Adaptive Discretization for Consistency Models
Authors: Jiayu Bai
Zhanbo Feng
Zhijie Deng
TianQi Hou
Robert C Qiu
Zenan Ling
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion models
Consistency models
One-step generation
Primary area: deep_learning

Abstract: Consistency Models (CMs) have shown promise for efficient one-step generation. However, most existing CMs rely on manually designed discretization schemes, which can cause repeated adjustments for different noise schedules and datasets. To address this, we propose a unified framework for the automatic and adaptive discretization of CMs, formulating it as an optimization problem with respect to the discretization step. Concretely, during the consistency training process, we propose using local consistency as the optimization objective to ensure trainability by avoiding excessive discretization, and taking global consistency as a constraint to ensure stability by controlling the denoising error in the training target. We establish the trade-off between local and global consistency with a Lagrange multiplier. Building on this framework, we achieve adaptive discretization for CMs using the Gauss-Newton method. We refer to our approach as ADCMs. Experiments demonstrate that ADCMs significantly improve the training efficiency of CMs, achieving superior generative performance with minimal training overhead on both CIFAR-10 and ImageNet. Moreover, ADCMs exhibit strong adaptability to more advanced DM variants. Code is available at \url{https://github.com/rainstonee/ADCM}.

Adaptive Discretization for Consistency Models
Jiayu Bai', Zhanbo Feng", Zhijie Deng', Tianqi Hou Robert C. Qiu', Zenan Ling'
School of EIC. Huazhong University of Science and Technology
School of Computer Science, Shanghai Jiao Tong University HHaawei
Abstract
Consistency Models (CMs) have shown promise for efficient one-step generation. However, existing CMs rely on manually designed discretization schemes, which can cause repeated adjustments for different noise schedules and datasets. To address this, we propose a unified framework for the automatic and adaptive discretization of CMs, formulating it as an optimization problem with respect to the discretization step. Concretely, during the consistency training process, we propose using local consistency the optimization objective to ensure trainability by avoiding excessive discretization, and taking global consistency as constraint to ensure stability by controlling the denoising error in the training target. We establish the trade-off between local and global consistency with a Lagrange multiplier. Building this framework, achieve adaptive discretization for CMs using the Gauss-Newton method. We refer to our approach as ADCMs. Experiments demonstrate that ADCMs significantly improve the training efficiency CMs, achieving superior generative performance with minimal training overhead both CIFAR- 10 and ImageNet. Moreover, ADCMs exhibit strong adaptability to more advanced DM variants. Code is available at atttp://githhe //ggtht.....///.// com/ rainstonee/ADDe ADCM
1 Introduction
Diffusion Models (DMs) 34 9
38 18
20 have achieved remarkable accomplishments in the field
of data generation, including images
4 29
30, videos [10] 2 41, audio [15] 28 [9] and 3D
contents [40] (2 [22] However, DMs require numerous iterations to achieve high-quality generation, significantly slowing sampling speed and making resource-intensive. Recently, many fast-sampling
methods for DMs have been proposed, including training-free methods 355 14
48 and distillation-
based approaches [25 31
26
45 32]. However, these methods often sacrifice quality for
faster sampling or incur substantial training overhead, which limits their practical application.
Consistency Models (CMs) 37 7 offer significant advantages in addressing these challenges. CMs sample trajectories from the PF-ODE of DMs and aim to map each point on these trajectories to their corresponding endpoint. Through this approach, CMs achieve single-step generation while preserving the advantage of DMs, which improve generation quality by performing more iterations. CMs achieve the mapping the endpoint by minimizing the distance between adjacent trajectory points. We refer to the selection of adjacent trajectory points as the discretization for CMs. Previous works 37 36 7 23 2T have shown that diseretization crucial for CMs' training. It determines the trainability and stability of CMs: poor trainability can impair final performance, while instability during training may slow convergence or even lead to divergence. A suboptimal discretization strategy may lead to an imbalance between trainability and stability [7]. It may also cause CMs to overly focus on training within specific time interval, leading challenges and ensure balanced training process, most existing works adopt empirical discretization strategies, which require manual adjustments based on different noise schedules and datasets 77).
loss of consistency [17]. To mitigate these
*Corresponding Author: Zenan Ling (lingzenan hhw.s.tu..sss.....
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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10
Früchet incepption Distaince (FID)
Frichee incepption Distaince (FID)
SCM
500
(Duns) ADCM (Ours)
(Durst
SCT
ADCM rours)
2.5
100
200
100
800 Training Images (Mings)
1700
Training Images (Mimgs)
(a) CIFAR-10
(b) ImageNet 64 x 64
Figure I: ADCMs significantly improve the training efficiency on both (a) unconditional CIFAR- 10 and (b) cans-conditional ImageNet 64 x 64. ADCMs achieve superior generation quality with only minimal amount of training data. indicates smaller model.
Our fundamental goal is to adaptively determine the discretization strategy for CMs' training consid- ering both trainability and stability, thereby improving the training efficiency and final performance of CMs. First, we propose that the discretization step should minimize the optimization objective of CMs, i.e., local consistency, to ensure their trainability. Second, the discretization step controls the denoising error in the training target of CMs, which affects the global consistency. Excessive denois- ing error can lead instability in CMs' training. thereby degrading the efficiency of CMs. Hence, we introduce global consistency as constraint to ensure stability. To adaptively balance trainability and stability, we formulate local and global consistency as a constrained optimization problem and relax it via the Lagrangian method, introducing Lagrange multiplier to express the trade-off between them. To achieve adaptive discretization, we propose using the GeasstNewton method obtain an analytical solution to the optimization problem. We refer to our method Adaptive Discretization for Consistency Models (ADCMs). Our analysis reveals that ADCMs adaptively adjust discretization steps by jointly considering local and global consistency, thus achieving balanced trade-off between trainability and stability.
In our experiments, ADCMs significantly improve the training efficiency and final performance of CMs. On unconditional CIFAR-10, as shown in Figure Ta ADCMs exhibit outstanding training efficiency. We achieve 1-step FID of 3.16 with only training budget of 12.8M images. In contrast, ECM 77. the most efficient CM from previous work, requires 51. 2M images to reach FID of 3.60. Moreover, we attain 1-step FID of 2.80 using only 76.8M images, outperforming iCT [36], which requires 409.66 6M images to achieve comparable performance. On class-conditional ImageNet 64 x 64, as shown in Figure 1b ADCMs significantly reduce the training overhead under the same model size. ADCMs achieve a 1-step FID of 3.49 with a training budget of only 12.8M images. When the training budget increases to 51.2M images, ADCMs achieve competitive 1-step FID of 3.04.
Contributions. Our contributions are summarized as follows.
We provide a unified framework for the discretization for CMs. Starting from local con- sistency and global consistency, we investigate the impact of discretization steps on the trainability and stability of CMs. Guided by these two principles, we formulate constrained optimization problem for selecting the discretization step. Previous discretization methods can be regarded as special cases of our framework.
Based on this framework, we propose Adaptive Discretization for Consistency Models (ADCMs). First, we relax the optimization problem using the Lagrangian method and establish a trade-off between the trainability and stability of CMs through the Lagrange multiplier. Then, employ the -ewus- method derive analytical solution to the optimization problem, enabling adaptive discretization steps that effectively balance
In particular, we focus on Consistency Training (CT) 37 over Consistency Distillation (CD) due to its superior empirical performance and its ability to bypass numerical solvers by directly leveraging training data for unbiased score estimation.

