR) of the Langevin proposal. The SNR controls the proposal step size 77 Larger SNR yields larger steps. With T 18, vary and SNR and report the resulting FID, NFE and Recall. As T decreases (i.e., closer to the data distribution), the distribution be- comes sharper, and MALA mixing deteriorates. Also, the SNR too small, proposals change little from the current sample, also leading to slow exploration. Given the constraints of limited number of generation, the choice of the target timestep 7 and the
Hyperparameter
Metrics
SNR
FID
NFE J
Recall
0.10
2.89
25.13
0.550
0.20
2.06
25.65
0.620
0.23
1.97
26.19
0.628
0.10
8.77
15:21
0.200
0.20
6.14
15:75
0.422
16.70
0.10
62.67
5.29
5:75
0.23
43.96
6.26
proposal SNR is crucial. A detailed analysis of hyperparameter provided in Appendix
Table 8: Hyperparameter analysis.
Discriminator NFE Representative studies that used time- dependent discriminator, such as DG (Kim et al. 2023) and DifiRS (Na et al.) (2024), provide effective correction capabilities. However, these methods rely on the discriminator throughout the whole sampling process, inevitably resulting in high Dis- criminator NFEs (D-NFE). Furthermore, DG requires gradient computation on the discriminator. In contrast, our method uti- lizes the discriminator only when constructing the MALA chain, thereby achieving significantly lower D-NFE compared exist- ing approaches. We analyze the D-NFE corresponding to the comparisons (Top) and visualize the results in Figure (Grad) denotes gradient computation. Compared to the baselines, our method requires significantly fewer D-NEE, which directly contributes to the wall-clock time advantages discussed earlier.
DeRS
134
DG
Discriminatoon NFE
AC (Durs)
2.1
Soare Scome Nit
Score NTE
Figure 9: Discriminator NFE.
CONCLUSION
We introduced AC-Sampler, which accelerates and corrects diffusion sampling via Metropo- lis usstiistsgs with Langevin proposal. By sampling at intermediate diffiusion timesteps and using only time-dependent discriminators for density-ratio estimation, our method improves sample quality and provides faster inference without requiring any fine-tuning of the base diffusion model. AC- Sampler theoretically achieves reduced NFE and decreases the KL divergence from the true marginal distribution each refinement step. Empirically, AC-Sampler attains better FID with fewer NFES across multiple datasets and is applicable to high-dimensional text-to-image generation. It also integrates seamlessly with various existing acceleration and correction methods.
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ACKNOWLEDGMENTS
This work was supported by the IITP (Institute of Information & Communications Technology Planning & Evaluation)-ITRC (Information Technology Research Center) grant funded by the Korea government (Ministry of Science and ICT) research was supported by Al Technology Development for Commonsense Extraction, Reasoning, and Inference from Heterogeneous Data(IITP) funded by the Ministry of Science and ICT 11220077) (50 %)
(50 %) This
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A PROOFS AND MATHEMATICAL EXPLANATIONS
In this section, we provide mathematical explanation and formal derivation of the theorems presented in the main body of the paper.
A.1 THEORETICAL ANALYSIS OF DLG KIM & YE 2023
Proposition A.1. Let q(x, t) be the true joint distribution over data x and diffusion timestep t, and p (x.t) t) be the model joint distribution.
Suppose that there is optimal time classifier xp", .ie., (1 q(l x), Vx,, If the model marginal distribution p" (x) does not match the true marginal distribution, i.e, x s.t. p (x) * q(x),
then the Markov chain defined by Gibbs sampling between p0(x and pd (t x) does not have q(x. t) as its stationary distribution.
Proof. Let's assume that the true joint distribution q(x,, f) is stationary distribution of Gibbs
Sampling, whose transition kernel is T(xx, t) - (x',t') p² (x' ttpp" (t' x'). Let (x, t) ((:-). and (x', (') is sample drawn from the transition kemne T((,, L) + (.,)). Since the stationary distribution should satisfy the invariance condition, for arbitrary x' the following equation holds:
// q(xx ((x, t) - (x,))) dx dt
q(x't' t')
(13)
/ʃ q(x, t)p (x') 1t)p (t' x') dxx dt // ((x,1)p (x (t)((t' xx) dx dt / a(t))((x' (x' (t)a(t' x() at
(14)
(15)
(16)
/ ((1) (x)) g(t' x)) dt
(17)
pt())
/ pO 12)x (x') xt)dt dt 12 (x')all' x) / p0(( xx)
(18)
(19)
<x'('')a(!1>>>>
(20)
Note that the marginal distribution ((t) pd (t) for every is uniform distribution. To satisfy the invariance, Vx', p (x') = g(x') so the proof holds due to the contradiction.
states that even if the time-classifier in DLG(Kim Ye 2023 is optimal, it is impossible to sample from the true distribution. So leverage Metropolis-Hasting Correction with this proposal distribution. Detailed explanation is given in Appendix
Proposition A.I
A.2 PROOF OF THEOREM
Theorem 4.1. Let X2 and X, be two arbitrary samples at diffusion timestep Then, for any fixed Xt-1, the density ratio of the true marginal distribution is given by:
(x()))
10j--1(5; Xx- (Re,.)
(1-1 x:)
(x.)
1xje-1(x; x(-1) (x(x,t)) ( xx)
where Le(x,,t) t) == ge (xx)/pp (xxx) denotes the likelihood ratio between the data and model marginal distributions timestep t.
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Proof.
We derive the marginal density ratio as follows:
(x,,
- (x,-1)
(x
X 1)
[t-1je(Xe- (Xt-1
xx.
(21)
ge (xxx))
9
9t/t-1 (xt
X(-1)
1t-1/t
(Xe-1 xx)
1 (Xt
Xx-1)
9x-1)1(x111
x+)
(22)
Q0/e-((X;
Xx-1) (0-11((X1))
x1)
(x
(X;
Xx-1)
(x-11((x-1
xx)
Pi 1|-1/11 (xt-1 xx)
Pi-1n(((111 (xt-1 xt)
(23)
(x,
X,-1)
Pi-1jo
(x,-)
X₂)
11-111 x
x2)
11- (x4-1 X,)
Following the derivation process of DiARS (Na et al.
2024). Eq
23 can be expressed as follows:
4 1
Xp-1) 1)
1x-1(x11,,t-
(x(x,tt t)
Pi-1 (x xx)
(24)
q:|t-1 (Xt X
Le( 8)
L 1x 1) p (xt-1 {x)
1
X(-1) 1) (x4
Lx(se, t)
Pe- 1/e
1 (X x()
(25)
9:/e-
X
(x(x,, t)
pl 1/e (x--1 Xx
Forward term
Likelibord ratio
Transition kernel
The Forward term and Transition kernel term are tractable since is a gaussian distribution. By approximating the Likelihood ratio by discriminator, we can derive a tractable form of the acceptance probability.
A.3 PROOF OF PROPOSITION
Proposition 4.2. Let the reverse diffusion process consist of a total number of timesteps T, and let the target fimestep in AC-Sampler be r. Suppose that I denotes the length of the Markov chain at timestep T. Let NFER denote the average reduction in NFE per sample achieved by AC-Sampler.