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local and global consistency, Additionally, we introduce an adaptive loss function to further improve performance.
Our experiments demonstrate that ADCMs significantly improve the training efficiency, while achieving competitive performance in one-step generation. On CIFAR-10, -0,, ADCMs achieve superior results using less than 25% of the training budget compared to previous works. On ImageNet, ADCMs also demonstrate strong performance with minimal training overhead. Furthermore, ADCMs adapt to advanced variants of DMs such as Flow Matching without manual adjustments.
1.1 Related Works
Consistency Models were first proposed by [37] achieving the distillation of Diffusion Models by mapping any point on the PF-ODE trajectory the endpoint of the trajectory. To accomplish this, it proposed sampling adjacent points on the trajectory and enforcing that the output near the noise end approximates the output categories: Consistency Distillation (CD) and Consistency Training (CT), corresponding to sampling trajectory points using pretrained DM and the forward diffusion process, respectively. iCT (36) explored the potential of CT, as it does not require a pretrained DM to sample trajectory points, thus supporting training from scratch. ECM 7 discovered that initializing the CM with a pretrained DM can effectively accelerate its training speed. TCM [7 discovered that CMs struggle to map the entire trajectory using a single model. Therefore, proposed a two-stage approach for CMs, enabling CMs to focus on learning tasks from different time intervals separately. CTM [13] and Shortcut Models [5] aimed to make CMs capable of mapping to any point on the trajectory, not just the endpoint, by introducing an additional time condition to assist the model's learning.
Consistency Models.
the data end. It divided CMs into two
Discretization for CMs. The training of CMs fundamentally relies the selection of adjacent trajectory points, process we refer to as the discretization for CMs. Various discretization strategies have been explored in previous works. iCT (36) proposed segmenting time based on the sampling steps of DMs M. ECM (7) introduced a decoupled approach, employing two functions: one to determine the overall magnitude of discretization steps and another regulate their distribution over time. Both iCT and ECM adopt exponentially decreasing time steps to enhance the stability of CMs' training. Alternatively, CCM [2I] introduced an adaptive discretization scheme by iteratively solving for the discretization step based on a Peak Signal -ggal-to-Noise Ratio (PSNR) threshold, ensuring a more balanced training across different times. sCM (3 formulated an "infinite" discretization approach, where adjacent trajectory points become infinitesimally close, transforming their distance into the first-order time derivative. However, sCM observed that this discretization scheme suffers from stability issues and proposed modifications to both the noise schedule and the neural network architecture, among other refinements, to ensure stable training.
2 Preliminaries
2.1 Diffusion Models
Given dataset with underlying distribution Pdatas DMs generate samples by learning to reverse noising process that progressively adds random Gaussian noise to the data, eventually transforming it into pure noise. Specifically, for data sample æp Pdata and noise sample ~ ~0,0 T), the diffusion process is defined as:
Dc Og + B,z
where t E [c, 7], and is a small value used to prevent numerical errors. 38 proposes that the diffusion process can be modeled as a forward SDE, which is then denoised step-by-step using the corresponding probability flow ODE (PF-ODE). DMs utilize a time-dependent neural network (NN) predict the unknown 2% in the PF-ODE. The optimization objective of DMs is given by:
min E (w.
(lf) - x
where fo(x;) = (t) + Coun (t) (Ct (t) Cnoise (t)), w(t) is weighting function and Fg is an
NN with parameters 0. We write [æx, 1) as (xx) for simplicity. Most DMs, including DDPM ( EDM [1] and Flow Matching [18), have training objectives that can be equivalently expressed as Eq. I through the design of precondition Cakip (t) and Cour (t).
3

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2.2
Consistency Models
CMs
37 23
aim map any point 2y on the PF-ODE trajectory the corresponding data zo. i.e.
fo( t) Zy. To achieve this, (1) at = O, CMs require that fo satisfy the boundary condition fo( zg. which implies that Csip
and (our 0: (2) for t >> CMs are trained to produce consistent outputs for any two adjacent points on the PF-ODE trajectory. Specifically, the optimization objective of CMs given by:
min Em.... [()) [[))( Se (2, 11
where 6 stands for stopgrad and At is the time interval that defines the adjacent time step corresponding to a given time i, which in turn determines the training target on the PF-ODE trajectory. When retrieving the adjacent point ø (t- we adopt Consistency Training (CT) paradigm [37] which enables unbiased estimation of the score function, expressed as æ=-a.Ro This approach eliminates
the numerical errors introduced by solvers required for Consistency Distillation (CD).
The choice of At, referred to as the discretization of CMs, plays a crucial role in their training [7 21 23
Previous works have proposed various discretization strategies, which can be categorized into two types, as outlined below.
Discrete-CMs. When At > is not infinitesimally small, CMs fall to discrete-CMs. Discrete-CMs require careful planning of the discretization schedule. iCT 36 divides the time interval [c. T] into multiple segments using the sampling time steps in DMs [1I, i.e., T = {fo, ..... tx) } where to = c and ix T, and samples adjacent time points within as and At. iCT proposes that the discretization of CMs requires meticulous planning and designs time steps that decrease progressively during training. ECM 7 also adopts dynamic time step scheduling. Unlike iCT, ECM samples in continuous time and then maps the corresponding time step through manually designed function. CCM [21] points out that the discretization step size affects the training difficulty of CMs at different times. CCM proposes setting PSNR threshold for CMs and solving the PF-ODE iteratively until the selected time steps satisfy this threshold.
Continuons-CMs. Taking the limit At + 0, CMs fall to continuous -Mrii, which can be considered equivalent to "infinite" discretization. 37 proves that continuous- CMMs can be optimized with:
dfe- (x:))
V Ve w())ff
(x:) (x;;)
dt
which is a continuous version of Eq. (2). Continuous CMs effectively avoid the discretization schedule of discrete-CMs. However, continuous-CMr often face significant instability challenges. sCM (3 addresses this instability for a specific DM with a specialized noise schedule, but it remains unclear that how to address the instability for continuous CMs under more general setting.
Overall, the choice of discretization step At is still challenging. If At is too large, CMs struggle to learn meaningful information, while if it is too small, instability issues arise (23]. Additionally, how At varies w.t.t determines the training emphasis at different times [36] 77. and suboptimal strategies can lead the model to focus specific time intervals, negatively impacting overall performance. Although various discretization strategies have been proposed, they often fail to identify the optimal At for each time step. On the one hand, existing discrete- CMs lack adaptive adjustment capabilities, requiring additional modeling and hyperparameter tuning for different noise schedules and datasets. On the other hand, continuous-CMM avoid discrete time steps by treating all time steps equally, but not all are equally important for effective training (36] 17 This limits training efficiency.
Methodology
3.1 ADCMs: Adaptive Discretization for Consistency Models
we illustrate the importance of discretization in training CMs. In this study, our fundamental goal is to determine which discretization strategy is most beneficial for CMs' training, ie.. the discretization step A for given time When we fix the NN's parameters 0- stopgrad and time I, we aim to find an optimal Al that contributes the most to the following training objective of CMs:
In Section 2.2
Exo (1lfo-(x)) - fo- (x) (x)-
(4)
4

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We define Eq. (4) as local consistency as it reflects the properties of CMs in local interval. First, we need that the objective in Eq. (4) trainable. To achieve this, we need choose an appropriate At such that the objective is as small as possible, thereby satisfying local consistency, namely:
min E [Ifo- (xx- fo- (Z)-
(5)
can be observed that when At = O, the local consistency in Eq. (5) is minimized. This implies that we need to choose At small as possible. However, previous works 37 7 23 have shown that when At is too small, CMs face severe stability issues, which slows down convergence and may even lead to divergence. The underlying reason is that the practical training target, i.c., fo (æz- fails to precisely denoise At to the ground-truth Vo, leading to the global denoising error quantified as:
Exp.. = [I (I)- zu
Remark 3.1. This denoising error is also an upper bound on the squared Wasserstein-2 distance between Pdsta and the data distribution generated by fe- at time At. Moreover, this error can be regarded as lower bound on the accumulated error from previous time steps, namely:
Emn.. [1.fo (21- Zo 1
Exo.s [l (IL.) fo (211- 112]
where At, is the discretization step corresponding to time 4, satisfying t, - Al; = f;-1-
We define Eq. (6) as global consistency because reflects the global properties of CMs. Excessive denoising error will cause CMs to optimize in the wrong direction, which leads to instability in CMs' training. Therefore, we propose that when selecting the discretization step At, it is necessary to ensure that the denoising error is constrained, namely:
Find AA, s.
Expa [1 (x:--)) zo || < 8
where 8 is upper bound that needs to be set manually. Clearly, when At takes the maximum value t - c, due to the boundary condition fo(x,,<) = Zg. the constraint in Eq. (7) will be satisfied regardless of the value of 8. This implies that need to choose the largest possible AV.
Notably, the optimization objective in Eq. 5 and the constraint in Eq. 7 respectively impose opposite guidance for At. When At is extremely small, the local consistency error in Eq. (5) is minimized, making it easy for CMs to optimize. However, this may cause the constraint in Eq. to exceed its upper bound. Conversely, when At is extremely large, the constraint in Eq. 7 will be easily satisfied, but it may cause the optimization objective in Eq. (5) to become too large and difficult to optimize. Therefore, we propose constrained optimization objective to achieve a trade-off between Eq. (5)
and Eq. (7). which is given by:
min 21
Exr.. [llfo (xx)) fo (1)= s.
Ex.. |Ifo (0:--r) 2011 < 6.
(8)
We denote the optimization objective E= [I]fo (a:) - So (x)- as Liscal, as it focuses on the local consistency information of CMs and controls the local consistency error for CMs. There- fore, minimizing Liecal effectively improves the effectiveness of CMs. We denote the constraint [ll.= (#r- 2o [ as Lghobul focuses on the global consistency and controls the denois- ing error in the training objective. Consequently, Lgiohal helps CMs effectively eliminate denoising error, find accurate optimization targets, and thus improve training stability and efficiency. Our goal is to ensure both the global consistency and the local consistency simultancously, enabling an adap- tive adjustment of CMs' discretization. However, directly optimizing the constrained optimization problem in Eq. (8) is challenging. To address this, we apply the Lagrange multiplier method to relax the problem. yielding the following formulation:
At" arg min E ( (t, At) + ACgloball (t, At)]. At
Here, the Lagrange multiplier acts as a weighting factor balancing the local consistency and the global consistency of CMs. We aim for À to be a constant independent of t, ensuring that the focus on trainability and stability remains consistent across different time scales. Typically, we set < 1, as ensuring whether CMs are trainable is of greater importance compared to their stability. We refer to our approach as Adaptive Discretization for Consistency Models (ADCMs), as shown in Figure We find that previous discretization strategies can be unified in ADCMs. We summarize this follows.
5