If the acceptance probability a satisfies a > T- then, as I + co, the expected NFE reduction remains strictly positive, while its variance converges to zero:
lim E[NFER] > O, lim Var(NFER) = 0. 1-000
(10)
1-000
Proof. Let's assume that the acceptance probability a is fixed in (0, 1], and that there is no burn-in process in the AC-Sampler for simplicity
Let R denote the total NFE required at timestep r for the MCMC step. At timestep7 Algorithm Tuuns I times. Since we make proposals until it is accepted, R follows the Negative Binomial distribution, i.e. R ~ NB(1, a). Let NFE, denote the total NFE required in the AC-Sampler sampling step. Then the following decomposition holds:
NFE,
R
[(T
(26)
Initial denoising stop
MALA step
Denoising step after MALA
Since we obtain the score value of each proposal at the MALA step and denoise total of I samples, the denoising step after MALA is 1(T 1). Since R follows negative binomial distribution,
have [[[ = - T + 1 + 1(T - 1) and Var(NFET) follows:
((1-a) Then, the mean of NFER is as
a
ENER = ENFE
(27)
=T T -
1
(T 1)
(28)
Q
1
T
T-T
T
(29)
We can get same result with burn-in process easily
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Taking limits on both sides yields
T - 1+1- 1
lim [FE
(30)
10000
a (: Boa>T-THE T-
>
(31)
Moreover, Var(NFER)
La at *. and thus taking limits gives lim Var (NFER) 0, which
concludes the proof.
A.4 PROOF OF THEOREM 4.3
Theorem 4.3. Let p.. Po 0. with optimal discriminator o², respectively. Then, the KL divergence between the true data distribu- tion and the refined distribution Po
denote the model distribution and refined distribution by AC-Sampler
0.40 is bounded by:
DxI(9o(xo))lp'* (xo)) S (xz((0(xo))ll)(xx))
(1I)
Proof. First, let T be timestep that MALA occurs in AC-Sampler framework. From (Ho al. 2020), the upper bound of KL divergence between the true data distribution and the model distribution can be written as follows:
20 (xo)
DRI(go 11 R.go
log
(32)
Pb (xo)
= E
- log Pu [xo)] 11 (40)
(33)
po.r (x0:))
= Engo
log 01:110(Xicr xo)
dxi:r
H(go)
(34)
91r (xI:T
x()
po:r (X0:T)
< go
91:7|0(X1:5
xx) log
dx1=7
Hg
(35)
(1 10 (1:r
(x0)
p0 P--11 Xii-
(xi)
q;(x;)
= Egg..
llog p. (x,)
log
H(g)
(36)
Di (1:111111111
xx:)
91
0i-1(x1-1)
9r (x7))
Pt-1/K
0
(Xi-1 xx))
= Eggoir
log pf (x+)
llog qo (x()))
log
H(q))
(37)
qi 111 (xi-1
x;)
1=1
T: 1/1 Xi-
xi)
= DRr.[Ur |1 + Expo..
log
(38)
pi-1 (x,-1
xx:
DKul(g ||
Egg
DKL (gi- (x--1
x;)
11 1/ (x:-1
x;)]
(39)
By substituting p with p the following bounded relation also holds:
DrI.(g 11 0.00
DKT.(gr 11 0.0
Eng [KRI xx) 1 (xi- xi)]] (40)
1=1
0.0" Since P;- (X,-1 x4)
(x [
x;) Vi it is sufficient show
DKL(gr 1 DKL -
(41)
When the discriminator is optimal and the burn-in process has been sufficiently performed, pa. 4r, DRi.gr (1+l |l " 0. Since the KI.-Divergence is non-negative, the proof is complete.
n
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The stationary distribution at timestep is L. The gap between the two distributions in Eq. 41 as follows:
DKL lar |1 DKL(gr ||
/ 9: log
9r
dx, - | gr log
Or 0
dx,
(42)
14
9r log /
97 dxc -
/
4
97 log
dx,
(43)
1
q, log L dx,
(44)
If the discriminator cannot distinguish between the two distributions pl: gr at all, i.e. Ly (x+, for all x;; 4, then the target distribution of Metropolis- algorithm becomes p As a result the gap between the two KL divergences in Eq. [4 becomes By training the discriminator, the gap converges to DKLla. Ip (2 0). indicating that the bound becomes tighter.
A.5 PROOF OF THEOREM 4.4
Theorem 4.4. Let T, be the transition kernel of MALA at timestep T. Also, pr 0,6",(1) denotes marginal
distribution at timestep т after the 1-th MALA transition from pt, and Pa 0.0".(1) (()) denotes the data distribution generated from pr
0.0" (() with denoising transition kernel Pi-1je L denotes a space [((=1)
of function which satisfies (Jz [[ <0.. 1f E Lo,,og
E E T2 LT,
(-(x+]
where a, B. E [1, co] satisfy + + 1, then the KL divergence between the true data distribution and the refined distribution Po
0.00 ((+1)) is bounded by:
DKL DxI(90(x))11p) 0.0" (1+1) (xo)) S DDL(90(x0)llp 0,000'000 (xo))
(12)
Proof. As in Theorem 4.3 it suffices to show that
DKLl ll 2ff.".0+1))
,(+1)] s DKLlat |l 1,.-00000
Then the below equation holds. We refer to the proof procedure in (ssvetkov et al. 2017).
DKL lgr |l 1 00.0".000 [o=.((+)]= = / 4r (x,) log
g- (x,))
(45)
0.0 (1+1) (x+)
/ dr (x+) log 0 (1+1) (x+) dx+-
(46)
/ or (x+,
Do 10.00.00(x, )x xx) dx
xx)dXx, } dx, H ((r)
(47)
0.00 (0000 (X+)
=- f-f (x+
Pr
9+ (x-)) +
xx) xx)ddix dx - H (9+)
dr (X,)
(48)
=- 1r (x((x+) {
Dr 0.000000 (1) (x7)
4r (x+) T7(x - (x) dx, - 1H4-HH(h -((+))
9: (X7)
(49)
0.8" 2000000 (Xx)
- 2r (x+) log (x((x)) /
Pr
T (xx - 87) dxc,
dx
H(g))
0: (Xt)
(50)
--f or (x((x)))og
(
pr
0.0 (000 To (x, 8r) d&c ) dxr
(51)
(x+)
0.0 1000 (8c)
$ s-ʃʃ or (x²)
log
Pr
T, Tx(x, - x2,)) dx,
(52)
ar (X,)
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Since + 3 + 1, apply Hölder's inequality (Hölder 1889) in Eq. 52
lf oi (xx)
log
pr
0 0.00000 (Xt,
Tr (x, x7) dx dx,dx,
(53)
47 (x,)
( ar (xx)) dše, dxr
$ (Il log
Pr
0.00 0010"
(Xct) (1())))()))
dx,
(J [T, + +z,I'' dāe, dx,,
Or (X,)
(54)
Since there exist a, B, and y such that 9- (x(xx) E L" log 10.00"
C and T, e L" holds, Eq. 52 is absolute convergence. Therefore, by Fubini's theorem, the order of integration can be interchanged.
(x,
- log
Pr
0.0 10000 (x+)
/ I+ (xx))) - xc,) xx, dā,
(55)
gr (X²)
- / log
Pr
0: 100010)) 11 ($cr
(xr))
gr (Xx dx,,
(56)
gr (86r
DKL(UT
|| || -0000:00)
(57)
The assumption in Theorem 4.4 is made solely to satisfy Fubini's theorem, and we note that the theorem is commonly adopted in prior works (De Bortoli et al. 2021 Lipman et al. 2023).