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Local consistency of CMs
Global consistency DMs
ADCMs balance local and global consistency
Figure 2: Discretization strategies of different models. CMs consider only local consistency during discretization, while DMs consider only global consistency. ADCMs balance local and global consistency and adaptively adjust the discretization step size based the information from both.
Remark 3.2. DMs, discrete-CMs and Continuous-CMr can be viewed as special cases of Eq. ( Specifically,
Choose the maximum optimization step At = t c. this corresponds to set À in our framework.
DMs 18
Discrete-CMs:
CC (37, iCT 36]. HCM 77: These methods consider the smoother trajectory changes near the noise end and empirically choose discretization step size that monotonically increases w.r.t. t. This is equivalent to estimating Eq. (9) empirically in our framework.
-CCM [21]: CCM ensures that for all 20, z, L. is always less than certain threshold 8. Since an analytical solution cannot be obtained directly. CCM requires iterative solving for all zo 2, This is equivalent making Liocal a constant in framework.
23 Choose the minimum optimization step - This equivalent to set X = 0 in our framework.
Continuous-
Analytical Solution. To achieve an adaptive solution for the discretization step, we propose using the Gauss-Newton method directly derive an analytical solution the optimization problem in Eq. (9). Since we assign higher weight to local consistency in the objective, we approximate for (i)):) using its first-order Taylor expansion, which is:
dæ
fo (x:- 2 fo (x,) vAt, v v= Va,
+ 0,
dt
where v can be efficiently computed the Jacobian-vector product (JVP) for S (.,)), evaluated at input vector (x,,t) and tangent vector (dar 1). following the method in [23]. Under this approxima- tion, the optimization problem is transformed into least-squares problem, whose optimal solution is given by:
A ER. (fo (xx) xo)]
At"
(10)
1+>
Exp,, (
From the expression of the discretization step, we have the following observations:
1. The optimal discretization step is inversely proportional to the magnitude of the Jacobian. This indicates that the output of the current network may vary significantly, and Liccal could be very large. Therefore, smaller step size is required to ensure effecttivnesss
2. The optimal discretization step is proportional to [ife (xx) æall2. which is effective estimate of Cglobal. This indicates that the denoising error may be very large at this time, and therefore, a larger step size is required to ensure stability.
3. The optimal discretization step is proportional to the linear correlation between and (o-( .p- This indicates that when and fe- (æx,) æt tend toward linearity, the direction local optimization aligns more closely with the direction of global optimization, allowing for the use of larger step size.

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Algorithm Adaptive Discretization for Consistency Models
Input: dataset D, diffusion parameter and Br, time range (e, T). network parameter 0, weighting factor X, update frequency mi batch size b
0- <- 0
repeat
Initialize empty set T and & <- T
repeat
Append to T
Sample mini-batch Zo ~ D, 2, DN,,N(
E₁ f- Cy + 3,2
Calculate A/" through Eq. 10
{ 1 A/"
until / S
Append to T
for k = to m do Sample mini-batch 2o ~ D, (0, I) and adjacent time points t, At T
2t f- 0, + 3,2, -t" f- 0r-Ar" 2o + -
Calculate loss L through Eq.
Update 9 using L
end for until Convergence
The above analysis demonstrates that the proposed discretization step can be adaptively adjusted based the current state of the NN, considering both Lglobal and Liocal As result, we achieve adaptive balance between the stability and trainability CMs at different times through Eq. (T)). Starting from T, we iteratively solve the optimization problem derive the adaptively optimal time segmentation T = [ti....,,y..
3.2 Putting ADCMs into Practice: Further Training Strategies
Adaptive Weighting Function. Through the analysis in Section 3.1 know that Ligobaal funda- mentally determines the training stability of CMs at the current time However, during the training of CMs, the NN only optimizes for Liocal- Therefore, further balance the impact of Lglabal time, we propose the following adaptive weighting function:
(l()
Eglobaal
I1 (2); Tol
When Lgtobal is very large, the training of CMs will suffer from instability. Therefore, a smaller weighting is needed, On the other hand, when Litobal is small, the CMs training objective aligns closely with the true target, and thus larger weighting is required.
Adaptive Distance Metric.
Previous works 7 36] have shown that compared to the squared L metric, Pseudo-Huber metric can effectively reduce training variance, which given by:
d(x y)
|| yo-yyyle + -
where is constant. ADCMs also Pseudo-Huber metric for training. At the same time, in order to ensure the consistency of the distance function, we have similarly modified the adaptive weighting function. The overall loss function of ADCMs can be expressed
(o) Se (æ.
+ 2
min Emo.
(11)
Vllffo (00-((()))))) 2D
1/
+ c -
where At" is obtained with Eq. (10 See Appendix B for more discussion loss function design.
Putting Together. We alternately optimize the time segmentation T and the NN's parameters during training, We typically update T after updating 9 for 25000 times, as the changes in the
7

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Table 1: Sample quality unconditional CIFAR-10 and class-conditional ImageNet 64 x 64. indicates additional training costs.
Table 2: Training efficiency on CIFAR-10.
Unconditional CIFAR-10
Method
CIFAR-10
ImageNet 64
Method
Training Budget (Mings)
1-Step FID (L)
(J) FID (1)
())
(
CD (LPIPS)
409.6
DDPM [9]
Diffasion Models
1000
3.17
250
11.0
iCT
409.6
2.83
<CT (TrigFlow)
204.8
2.85
DPM-Nolver
10
4.70
3.42
SCT (VE)
51.2
4.18
1. Rectified Flow
20
127
2.58
BCM
12.8
4,54
BCM
$1.2
EDM2-S
P
1.58
EDM2-XL [0
63
1.33
ADCM (Ours)
12.8
3.16
ADCM (Ours)
76.8
2.80
Joint Truining
MtyleGAN-XI,
1.52
SiD X
1.92
1.52
CTM 1
1,87
1.92
Consisicncy-
1.69
1.62
Table 3: Training efficiency on ImageNet 64 x
DMD2 44
1.28
64.
Diffesion Distillation
DFNO (LPIPS)
3,78
7.83
PID (LPIPS) 39
3.92
9.49
Clas--onditional ImageNet
TRACT
3.78
7.43
Model
Training Budget (Mimgs)
1-Step
10.70
(LPIPS)
6.20
2. Rectified Flow
4.85
Consistency Models
iCT-deep
(LPI5)
3,55
6.20
sCT (TrgFlkww)
2x
2:25
37
8.70
13.0
ECM
1x
12.8
5.51
:CT 10
2,83
4.02
FCM
2%
12.8
3,67
i-Teee
86
2.51
3.25
ADCM (Ours)
1x
12.8
5.12
ADCM (Chrr)
18
25.6
4.65
TCM
77
2,46
2.20
ADCM (Ouns)
4,23
sCD
33
3.66
2
ADCM (Uurs)
ADCM (Ouns)
3.28
ADCM (ours)
2,80
3.04
(Ouns)
NN are relatively slow. Before training the network, we fix its parameters and perform simulation- based optimization starting from = T. We iteratively update t using Eq. (TO) until = recording the optimization process as T { " 1. We observe that during the optimization process, the expectation in Eq. (TO) is well approximated using single mini-batch. This is because we do not require precise step sizes, only the trend of their change over time t. Subsequently, we fix the time segmentation and optimize the NN. The detailed process is illustrated in Algorithm
4
Experiments
To validate the effectiveness of ADCMs, we perform unconditional and class-conditional generation experiments CIFAR-10 (16) and ImageNet 64 64 3 respectively. For CIFAR-10, initialize CMs with pretrained DM from [.] For ImageNet 64 x 64, we adopt the pretrained DM from [12] otherwise specified, our experiments are conducted under VE SDE (38) settings. We evaluate the sample quality using FID 8 and measure the generation speed using the number of function evaluations (NFEs).
We compare ADCMs with different generative models, as shown in Table FIDs with * indicate that they have additional training costs compared to other CMs, such a larger model auxiliary model used during training. Experiments show that ADCMs achieve high-quality single- step generation without additional training costs. See Appendix C for multi-step generation results.
4.1 Efficiency ADCMs
We evaluate the training efficiency of ADCMs on both unconditional CIFAR-10 and class-conditional ImageNet 64 x 64. For CIFAR-10, we use a unified model size and measure computational cost by the total number of training images. For ImageNet are taken into consideration. TCM [17 is excluded from the comparison since its two-stage strategy introduces significant training overhead. For a fair comparison, we reproduce some baseline results, as detailed in Appendix A.3
x 64, both model size and training budgets