B METROPOLIS HASTINGS ALGORITHM IN JOINT SPACE
In DLG (Kim Ye 2023), a time classifier was proposed to detect whether a sample had left the manifold after Langevin dynamics at timestep t. The proposal distribution of DLG is as follows:
0.4 P proposal (x, i t) = P Pproposal [x]
x, t) .) Prropossal x, t, x) = Pproppssal {x x, t) Ppropossal (ē ()) (58)
First, given (x, t). sample x using one step of Langevin dynamics. After that, sample i using the time classifier conditioned on The proposal distribution in the joint space depends not only on the score network but also on the time classifier.
However, as we showed in Proposition A.I this approach cannot converge to the true joint distribution even when the time classifier is optimal, i.c., 1d (t|x) = qt()x). To address this issue, we perform the Metropolis- Hastting algorithm in the joint space of time and data. To compute the acceptance probability in the joint distribution, we extend the density ratio formulation presented in Theorem 4.1
This extension is proposed in the following corollary.
Corollary B.1. Let x, X be arbitrary samples at diffusion timesteps T, respectively, and let x be any fixed point at timestep If min(t,t) t) is satisfied, the density ratio of the true joint distribution ((...) ) is given by:
( 2)
lir (x/(+)
(x+1xxxx)
(59)
q(x,
9:lr (x|(x)
(x(x)t pe (x+
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Algorithm 2 www..tttaaannneeeettessssssssss/
Input: x, 1, Lt
Output: Accepted sample X,
repeat
2: Sample X ~ Pproposal, x, t)
3: Sample ~ Proopumaa
4: Compute 50 (x, 1) and T=== (x,
5: Compute acceptance probability: åzoint (x, 8, La, L2,
6: Sample ~ 10)
7: until
8:
return 8,1,8,1 L
Proof. For Vx, with 7 < t, the below equation holds.
g 2)
g(
g(t)
(60)
(xx t)
d(x
((t)
(X)
(61)
@r (x)
(X x,,)
(+1e(x, (x)
Or (x,)
(62)
(x/((x xx,
(-1(xx/x))
9: (xx,)
()) (8 x,) (x](xx
(63)
(x/((x xx,) (x1(xx/8)
9// (x x+) (xxx(x/>x)
p 1x (8)
1 1 (x)
(64)
(x)((x
xx,)
p (x+|x)
9-11(x, (x)
1 (x x))
(8)
x,,) L, (x,,
T,(X,,) 8)
p411 (x, (x) 1n (x~|*²
(65)
9//- (x |x,)
L, (x,,))
Ly (x+.
4//- (x xx))
(,(,, E)
p. (x+1x)
(66)
/// (x x,)
Lx(x,t t)
(x, (32)
Ppraposal x), is tractable. In detail, ph Pproposal (x x, is Langevin proposal, which follows Gaussian distribution and Pproposal (ł x) can be evaluated using the output of the time classifier. Therefore, the acceptance probability in the joint space can be computed. The acceptance probability is given as follows:
The density of the proposal distribution, Pfruposaa (x) x, t) %
egoine (x, L,
li
(x
X
i L
p
(x (x)
proposal (x
e
X. f)
Ppropossal (t
x)
min
1,
(67)
Qt
(x
(x+1x)
proposal
(x x, t)
P %
Pproposaa (t x)
Likelibood ratio
Forward
Transition kernel term
Proposal term
Xy can be any point. We choose the value of 7 such that it does not deviate significantly from the original timestep t. In our experiments, we empirically set 7 so that the standard deviation of (=)o(x+ Xo) differs from that of Imin(t. (Xmin(e,)) (X xx) by 0.1, based on the VESDE(SSng et al. 2021b) parameterization. We set &, = (H4(x, 5 (x, t)) + p s (x, t))) The detailed process is Algorithm
Since the timestep is proposed for every update of Alg. 2 we need to reassign the starting timestep I for denoising. We first fix the total number of steps T. and perform T steps of denoising
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Table 9: FID and NFE on unconditional CelebA-HQ 256x256 generation with ScoreSDE (Song et al. 2021b DLG (Kim & Ye. 2023), AC (marginal), and AC (joint).
FID|
NFEL
FID1
NFEL
FIDL
NFE
ScoreSDE (KARI)
121.27
40
122.74
125.15
198
+DLG
20.19
21.21
107.21
(marginal))
103.81
84.13
+AC (joint)
15.13
15.94
22.55
44.06
15.69
87.26
ScoreSDE: (KAR2)
83.21
40
57.28
98
29.74
198
DLG
17.92
21.21
12.12
47.21
8.14
107.21
+AC (marginnl))
45.74
19.55
19.48
42.97
9.45
94.34
+AC (joinl)
8.45
20.05
9.55
40.05
6.60
988.27
from down to the proposed timestep Then, we perform steps of denoising from to O. Our methodology generalizes the approach of DLG (Kim & Yel 2023). While DLG generates samples from the joint space time and data using Gibbs sampling. we introduce the Metropolis-Hastinngs algorithm correct samples toward the true data distribution by additionally training time- time-dependent discriminator. We adopt the time classifier from the official code of DLG and use the argmaa of the classifier output the proposed timestep, following their original approach. Since using the argmaz results in a deterministic time proposal distribution, we set p (t. x) when computing the acceptance probability.
For fair comparison, we reproduced the experimental setting of DLG. We first obtained the best parameters for both the KARI and KAR2 samplers reported in DLG, and then reproduced their performance using these optimal settings. Subsequently, we increased the number of denoising steps while keeping the remaining parameters unchanged.
C HYPERPARAMETER DETAIL
As described in the main text, our method treats the MH target diffusion timestep T and the Langevin step size (controlled by the signal-to-noiss ratio, SNR) as the primary parameters. In addition, we employ several auxiliary hyperparameters: the number of skipped steps nakip, the burn-in length Thumm-ins and the number of parallel chains Tichain Their roles are summarized as follows:
nskip: Controls how many intermediate steps are skipped between proposals.
Tbumm in: Specifies the number of initial iterations discarded to reduce initialization bias.
Rcchiin: Denotes the length of MCMC chains. With one initial point, can get Nehain samples.
Among the hyperparameters, regard the choice of 7 as the most critical. As r decreases-i.e. the state approaches the data distribution- marginal distribution becomes sharper. This sharpness increases the computational burden of moving across the space via MCMC. While smaller brings the chain closer to the true data distribution (see proof Thcorem 4.2), also requires a larger number of samples to sufficiently cover the support. Conversely, if 7 is set too low, the effective reduction in NFE diminishes and distributional alignment becomes less pronounced. Therefore, selecting appropriate 7 is essential. Empirically, found that setting between 1T and S achieves the most effective trade-off.
The second key parameter is the SNR, which controls the step size y) of the Langevin proposal:
VT = X (211)
(68)
tmolsmall SNR yields excessively small step sizes, limiting sample diversity, while a too-large SNR hampers convergence of the MH correction. Based on prior works that adopted Langevin sampling in diffusion models (c.g., Song & Ermon (2019); [Song et al. (2021b)), we set the SNR in the range of 0. 1.25...5.