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wi Adephhie Stemetartion
Imagalist 64:64
402
5.
Adaphale Elsceezzaiion
531
200
200
CIPAR-10
DCN
AXcMbrand
10
20
100
100
17.21
Training
Traiiig steps
(a)
(b)
Figure 3: (a) Training time cost of ADCMs. (b) Training dynamics of different discretization methods. Compared to other CMs approaches, ADCMs have a faster convergence rate and better final performance. (c) Training dynamics for different A. A large À improves stability but hurts final performance, while too-small reduces stability and hinders convergence.
10
ECM
ADCM toues!
10
ADC Jauns)
10-2
10:4
10-4
10
10
10
10-2
10-1
10'
(a) EDM (VE)
(b) Flow Matching
Figure 4: Adaptive Discretization on (a) EDM (VE) and (b) Flow Matching.
Data Efficiency. On unconditional CIFAR-10 10. as shown in Table 2 ADCMs achieve high-quality one-step generation results with training budget of only 12.8M images. Compared with ECM M the most efficient CM to date, ADCMs achieve better generation quality with only 25% of its training budget. Moreover, ADCMs surpass all previous CMs in one-step generation performance with only 76.8M training images. On class- faass-onndiionall ImageNet 64 x 64, as shown in Table 3 ADCMs significantly reduce the training budgets of CMs. Compared the most efficient ECM ADCMs can achieve better 1-step FID with the same model size and training budget. Moreover, ADCMs exhibit notable improvements as both the model size and training budget increase. With a model size, ADCMs achieve 1-step FID of 3.49 with training budget of only 12.8M images. Remarkably, ADCMs are able to surpass iCT-deep (36 with only 3% of its training budget.
Computational Efficiency. We first compare the training time cost of ADCMs with other CMs. as shown in Figure 3a It can be observed that ADCMs introduce only about 4% additional time cost under the same training epochs while improving the final performance. We also explore the convergence speed of ADCMs on unconditional CIFAR-10 10 with different CM approaches, as shown Figure 3b can be observed that ADCMs converge significantly faster than other CM approaches, while also achieving better final performance.
4.2 More Results
Adaptive Discretization Step of ADCMs. We explore the relationship between the adaptive discretization step At of ADCMs and time under different noise schedules. and compare it with existing discretization methods. We modify the discretization strategies of iCT and ECM under Flow Matching setting to be functions of SNR in order to enhance their performance. The results are shown in Figure 4 ADCMs are able to adaptively learn discretization strategies that are similar in trend to empirical ones without manual adjustments. addition, compared to other discretization schemes, ADCMs adopt finer discretization at smaller and coarser discretization at larger I. As result, ADCMs place greater emphasis on time intervals closer the data during training, which aligns with empirical practices in both DMs and CMs [1] 36 7