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Experimental results varying these two key parameters reported in Table I0 We conducted experiments with the total number of timesteps fixed at 18. while keeping skppp 7iuum-in- and Mehain constant. As shown in the Table, when is set too small, the MCMC chain tends to remain in a limited region of the space for a long time. Consequently, covering the entire distribution requires significantly higher computational cost, which is reflected in the degraded FII and Recall metries. Furthermore, as the SNR increases, the acceptance probability gradually decreases. Across our overall experimental setup, the SNR satisfies SNR < 0.25, which allows us to maintain an acceptance probability of approximately a 2 0.25 (see Table I0 Because a is sufficiently large, as argued in the main text, substantial NFE reduction can be achieved even with small chain length l. These experimental results support the preceding analysis in Proposition 4.2
Method
SNR
FID
NFE,
Recall
Cf
EDM (Heun)
2.01
35
0.627
0.1
2.89
25.13
0.550
0.9150 + 0.0023
0.2
2.06
25.65
0.620
0.6248 + 0.0033
13
+AC
0.23
1.97
26.19
0.628
0.4703 + 0.0030
0.27
2.09
28.11
0.625
0.2493 = 0.0019
8.77
15.21
0.200
0.8689 + 0.0027
0.2
6.14
15.75
0.422
0.6560 + 0.0039
8
+AC
0.23
6.73
16.70
0.441
0.4671 + 0.0034
0.27
9.45
18.66
0.448
0.2733 + 0.0022
0.1
62.67
5.29
0.8525 + 0.0046
0.2
46.60
5.75
0.6590 + 0.0054
3
+AC
0.23
43.96
6.26
0.4812 0.0054
0.27
39.44
9.13
0.2703 1 0.0022
Table 10: Results for different values and SNR settings, including FID, NFE, Recall, and acceptance probabilities with 95% confidence intervals.
FDM (Hem)
Mothod
ACA
8
2
FIDI
2,00
200
1.97
1.34
2.01
1.98
1.97
34.39
NFEL
32.30
28.25
30.28
26.19
0.640
0.638
0.623
0.627
0.0125
1.634
0.631
Method EDM (Heun)
AC
Vitan
500
300
100
50
FID
2.00
2,02
2.07
2.01
2.03
1.97
NFE]
26.69
26.19
26.12
19.28
26.38
35
Recall?
0.627
0.630
0.624
0.622
0.629
0.625
Malhel HIM (Haun)
20
50
10
2.01
2.03
2.01
1.97
1.99
1.90
1.99
1.98
NPH]
97.58
60.81
42.57
48.54
38.09
57.46
16.23
Knoalf
0.629
0,632
0,632
0.633
0.627
0.640
0.624
0,629
Table 11: FID and NFE for dif- Table 12: FID and NFE for differ- Table 13: FID and NFE for dif- ferent skip steps.
ent chain steps.
ferent burn-in steps.
The following reports the results of varying each auxiliary parameter. Tables II 12 and present the outcomes for changing Nskkip Plchain. and Thumm-in, respectively. Unless otherwise noted, all experiments are conducted with T 18, T = 13.
Table TT reports the effect of varying nskip while fixing SNR 0.23, Rehmin = 300, and 7ham = 10. The parameter Rskip helps reduce autocorrelation between samples; however, excessively large values increase the NFE, limiting the achievable acceleration gain. Empirically, we set Neskp 0 1 for CIFAR-10 and ImageNet, and maximum 4 for CelebA-HQ 256 x 256.
Table 12 shows the results obtained by varying Achain while fixing SNR 0.23 and 7bumm-n 10. When SNR is too small, recall may vary with 77hhain but under reasonable SNR values the recall is largely insensitive Plahain Nevertheless, setting Fichain too small can hinder effective NFE reduction.
Table 13
investigates the role the burn-in process by varying Thurn while fixing SNR = 0.16 and Mahain 1. We set Nehaiin = 1 in order to isolate and examine the effect of correction on each sample.
We observe that after about 10 burn-in steps, the chain sufficiently converges, indicating that the score-based proposal distribution indeed allows proper convergence. In practice, we 70burn-i < 10.
D MCMC MIXING
Images are figh-dimensional data, which makes direct statistical evaluation of Markov chain mixing challenging. To address this, assess mixing indirectly by analyzing the class labels of generated
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images. We trained a ResNet based classifier that achieves 95% accuracy the CIFAR-10 10 test set, and used it to assign class labels to each generated image, thus forming a class sequence along the MCMC chain. We constructed Markov chains of length 300.
We measured Integrated Autocorrelation Time(IACT) (Birdsall al. 1994) 30 times with a maximum lag of 100. With Table 13 we observed trends consistent with Recall metries. Lower IACT values indicate better mixing, suggesting that our method yields well-mixed samples. However, we note that IACT originally defined for continuous variables, and applying to categorical class labels can be limiting.
To complement IACT, we also computed Cramér's $ V (Akoglu 2018) 30 times to assess class autocorrelation in the discrete label space in Table [4. Under the best-performing setting (T 5, SNR 0.23), the value at lag shows relatively strong correlation (Akoglu 2018), which is expected since our sampler proposes candidates based on local gradients. Nevertheless, both improvements in the Recall metric and our toy experiment [5.3 demonstrate that, despite such correlations, the chain able to generate sufficiently diverse samples.
Lag
Cramér's V d sid
0.035
2
0.259 +: 0.037
0.214 0,044
0.022
5
0.172 1: 0.036
Table 14: Cramér's V across lags.
Method
SNR
FID 1
Recall 1
IACT of class sequence
EDM (Heun)
2.01
0.627
0.1
2.89
0.550
8.71 + 10.02
0.2
2.06
0.620
2.45 1: 3.23
+AC
13
1.97
0.628
1,53 +: 1.70
0.27
2.09
0.625
1.89 +: 1.35
0.1
8.77
0.200
29.17
1 26.86
0.2
6.14
0.422
21.65 + 24.84
8
+AC
0.23
6.73
0.441
+: 24.50
0.27
9.45
0.448
24.39 + 25.55
0.1
62.67
38. + 27.39
34.93 +: 27.83
3
+AC
0.23
43.96
22.51 + 23.15
0.27
39.44
24.45 +: 32,90
Table 15: IACT of class sequence for different values and SNR settings.
E METROPOLIS-HASTING ALGORITHM AND ALGORITHM
Algorithm D jemploys a propose-umtil-aceee update: at each step, proposals are repeatedly drawn and subjected to the MH accept-rejed reject test until one is accepted, and the accepted proposal is then emitted as the next sample. We adopted this design for empirical reasons, namely to mitigate stagnation and preserve sample diversity. In canonical Metropolis- Hastings, however, a rejection corresponds to self-transition, which is essential for preserving detailed balance. Eliminating proposing until acceptance alters the transition kernel and can introduce stationary bias.
by
This variant be interpreted Jump Markov chain (Rosenthal et al. 2021). In such chains, the target distribution is implicitly modified because the rejection mechanism no longer permits self-transitions. Following Rosenthal et al., (2021), the stationary distribution of the jump chain, denoted ft, can be expressed in terms of the original stationary distribution π as
+x)) co(())(<)),
(69)
where axx) := 1 - P Paramstiio is the escape probability at state x, and c = Ey- [a())((y)) is normalizing constant. Here Piramsition-1] (-I-) denotes the transition probability of the original MH chain. The KI. divergence between and स is then
T(x) ((x)
1
E By><[a(u)]
DKL. [-]]
Expa
log
= Exaa
log
= Exaw
log
(70)
co(())
This formulation shows that the jump chain introduces a KL divergence bias. When (xx) is constant over the support of π. no bias arises; otherwise, the deviation can be non-negligible Despite the strong empirical performance of our method, a distributional gap remains. We leave a rigorous theoretical analysis of this gap future work.
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F ADDITIONAL EXPERIMENT
F.1 EXTEND AC-SAMPLER CORRECT EACH SAMPLES MULTI STEP CORRECTION REFINED PROPOSAL WITH A DISCRIMINATOR
After the burn-in process of the Metropolis-Hastingg method can be regarded samples drawn from the true distribution. This demonstrates that the Satropolis-Hasttnge algorithm can be used not only to accelerate sampling, but also to correct intermediate samples to better match the target distribution.
algorithm, the samples generated from our
Focusing solely on the correction perspective, our proposed framework naturally incor- porates the following methodological components: multi-step correction and refined pro- posal with a discriminator. As discussed in the main text, we initially present our al- gorithm using single-step formulation for simplicity. However, applying our method in a multi-step setting is straightforward and does not pose any conceptual or techni- cal difficulties. Therefore, we also conducted experiments under the multi-step setting.
Furthermore, DG (Kim al, 2023) proposed correcting the score network using the gradient information from discrimina- tor. Since we adopt exactly the same training scheme for the dis- criminator as in DG, is reason- able to apply refined proposal based on the corrected score net- work. This implies that the dis- criminator trained at timestep т only provides a likelihood tio estimate, but also enables re-
Table 16: Comparison of FID and sampling settings under differ- ent configurations
Sampling
FID
NFEL
T
SNR
Nizamin
Tidbie
EDM (Heun)
1.97
18
2.18
18
3.3,57
0.16
PC
2.13
51
18
7.9,
13
0.16
2.00
18
11.
15. 13,
0.16
1.94
18
3.
0.16
+AC
1.96
55.33
18
7.9.
13
0.16
1.93
54,91
18
11,
13,
17
0.16
1.98
$4.65
18
3,3,5,
0.16
+AAC with DG,
1.87
54.56
18
7.9. 11.
0.16
1.92
54.17
18
11,
15,
0.16
fining the proposal distribution @ P It is possible to use
DG, in accelerating, but DG, needs gradient calculation and this made sampling speed slow.
We present the results of both extensions in Table 16 We denote this discrininator-guided proposal scheme as DG,. The result demonstrates the effect of MALA correction across various choices of the correction timestep т. Although the correction timestep increases while maintaining the same SNR, AC-Sampler either improves or maintains the baseline FID score. Furthermore, we observe that incorporating the DG scheme into AC-Sampler leads meaningful reduction in FID with shorter length of chain. In contrast, PC-Sampler (Song et al. 2021b) often fails to correct samples at large correction timesteps.
In the table, setting 7lebain = indicates that Metropolis Haittngsass correction is applied for each individual sample. While this setting does not reduce the number of function evaluations (NFE), effectively corrects each intermediate sample.
F.2 DISCUSSION ABOUT TRADE-OFF IN CLASS-CONDITIONAL GENERATION
When performing conditonn generation, the number of inde- pendent class samples plays an important role. For FID evaluation, we use 50K images. With Mechiin 10, the baseline involves 50,000 independent class samplings, whereas only 5,000 samplings occur with our method, which may lead to class imbalance. If this effect did not exist, the NFE reduction could be even more effective. To evaluate our method fairly under this setting, we generate 250K samples with Nekain 5 (this setting makes total of 50K independent class samplings) and compute the FID five times using randomly selected subsets of 50K samples. We report the mean and standard deviation of the resulting FID. Ta-
Table 17: FID and NFE results on ImageNet 64x64 across different (SNR, 7) settings.
Methed
SNR
NLEL
EUM dilenz
2.30
61
0.12
228::005
$2.9
0.22
2.20-001
51.30
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ble 17 presents this analysis, showing that method can significantly reduce NFE while maintaining a comparable FID.
F.3 CIFAR-10 WITH SCORESDE
We also conduct experiments with Score-SDE (Song et al. 2021b) and DLG (Kim & Ye 2023) Table and [ our reproductions of the base models -Scor-SDDE and DLG) alongside our method. Because the publicly released Score-SDE checkpoint is configured for sampling with roughly 1,000 NFE, achieving strong performance substantially lower NFE is inherently challenging. Following the evaluation protocol described in the DLG paper, our reproduced results improved over the base checkpoint but did not exactly match the values reported in the original work. Under the same setting, applying method yielded consistent distribution correction and quality improvements, even in the low-NFE regime.
FID4
NFF.
FID_
NFF,
FIDJ
NFF
ScoreSDE (Base)
27.35
26.58
26
26.72
36
ScrechOlt (laxe)
+DLG
25.95
11.23
24.86
21.23
24.25
52.23
DLG
+AC (Ours)
25.18
10.95
23.83
19.13
23.14
29.07
(Oun)
Table 18: Experiment results on CIFAR-10 10 with ScoreSDE
NFE
Figure 10: ScoreSDE base ex- periment on CIFAR-10
F.4 ADDITIONAL ABLATION STUDIES
Feature extractor ablation To verify whether our pro- posed method relies on the semantic guidance of pre-
Table 19: Ablation study on classifier dependency.
trained classifier which is used in discriminator to extract feature, we conducted an ablation study using randomly initialized classifier. In this setting. the discriminator pos- sesses no classification ability or prior semantic knowl- edge. As shown in Table 19| although utilizing pre-
Method
FID +
NFE. J
EDM (Heun)
2.01
35.00
(w/ Pre-trained)
1.97
26.19
(w/ Random Init.)
1.99
28.19
trained ADM classifier yields some performance gain (FID
1.97), our method with randomly initialized classifier ("AC w/o pre-trained") still achieves an FID of 1.99, outperforming the Base model (2.01) while maintaining improved sampling efficiency (NFE 28.19). This result demonstrates that our approach does not fundamentally depend on the availability of pre-trained classifier or feature extractor. Note that the use of the ADM classifier as feature extractor follows standard practices established in prior time-dependent discriminator frameworks, such as DG (Kim et al. 2023) and DittRS (Na et al. (024)). We further extended this analysis to the Texl-to-Image (T2I) model under the same randomly initialized setting. The results for the T2I experiments exhibit similar trend and detailed in Table 5
Generating a small number of samples Our method achieves acceleration gains Techain increases; specifically, a larger Teehin extends the MALA chain, which reduces the average NFE. However, when generating a small number of samples, neetain may not be sufficiently long. Therefore in this setting, our method may not accelerate sampling processes. To empirically quantify the impact of small Mehain this setting, we generated a small number of samples (4 for SD-v1.5 and 10 for CIFAR- with batch size of using various Rchain values. As shown in Table 20) our method requires more time than the baseline when 7echain is very small. However, once 7ehain exceeds a certain threshold, generates samples faster than the baselines. Specifically, we achieved faster speeds than the baseline for CIFAR- 10 when Necaiin 2 3, and for Stable Diffusion when Rehhin 2 4. While our method may not offer speed advantages for single or dual image generation, it delivers both acceleration and correctness as the number of samples increases (2 o 3). This highlights its practical applicability for efficient and correct sampling in real-world scenarios.
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Table 20: Wall-clock time and NFE analysis in the small sample, low-nhchain regime. We compare the efficiency of our method (+AC) against baselines when generating small number of images (4 for SD-v1.5, 10 for EDM CIFAR-10).