---

Table 4: Generalization to Flow Matching.
indicates additional training costs.
Table 5: Scalability to ImageNet 512 x 512.
Method
NFE (
FID (L)
Clas-Conditional ImageNNt 512 x 612
1- R-Rectified Flow 20
378
Method
Model Size
Training Budget (Mimgs)
1-Step PID
2- R-Rectified Flow
20
4.85
SCT
1x
204,8
10.13
CCM 21 (w GAN)
1.64
SCT
2x
204,8
5.84
Consisteney- FM (w/o GAN)
5.34
BCM
1x
6.4
25.69
2x
ECM 7
5.82
sCT
23
1
88.52
ADCM (Ous)
1x
6.4
23.12
ADCM (Ours)
2x
6,4
10.53
ADCM (Ours)
1 5.14
À as Trade-off between Stability and Effectiveness. We control the trade-off between the trainability and stability of ADCMs through the Lagrange multiplier À according to Eq. (9). We perform ablation study on by examining the training dynamics of ADCMs CIFAR-10, as shown in Figure 3c We find that when À is small, i.e., more emphasis is placed on figlobal: CMs converge quickly, but the final generation quality is relatively poor. When À is large, i.e.,
unconditional
more emphasis is placed on Liocal. CMs become more unstable, making them difficult to converge and reach the optimal solution. Ablation study on loss function are deferred to Appendix B
ADCMs Adapt to Different Variants of DMs. We conduct experiments on Flow Matching 18 20 an advanced variant DMs. We initialize CMs with pretrained DMs from 20) and compare ADCMs with other Flow Matching-basse ECM and SCT, two state-of-the-ans CMs. All CMs are trained under training budget of 12.8M images. As shown in Table ADCMs achieve superior performance over other C.Ms without manual adjustments, which demonstrates the strong adaptability.
distillation methods. Additionally, we conduct experiments on
Scalability to High-Resolution Images. To further assess the scalability of ADCM, we conduct experiments on ImageNet 512 x 512. We adopt EDM2 [12 as the base latent diffusion model, which employs SD VAE for image encoding and decoding. We compare ADCM with sCT [23 and ECM 77, the two most efficient prior CMs. The detailed results are reported Table It can be observed that ADCM scales effectively to large-scale datasets. As the model size increase, its performance improves substantially. Moreover, ADCM consistently outperforms ECM under the same training cost, further demonstrating its empirical effectiveness and training efficiency.
5 Conclusion
In this paper, we propose ADCMs, a unified framework for adaptive discretization in CMs. By formulating discretization as optimization problem, we introduce local consistency the opti- mization objective and global consistency as constraint, establishing a trade-off using the Lagrange multiplier. Leveraging the Gauss-Newton method, ADCMs enable adaptive discretization, improving both trainability and stability. Experimental results show that ADCMs significantly improve training efficiency and final performance of CMs on different datasets while demonstrating strong adaptability to different variants of DMs.
Acknowledgments and Disclosure of Funding
Z. Ling is partially supported by the National Natural Science Foundation of China (via NSFC-62406119), the Natural Science Foundation of Hubei Province (2024AFB074), and the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence (2033B121201000) Z. Deng partially supported by the National Natural Science Founda- tion of China (via CSFC-92470113 18 and NSFC-62306176 and the Natural Science Foundation of Shanghai (23ZR 428700). R. C. Qiu is partially supported in part by the National Natural Sci- ence Foundation of China (via NSFC-12141107), the Key Research and Development Program of Wuhan (2024050002030100), and the Key Research and Development Program of Guangxi (Guike-AB21190034)
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A Experiments Details
A.1 Precondition
For VE-based ADCMs, follow the parameterization of EDM. Specifically, set:
dana
ddnta
1 Choise (t) = log t.
Cskip(t)
data + 12 Cour Cour(t)
Cin (t)
V
12
Volu
4
For ADCMs the Flow Matching setting, using EDM's precondition causes the model output to deviate from Zn. which contradicts the objective of CMs to estimate zo- We use a pretrained model from rectified flow 20 whose output is:
St 20
Fo
t
which implies (t) and Croise (t) t. To ensure the model's final output matches æu, we accordingly modify the preconditioning to:
Cakip (t) = 1, Cour (f) = -t.
A.2 Hyperparameters
Batch Size and EMA. For unconditional CIFAR-10, we use a batch size of 128 with an EMA decay rate of 0.9999 for training budget of 12.8M images. We use batch size of 1024 with an EMA decay rate of 0.99993 for training budget of 76.8M images. For class-conditions ImageNet 64 x 64, we set the batch size to 128, 256, and 512, corresponding to training budgets of 12.8M, 25.6M, and 51.2M images, respectively. We use Power function EMA for class-conditional ImageNet 64 x 64 following 11)
Time Sampling. For unconditional CIFAR-10, we follow previous works 336 log-normal SNR distribution for time sampling, which can be expressed as:
7
23
and use
log(SN((t)) ~ (P W sud
where SNR(t)= = 2=, P'mann = -.11, Pad = 2.0. Since the time segment T discrete, we also apply
a,' discretization to the sampling results following [36]. For class-conditional ImageNet 64 x 64, we sample uniformly within the time segment T.
Lagrange Multiplier A. For unconditional CIFAR-10, we set À = 0.01. For class-conditional ImageNet 64 64, we find that starting with small À led training instability on ImageNet 64 64. Therefore, we follow previous work [36] 7 and select À = 0.64 for warm-up, gradually decreasing it 0.01. We summarize the hyperparameter settings in Table
A.3 Baseline Reimplementation.
Some of the baseline results in this paper, including those in Figure 3b Table and sCM 23 under VE settings, are obtained from our own reproductions. Under the VE SDE setting, for a fair comparison, initialize all neural networks using the pretrained DM provided by EDM (TT. We also adopt a consistent EMA decay rate of 0.9999 and a dropout probability of 30% (except for CD where dropout is set to as dropout can lead decline in CD's performance). We do not make further modifications to other parameters. For sCM under the VE SDE and Flow Matching setting, we apply the advanced training techniques from 23 except for the network architecture changes, allowing sCM to utilize pretrained DMs. We do not use the adaptive weighting and tangent warmup techniques, as we find that they degrade the performance of sCM. For all baselines under the Flow Matching setting, we replace their original discretization scheme, time sampling, and weighting function-from being functions time / to being functions SNR. is important to note that without manual adjustments, the performance of these baselines degrades significantly (e.g., the 1-step FID of FCM drops from 5.82 15.55). The implementation code is available the supplementary material.
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Table 6: Hyperparameter Settings
Unconditional
Class-conditional ImageNet 64 x 64
CIFAR-10
Base model
EDM 11
EDM2-S [2]
EDM2-M [2]
Model capacity (Mparams)
55.7
280.2
497.8
Model complexity (GFLOPS)
21.3
101.9
180.8
GPU types
RTX3090
RTX3090
A100
GPU memory
24G
24G
40G
Number of GPUs
1
8
4
Dropout probability
30%
40%
50%
Optimizer
RAdam
Adam
Adam
Learning rate schedule Learning rate max
fixed
square root 0.001
square root 0.0009
0.0001
Pseudo-Hute Huber c
0.03
0
0
Time sampling
log-normal SNR -1.1
uniform
uniform
Pmean
Pad
2.0
B Ablation Study
We investigate the impact of adaptive loss function in ADCMs, including the choice of weighting function and distance metric. We perform ablation study unconditional CIFAR-10 under the same training budget 12.8M images.
Table 7: Ablation Study Weighting Function.
Weighting Function
1-Step FID (L)
5.70
4.09
4-1-1
3.84
Adaptive weighting (Ours)
3.16
Weighting Function. The choice of weighting function crucial for training CMs. An inap- propriate weighting function can lead to imbalanced optimization over time, ultimately degrading performance. We investigate the impact of different weighting functions on ADCMs. The detailed results presented in Table The results show that our designed adaptive weighting function effectively enhance the generation capability ADCMs. Notably, even without the loss function improvement, ADCMs still outperform ECM's I-step FID (4.54). This demonstrates that the im- provement of ADCMs mainly comes from our designed adaptive discretization strategy while our proposed adaptive weighting function further enhances the performance of ADCMs.
Distance Metric. We investigate the effect of different distance metrics on the performance of ADCMs. Following common practice in prior works 36 17.. we adopt the Pseudo-Hulle Huber metric due to its robustness to outliers 36 The Pseudo- metric is defined as
d(x, v) VII v12 2
which provides a smooth interpolation between the L2 and squared Ly metrics. Specifically, when e = 0, it reduces the standard La distance, while as e - se, approaches the squared L2 distance. smoothly bridges the gap between the L22 and squared L2 metric. When 0, the Pseudo-Huber metric degenerates to the standard Ta metric. When e - oc, becomes equivalent to the squared Le metric. To address this phenomenon, we conduct experiments with different values of c, and the
15