(a) Stable Diffusion v1.5 (Generating 4 images)
Method
Thchain
NFE
Total Time (s)
Images
SD-v15 (DDIM)
50
34
4
AC
1
58.25
48
4
AC
4
42.25
31
4
(b) CIFAR-10 (Generating 10 images)
Method
Vcbain
NFE
Total Time (s)
# Images
EDM (Heun)
6.82
10
AC
37.14
7.44
10
+ AC
35.89
7.13
10
AC
33.98
6,81
10
AC
32.52
6.47
10
G ADDITIONAL RELATED WORK
G.1 SDE-BASED SAMPLING AND SOLVER RELATED CORRECTIONS.
Several SDE-based methods mitigate numerical errors by improving the accuracy of the sampler itself: adaptive adjustment of SDE solver hyperparameters to reduce discretization error (Kong et al.) 2021). controlled-variance reverse SDEs that share the same marginals for more efficient sampling (Zhagg et al. 2022), and higher-order SDE solvers to further reduce solver error. (Qiao et al. 2023 These works focus on improving the numerical integration quality of the diffusion trajectory.
G.2 APPROACHES ADDRESSING TRAINING- MISMATCH.
Other works instead target the mismatch between the training distribution and the inputs encountered during inference. Examples include: training diffusion models with input perturbations under an assumed Gaussian mismatch (Wang et al. 2023), and adversarially optimizing perturbations and model parameters to achieve distributionally robust diffusion training. (Liang et al. 2023) These approaches retrain the model to better handle off-manifold inputs during sampling.
G.3 METROPOLIS-HASTIOE
ALGORITHM IN DIFFUSION SAMPLING
The work in Aloui et al. 2024) applied the Metropolis-Hastinng (MH) algorithm to the diffusion sampling process. the paper, an acceptance network was trained via the loss based on detailed balance equation to calculate the acceptance probability, which was then utilized during the sampling procedure. However, their approach suffers from several key limitations:
1. No Distribution Correction: The acceptance probability calculation in Aloui al. 2024) involves a target distribution score term within its loss function. This term was approximated using the pre-trained score network. Consequently, the target distribution effectively be- comes the model distribution itself, meaning that no true distribution correction is performed.
2. Dependency on Proposal Distribution: The loss function used in (Aloui et al. 2024) is dependent on the proposal distribution. This introduces major practical drawback: if the proposal distribution is changed (e.g., switching to a different sampler step size or schedule), the acceptance function must be re- re-trained entirely.
Our proposed method addresses these limitations. By leveraging discriminator and Theorem 4.1, ensure that the acceptance probability can be computed regardless of the specific proposal distribution. This is achieved solely by training the discriminator, eliminating the need to re-learn the acceptance
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function when the proposal distribution changes. Our method is applied top of base sampler, and therefore remains orthogonal to SDE-based solver improvements while preserving their advantages. Moreover, because approach corrects generated samples solely through discriminator training, mitigates the training-inferent mismatch without requiring any additional training of the underlying diffusion model.
H EXPERIMENT SETTING
H.1 EXPERIMENTAL SETUP
Setups. We evaluate on CIFAR-IO, CelebA-HQ 256x256, ImageNet 64x64, and Ima-
geNet 256x 256. On CIFAR-10, we assess EDM (Karras et al. 2022) and DDO (Zheng et al.
2025) using the Heun sampler as EDM, ScoreSDE (Song et al. 2021 adopting samplers KARI (deterministic) and KAR2 (stochastic) (Kim & Ye 2023). For ScOreSDE, refer Appendix
F.3 On
CelebA-HQ. we use the ScoreSDE (Song et al. (021b) checkpoint within the DLG codebase (Kim
Ye 2023) with KARI, KAR2 sampler. On ImageNet 64 64, we use the EDM checkpoint with the
SDE sampler from (arras et al. 2022): on ImageNet 256x256, we use the DiT checkpoint (Peebles)
& Xie 2023) with DDPM sampler (Ho et al.) 2020). For text-to-image generation,
the SD
v1.5 checkpoint (Peebles & Xie
2023 with DDIM sampler (Song et al. 2021a).
Codebases and checkpoints. Our experiments use the official repositories of EDM DL DC DDC ScoreSDE D and SD-v1.5
Discriminator training. We train time-dependent discriminator per network following DG (Kim ct al. 2023) and DifRS (Na et al. 2024), using the pre-trained ADM classifier (Dhariwal & Nichol 2021) the feature extractor. We use random initialized feature extractor with no training (same structure, no classifying ability at all) for T2I case since there is no ADM classifier for latent diffusion model. We only train discriminator. Compared to training diffusion model, discriminator training is substantially cheaper.
Metrics. We report FID and the mean number of function evaluations (NFE) of the score network (as in DLG (Kim & Ye 2023)), ) since NFE varies across samples in our method. FID is computed on 50K generated samples against the 50K test images; for CelebA-HQ 256x256, For COCO (Lin et al. 2014) generation task, we report 5K FID. Also we report Precision Recall metric assess both the fidelity and diversity of generated images. The computation of FID follows the official implementation provided by DG (Kim et al. 2023). We measure the Precision and Recall using the ADM codebase (Dhariwal & Nichol 2021) l). For text-to-image generation, we evaluate Clip Score (essell et al., 20211. ImageReward (Xu et al. 2023a and GenEval score (Ghosh et al. 2023).
report 10K FID.
All experiments were conducted on NVIDIA RTX 3090, 4090 GPU and A.00 GPU using Python 3.8, PyTorch 1.12, and CUDA 11.4.
H.2 DISCRIMINATOR DETAILS
To implement the dine-dependent discriminator, we directly used the official DG codebase and followed their approach. On CIFAR-10. used the discriminator checkpoint provided by DG only when the base diffusion model was EDM and the NFE of the EDM (Heun) sampler was set to 35. For all other cases, trained the discriminator ourselves using the DG codebase.
hutpss//ggithbbb.cmm////aaa
httr:://qtinob.ccm22220/////
httoss//iLihbb.ccmmaaall.b.....
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Table 21: Sampling configuration and performance metrics (FID/ NFE) with various diffusion and sampler combinations.
Dataset
Tisk
Base Model
Base Sumpler
T
SNR
FTD
NPE
CIFAR-10
Uncond.
EDM
EDM (Heun)
0.23
11
2.10
22.78
0.23
300
0
13
1.97
26.19
0.23
500
13
2.02
26.12
0.23
3,4.5
4
13,14,15
1.93
44,40
6
2.38
15.82
11
5
3.24
10.57
CIFAR-10
Uncoad.
DDO
EDM (Heun)
16
0.175
2
2
13
1.41
29.41
CIFAR-10
Uncoad.
D
8
0.16
10
0
7
3.09
7.54
10
0.15
3
0
9
2.55
9.93
Uncond.
SooreSDD
0.23
250
11
23.80
25.18
0.23
250
14
23.83
19.13
ImageNett 64x64
Cond.
EDM
EDM (SDE)
0.16
26
2.25
58.75
64
0.18
2
50
1.77
121.98
256
2
225
1.42
483.86
CelehA-HQ 256x2256
Uncond.
ScoreSDXE
KAR1
0.16
25
0
13
15:13
15.94
99
0.16
25
10
3
60
15.69
87.26
CelehA-HQ 256x256
Uncond.
ScoreSDE
KAR2
49
0.16
50
10
4
33
9.58
40.05
ImageNet 256x256
Cond.
DIF
DDPM
250
0.12
2
190
0
10
2.31
234.38
COCO
Cond.
Stable Diffusion
DDIM
0.18
5 22
23.16
45.24
Our discriminators were trained on single NVIDIA RTX 3090 GPU. For the feature extractor, used commonly adopted (Na et al. 2024 Kim et al. 2023 pre-trained classifier from ADM (Dhari- wal & Nichol 2021). The discriminator takes the features extracted by this network as input, and during training, we only updated the parameters of the discriminator network. Detailed training settings are provided in Table 22
Indeed, while our method requires training an additional discriminator, we would like to emphasize that the training cost is significantly lower compared to that of the score model. As summarized in the table 23 our diseriminator much smaller and faster to train than the pre-trained score network. Compared to fine-tuning a pre-trained diffusion model, our approach introduces substantially lower computational overhead and does not modify the pre- pre-trained model in any way. Importantly, enable both acceleration and correction purely through discriminator training only. Table 24 shows the robustness of our -ime-dependent discriminator. To evaluate the performance of our discriminator at different timesteps, conducted experiments with Nehain ensuring that each sample is corrected independently. (No acceleration was applied, as our goal was isolate the effect of the discriminator
Table 24: FID and NFE comparison of AC with different
Method
FIDI
NFE]
+AC
2.02
48.51
1.97
48.53
1.99
48.54
2.00
48.72
1.92
49.89
2.00
49.03
timesteps) Using fixed SNR, we applied the AC-Sampler at various 7 values with the same random seed. The results consistently showed improvements in FID across timesteps, suggesting that the discriminator effectively approximates the density ratio 2 at multiple temporal locations.
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Table 22: Configurations of the discriminator.
CIFAR-10
ImageNot64
CelebA-HQ256
ImageNNe226
COCO
Diffusion Backhone
Model
EDM
DDO
ScoreSDE
EDM
Score
DIT-XL2
Stable Diffusion
Feature Extractor
Model
ADM U-Ner encader
ADM
ADM U-Net encoder V
ADM U-Nes enerder
ADM U-Net mncoder
ADM U-Net encoder
ADM U- eneader
Architecture
U-Net encoder
Pre-trained
Depih
4
4
Width
128
128
128
128
128
128
128
Resolutions
32,16,8
Ioput shupe (data)
(8.32,32,3) (3.8.8,512)
(B.32.32.3)
(8.32,32,3)
(B, (8.64.64.3)
(8,256.256.3) (B.8,8.512)
(8.32,52.3)
(B.64.64.4)
Output shape (feature)
(B,8.8.512)
(B.8.8.512)
(B.8,8,512)
(B.8.8,384)
(B.8.8.512)
Discriminator
ADM U-Net encoder
ADM U-Net encoder
Architecture
U-Net encoder
U-Net encoder 2
U-Net encoder
U-Net encoder
U-Net encoder
Depth
2
2
Width
128
128
128
128
128
128
128
Attention Resolutions
32,16,8
Inpur shope (feature)
(8,8&8552) (B,1)
(B.8.8.512) (B.3)
(8,8.8,512) (B.1)
(B,8.85512) (B.1)
(8.8.8552) (B.1)
(R.8.8,84) (B.1)
(8.8.8,512)
Ourpat shape (logit)
(B.1)
Discriminator Training
Time scheduling
VP Importance
Cosine Importance
VP
Time sampling
Importance
Importance
Impertance 20
Importance
Importance
Time weighting
2
Barch size
128
128
128
128
512
128
data samples
$0,000 $0,000
50,000
generated samples
50,000
50,000
50,000
30,000
50,000
5000
Epoch
60
70
60
30
Table 23: Training cost comparison of the score model and discriminator. CIFAR-10
Training
Parameter Size
Training GPU
Training Time
Score (EDM)
55.7M
8xV100 GPUs
22 days <2 hours
Discriminator
2.9M
1xRIX 3090
To evaluate the robustness of our method under an imperfect discrim- inator, we conducted experiments using partially trained discrimi- nators in table 25 We observed that as the discriminator training progressed, the quality of the generated samples consistently im- proved. Moreover, our method significantly outperformed where all proposals accepted without discriminator. These results indicate that even an imperfectly trained discriminator can still yield meaningful performance gains. This supports the theoretical claim in Appendix [A.4] where we show that continued discriminator training leads to improvements in KL divergence.
Table 25: Ablation on discrim- inator epoch.
Method
Epoch
FID
EDM (Heun)
3.23
17.00
3.40
AC
2.56
15.80
2.59
15.80
10
2.39
15.81
20
2.43
15.82
2.38
I THE USE OF LARGE LANGEUAGE MODELS (LLMs)
We acknowledge the use of Large Language Model (LLM) during the preparation of this manuscript. The L.LM was employed solely as a general-purposs writing assistant to improve readability, grammar, and clarity of exposition. It was not involved in the idcation of research questions, the design of experiments, the development of methods, or the interpretation of results. The scientific contributions of this work, including problem formulation, methodology, theoretical analysis, and empirical evaluation, were conceived and carried out entirely by the authors. The role of the LLM was limited helping refine the presentation of the text, and did contribute substantively the research process itself.
J GENERATED IMAGES
We provide sample images at Figure 11 generated by applying our algorithm the baseline models and samplers, used in our experiments.
12
13
14
15
generated by our sampler. These are samples
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Cols
Figure 11: The uncurated generated images of AC-sampler unconditional CIFAR-10 with EDM (EDM(Heun) sampler, NFE=26.19, FID=1.97).
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Figure 12: The uncurated generated images of AC-sampler conditional ImageNet 64x64 with EDM (EDM(SDE) sampler, NFE=59.30, FID=2.27).
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Figure 13: The uncurated generated images of Aamsmmeeer unconditional CelebA-HQ HQ 256x256
with ScoreSDE (KARI sampler, NFE=15.94 5,44, FID=15.13).
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25
Figure 14: The uncurated generated images of sammame unconditional CelebA-Ho 256x256 with ScoreSDE(KAR2 sampler, NFE=20.05, FID=8.45).
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(a) Vanilla, NFE=50, GenEval Scorc=0.4219
(b) + AC (Ours), NFE=47.10, GenEval Scorc=0.4453
Figure 15: Generated images from Stable Diffusion v1.5 and AC-sampler using the prompt "a photo of wine glass and handbag" (DDIM sampler).