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Table 8: Ablation Study on Distance Metric.
Distance Metric
1-Step FID (v)
L
3,54
Pseudo- Preudo-Huber (c=0.0003)
3,44
Rseudo-Huber (c=0.003)
3,42
Pseudo-Huber (c=0.03)
3.16
Pseudo- Huber (c=0.3)
4.42
Pseudo- Huber (c=3)
5.23
Squared 1.2
5.33
results are presented in Table 8 It can be observed that Pseudo-Huber metric smoothly interpolates between L2 and squared L2 through the control of the parameter c, thus achieving performance that surpasses both extremes. These results clearly demonstrate the significance of choosing Pseudo-Huber our distance metric.
We also examine the impact of mismatched distance metrics between the original CM loss function and the weighting function. We fix the distance metric applied to the original CM loss function as Pscudo-Huber with c = 0.03, while applying different distance metrics the weighting function. As shown in Table mismatched distance metric leads degraded performance of ADCMs.
Table 9: Impact Mismatched Distance Metric.
Distance Metric on Weighting Function
1-Step FID (L)
Squared L2
4.09
L2
3.36
Pseudo-Huber (c=0.03)
3.16
C Two Step Generation
Compared to other distillation methods for DMs, CMs have the significant advantage of preserving the inherent characteristics of DMs. specifically, the ability improve generation quality through multi-step sampling. We investigate the two-step generation performance of ADCMs on unconditional CIFAR- 10, with results shown Table IO We set the intermediate 0.420. that ADCMs not only maintain optimal single-step step generation performance but also demonstrate strong two-step generation capabilities, second only ECM 7 which is specifically designed for two-step generation.
be observed
Table 10: 2-step generation results on unconditional CIFAR-10. ADCMs achieve the best 1-step FID while also attaining the second-best 2-step FID.
Method
1-Step FID (Q)
2-Step FID ())
iCT
4.18
2.58
sCM (VE)
5.62
2.73
ECM
4.54
2.20
ADCM (Ours)
3.16
2.44
D Limitations and Broader Impacts
In this paper, we introduce ADCMs, an adaptive discretization method for CMs. Our approach effectively improves both the training efficiency and generation quality of CMs, and demonstrates adaptability to different variants of DMs. However, ADCMs focus on Consistency Training (CT), as it generally yields better performance. In the case of Consistency Distillation (CD), estimating
16

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Gglobal significantly increases training costs due to the need for iterative solving of the endpoint of the PF-ODE. We leave this issue for future work. ADCMs enable efficient content generation for creators while significantly reducing the computational cost of obtaining similar models. Additionally, similar other deep generative models, ADCMs could be misused to generate harmful fake content, and the proposed method may further exacerbate the potential risks associated with malicious applications of such models.
License
We list the used datasets, models and their citations, URLs, and licenses in Table 11
Table 11: Licenses and citations for existing assets.
Name
URL
Citation
License
CIFAR-10
https: Setps://www.co toronto. edu/
'kriz/cifar.. htmml
ImageNet
attps://www.n wmwge//wwwimag....mmmmm/mm net org
EDM
https://githut: com/ NVlabs/
Creative Commons Attribution-
edm
CaloommrrcialSharrAAieee 4.0
International License
EDM2
https http::///gttuuu.cooo... edm2
I2
Creative Commons Auribution-
NonCommcreial ShireAAkkk 4.0
International License
Rectified Flow
https: https://github. com/ bebbitb/Becttfiedd22
20
F
Additional Samples
We provide additional samples of ADCMs from unconditional CIFAR-10 and class-conditional ImageNet 64 x 64 in Figures 3
17

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Figure 5: 1-step samples from unconditional CIFAR-10 trained with budget of 76.8M images.
18

---

Figure 6: 1-step samples from unconditional CIFAR- 10 trained with budget of 12.8M images.
19

---

Figure 7: 1-step samples from class-conditional ImageNet 64 x 64 trained with a budget of 12.8M images.
20