35


Reviews:
[{"review_id": "WL3lfb3jFc", "reviewer": "Reviewer_MQvF", "summary": "Dear authors, I am the AC. Since two reviewers ghosted the paper or wrote last-minute that they wouldn't be able to submit a review, and since I was unable to find emergency reviewers, I have now written an emergency review of the paper. \n\nThe paper proposes a Metropolis-Hastings correction at an intermediate diffusion step \\tau to accelerate the sampling process of diffusion models. The idea of integrating MCMC updates into the reverse process is conceptually interesting, and the theoretical analysis is clearly written. The experimental results show modest improvements in FID with fewer or comparable NFEs.", "questions": "Is my understanding correct that the improved efficiency is only observed for the (imo unrealistic assumption) that multiple images are generated for the same prior noise?\n\nWhat is the exact goal of the theorems and how do they relate to practical settings? E.g., Prop 4.2: how are all the assumptions made implicitly here realistic?\n\nWhy are you not comparing to other methods for reducing the step size without having to train a new diffusion model?\n\nWhy did you not also make a comparison for generating a single (or a small number of) images?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Bringing an explicit Metropolis-Hastings correction into the diffusion sampling loop is an interesting idea to integrate score-based generative modeling and classical MCMC. Prior papers have explored MH with diffusion in other contexts, such as MCMC correction for model composition and Metropolis sampling for constrained diffusion, but using an MH step specifically as a generic accelerator within the standard image-synthesis pipeline is still relatively underexplored.\n\nThe paper not only proposes a new method but also analyzes the expected NFE under acceptance/rejection dynamics (e.g., conditions under which truncating at \\tau, followed by a short MH chain, reduces per-sample NFEs). \n\nThe empirical results, while limited in their scope (see below), are promising and show consistent improvements in both terms of FID scores and efficiency.", "weaknesses": "While the MH-corrected acceleration mechanism is interesting, the claimed efficiency improvement is not convincing. The method truncates the reverse diffusion at an intermediate noise level \\tau runs a short chain there, and then further denoises each accepted sample from  \\tau -> 0. Thus, every accepted sample still requires a full denoising segment, meaning there is no intrinsic saving *per sample* unless multiple final samples share the same denoising sequence from T to \\tau. Again, if the goal is to generate one image at inference time (in the realistic setting), there is no saving, as far as I understand. \n\nIn the \"amortized\" setting, where one truncated trajectory is used for several final samples, the expected NFE per sample can indeed drop, as shown in Proposition 4.2. However, if only one sample is drawn per trajectory, the method would be slower, not faster, due to the added proposal evaluations. The paper would benefit from explicitly stating this amortization assumption or correcting my understanding of the paper. \n\nWhile the paper emphasizes improved efficiency and reduced NFEs, it does not adequately situate the proposed MH-based acceleration among existing approaches explicitly designed for fast diffusion sampling. In particular, recent methods such as Tong et al., “Learning to Discretize Denoising Diffusion ODEs (LD3)” (ICLR 2025) and the references mentioned herein (which also work with a trained diffusion model) should be compared to. Relating to my point above, the comparison should be done for generating one single sample for the same number of time steps (plus the overhead of your discriminator), not averaged over a generation of a batch of l images.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GFCGAPdMj0", "reviewer": "Reviewer_jfbK", "summary": "The paper proposes AC-Sampler, a “accelerator-corrector” for diffusion models that jumps to an intermediate timestep (instead of starting at pure noise) and then applies Metropolis–Hastings (MH) with a MALA proposal built from the pretrained score network. This both shortens the reverse trajectory (speedup) and, via MH acceptance, corrects samples so their marginal at targets the true distribution. A time-dependent discriminator provides an estimate of the density ratio so the MH acceptance probability is tractable.", "questions": "1. Is the proposed method compatible with classifier-free guidance (CFG)? While CFG has known theoretical issues, modern large-scale systems rely on it heavily.\n\n2. Could the authors add a discussion in related work comparing their approach with SDE-based sampling methods [1–3] (which can be viewed as Langevin-type correctors) and with work that addresses training–inference mismatch [4,5]?\n\n\n[1] Gotta go fast when generating data with score-based models.\n\n[2] SA-solver: Stochastic adams solver for fast sampling of diffusion models.\n\n[3] Seeds: Exponential sde solvers for fast high-quality sampling from diffusion models.\n\n[4] Input perturbation reduces exposure bias in diffusion models.\n\n[5] Improved Diffusion-based Generative Model with Better Adversarial Robustness.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Clever decomposition of the acceptance ratio and use of a time-dependent discriminator make the MH step closed-form and cheap to evaluate。\n\n2. Theoretical guarantees: Expected NFE reduction when the acceptance rate exceeds a mild threshold; KL to the data distribution does not worsen and improves with more MALA steps (under stated integrability conditions).\n\n3. Designed to sit atop existing accelerators/correctors (e.g., DPM-v3, DG), often improving their FID at fewer steps.", "weaknesses": "1. The proposed method does not appear to provide acceleration when the batch size is 1; in this regime, the acceleration gain vanishes.\n\n2. Hyperparameter sensitivity & chain design: Performance depends on the choice of target timestep 𝜏 and the proposal step size/SNR, as well as burn-in/chain length—these trade speed for acceptance/mixing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8QE9utwvt8", "reviewer": "Reviewer_pPCw", "summary": "This paper proposes a sampling algorithm called Accelerator-Corrector Sampler (AC Sampler) that both accelerates and corrects diffusion sampling. This sampler is based on Metropolis-Hastings (MH) algorithm. The acceleration is achieved via a “warm-start” by performing denoising from prior distribution to a target tilmestep $\\tau$ which serves as the initial sample of MCMC. The MH acceptance probability is computed with the help of score function and also an additional time-dependent discriminator. This time-dependent discriminator is trained to predict the likelihood ratio of the unknown target data distribution and model’s marginal distribution at time $t$. The other terms in the expression for acceptance probability are Gaussian distributions and can be easily computed. \n\nThe advantages of AC-Sampler have been shown on CIFAR-10, CelebA-HQ 256 and ImageNet- 64 and 256. AC-Sampler can also be composed with other SOTA samplers such as EDM’s Heun sampler, discriminator guidance, DPM-v3 etc. and it results in improved FID score. In many cases, the average NFEs are also comparable or less, which is ideal.", "questions": "1. This work  is probably relevant and could be included in related works:\nScore-Based Metropolis-Hastings Algorithms, Ahmed Aloui, Ali Hasan, Juncheng Dong, Zihao Wu, Vahid Tarokh, 2024\n3. What is the typical length of MCMC chain i.e. how main times do we need to repeat Algorithm 1 before generating the final sample?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The proposed method is orthogonal to many existing samplers and can be combined with them as indicated in Table 1 and Table 2. Further, this composition results in improved FID in general.\n2.  The proposed algorithm seems to have better mode coverage than  EDM as indicated by Recall metrics. This also has been qualitatively demonstrated against DDPM solver in Figure 6.\n3. The paper provides theoretical proof (Theorem 4.3 and 4.4) which shows that the generated sample distribution with AC-Sampler is closer to the true data distribution in terms of KL divergence. The paper also provide results on the expected reduction in the number of NFEs.", "weaknesses": "1. The method requires training an additional time-dependent discriminator however it needs to be done only one time. There will  also be additional overhead from this discriminator during sampling. It is unclear if the forward pass through the discriminator is accounted for in NFEs. The discriminator also uses a pre-trained ADM classifier as a feature extractor which might not be readily available for all datasets. \n2. There is a potential mismatch between theory and practice. The practical implementation of MH in Algorithm 1 employs “propose-until-accept” design. This can introduce stationary bias as mentioned in Appendix E. This should be highlighted in the main paper. This also means that in this case, the chain wouldn’t converge to the desired target distribution $q_T$ but would rather converge to a different distribution due to the bias. \n3. Appendix B reports poor performance on CelebA-HQ 256x256 where the method in the main paper doesn’t scale to high dimensional data. Appendix B proposes to do MH algorithm in the joint space of time and data. This suggests that the primary algorithm from the main paper is not robust. It is also unclear if this method can be applied to Text-to-image models. \n4. Additional overhead in terms of wall clock time over many samplers such as DDIM, DPM-v3 etc. is unclear from the paper. There is only a  comparison against EDM’s Heun sampler in the paper.  In some cases, the NFE reduction is not very significant and therefore confidence intervals or the standard deviation needs to be reported.\n5. The performance of the method is quite sensitive to the choice of $\\tau$ as indicated in Table 9. In addition, there are many hyper parameters to tune such as burn-in length, number of steps to skip, number of parallel chains etc.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)