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Reviews:
[{"review_id": "a566X3a6GQ", "reviewer": "Reviewer_q5Qh", "summary": "This paper focuses on the problem of choosing discretization schedules in Consistency Models (CMs). Instead of relying on manually designed time-step schedules, the authors formulate an adaptive discretization strategy by balancing two objectives: (1) local consistency (ensuring small errors between adjacent diffusion states for trainability) and (2) global consistency (limiting accumulated error over the trajectory for stability). Using a first order taylor expansion, the authors proposed a way to determine $\\Delta t$ adatively (which is the proposed ADCM method).\n\nEmpirically, ADCM yields significantly improved sample efficiency – achieving comparable or better generation quality with far fewer training iterations – on CIFAR-10 and ImageNet $64\\times64$. It also demonstrates the ability to generalize to different diffusion model variants (e.g. flow matching) without manual retuning.", "questions": "See Cons in the \"Strengths And Weaknesses\".", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Pros\n\n1. The paper proposes an unified framework for discretization in CMs by casting the selection of time-step $\\Delta t$ as an optimization problem balancing local consistency and global consistency (via a linear combination).\n\n2. The proposed method is both intuitive and effective. The linear combination of the global and local consistency is pretty natural and easy to follow (though I'm not sure what's the significance of framing it as an \"constrained optimization\" problem). The empirical results suggest that it's effective with proper choice of the combination coefficient.\n\n3. The presentation is clear. Section 1 and 2 give pretty clear introduction to the background and the importance of the problem studied in this paper.\n\n\n\nCons\n\n1. I'm mainly concerned about the technical significance of this paper. On the theoretical side, I don't think framing equation (5) and (6) as a constrained optimization and later converting that to a linear combination of the two is of great significance. I think just presenting equation (8) directly without the wrapper of optimization is equally well motivated. It is OK to present something simple and intuitive, as long as it's proven to be empirical effective -- which I think this paper also lacks (see details below).\n\n2. Insufficient empirical evaluation. The presented experiments results are positive, but have limited scope. To name a few missing aspects that I think people will benefit from: (a) all the results are on relatively low-resolution image datasets, and it remains unclear how this method will perform on higher-resolution generation problems. (b) how sensitive is the optimal hyper-parameter to the specific domain & DM variants? Is it mostly a constant across all or do people need to tune it for every task?\n\n3. The empirical results can be presented better. I think many empirical studies settings, ablation studies, etc. are deferred to appendix. I think there are many interesting questions that worth more discussion in the main paper -- e.g., the significance of huber loss, why huber with large C has worse performance than L_2 (in Table 7).", "was_revised": true, "final_justification": "I raise my score from 3 to 4 for the additional empirical evaluation. The theoretical contribution is not clear to me, though I don't hold a strong opinion.", "rebuttal": "We thank the reviewer for your careful consideration. We greatly appreciate the positive comments and address your concerns below.  \n\n**Technical Significance:**\nOur theoretical contributions lie in\n(1) Novel theoretical formulation: To the best of our knowledge, this is the first work to introduce **global consistency** into Consistency Training, in order to control accumulated errors across discretization intervals and improve the stability of training.\n\nWe respectfully disagree with the comment that \"just presenting Eq.(8) directly without the wrapper of optimization\" would suffice.  The constrained optimization formulation is not merely a wrapper; it plays a central role in both the theoretical motivation and the practical design of our method. In our formulation (Eq.(7)), local consistency is cast as the primary training objective, while global consistency is introduced as a constraint. This separation is intentional and meaningful: local consistency drives accurate stepwise transitions (the core goal of CM training), whereas global consistency captures desirable trajectory-level behavior. Framing this as a constrained problem highlights their asymmetric roles and directly motivates the design of the relaxation in Eq.(8), where a small trade-off coefficient ensures that local consistency remains dominant. Without this optimization-based grounding, Eq.(8) would lack a clear theoretical justification, and the choice of trade-off weights would appear arbitrary.\n\nMoreover, we kindly note that, as discussed in Remark 3.1, our formulation is not limited to deriving Eq.(8); rather, it provides a **unified theoretical framework** that encompasses previous heuristic discretization strategies as special cases (via different constraint formulations or trade-off weights).  This unification not only clarifies existing methods but also enables principled extensions going forward.\n\n(2) Closed-form and efficient solution: Another key aspect of our theoretical contribution lies in the solution of Eq.(8), which is nontrivial due to the inherent complexity and nonlinearity of the NN. We address this challenge by deriving a **closed-form** solution through a principled approach based on Gauss-Newton and first-order approximations, making the approach both **efficient** and **theoretically grounded**. As noted by other reviewers, this illustrates a \"neat\" example of the interplay between theoretical formulation and practical machine learning.\n\n \n**Higher Resolution Generation Results:** We kindly note that most seminal baseline works [^1][^2][^3] conduct experiments exclusively on CIFAR-10 and ImageNet at  $64 \\times 64$ resolution, which is consistent with our experimental setup. Therefore, our choice of datasets and resolutions is not a limitation but aligns with prior art. Even on these lower-resolution datasets, previous methods exhibit training instability and rely on manually designed discretization schemes, incurring substantial training overhead. In contrast, our method effectively addresses these challenges, as demonstrated in our experiments, which provides strong support for our claims. \n\nTo further address your concern, we conduct additional experiments on ImageNet at $512 \\times 512$ resolution.  Training at this scale is time-consuming, which limits  the extent of experiments we can perform during the rebuttal period. Moreover, we note that only sCT[^4] reports results at this resolution. Comprehensive comparisons with other baselines are difficult, as training and re-implementing these methods demand significant time. Thus, we choose to compare our method with sCT and ECM[^1], the most efficient prior CM.\n\nWe choose EDM2 [^5] as the base latent diffusion model which employs SD-VAE for image encoding and decoding. The detailed results are presented in the following table. Our results are promising. We outperform ECM,  with a similar training budget. While we do not yet match sCT’s performance, their method involves extremely higher training costs  and specialized architecture. The results demonstrate that ADCM is capable of scaling to high-resolution images while maintaining high efficiency.\n\n| Method | Training Budgets (Mimgs) | 1-step FID |\n|:---:|:---:|:---:|\n|sCT-S                  |204.8|  10.13 |\n|ECT-S                  |6.4|  25.69 |\n|ADCM-S (Ours)          | 6.4 | 23.12 |\n\n\n**Optimal Hyper-parameter $\\lambda$:** Following your suggestion, we examine the sensitivity of our proposed hyperparameter $\\lambda$. Specifically, we investigate the difference between the optimal $\\lambda$ under the Flow Matching setting and that under the VE setting. The detailed results are presented in the following table. Notably, the choice of $\\lambda$ shows consistent behavior across settings: in both the VE and Flow Matching cases, setting $\\lambda = 0.01$ achieves the optimal 1-Step FID. This consistent trend highlights the robustness of ADCM and substantially reduces the need for hyperparameter tuning.\n\n| $\\lambda$ | 1-Step FID(VE) | 1-step FID(Flow Matching) |\n|:---:|:---:|:---:|\n|0.05          |6.20 |  4.24 |\n|0.02          | $\\underline{5.20}$ | $\\underline{3.59}$ |\n|0.01          | **5.14** | **3.16** |\n|0.005          | 6.14 | 3.75 |\n\n\n\n**Huber Loss:** We adopt the pseudo-Huber loss, similar to standard practice in prior works [^1][^2][^3], as the pseudo-Huber metric is more robust to outliers [^2].  The Pseudo-Huber loss, expressed as $ d(x, y) = \\sqrt { \\| x-y \\|_ 2^2 + c^ 2} - c $, smoothly bridges the gap between the L2 loss and the squared L2 loss. When $c=0$, the Pseudo-Huber loss degenerates to the standard L2 loss. When $c \\to \\infty$, it becomes equivalent to the squared L2 loss as $\\lim_{c \\to \\infty} \\sqrt{\\|x-y\\|_2^2 + c^2} - c = \\frac{\\|x-y\\|_2^2}{2c}$. \n\nTo address this phenomenon, we conduct more ablation studies on the effect of c based on Table 7, and the results are presented in the table below. It can be observed that Pseudo-Huber smoothly interpolates between L2 and squared L2 through the control of the parameter c, thus achieving performance that surpasses both extremes. When c becomes too large, the Pseudo-Huber loss asymptotically approaches the squared L2 loss, leading to a decline in performance. These results clearly demonstrate the significance of choosing Pseudo-Huber as our distance metric.\n\n| Loss Function | 1-Step FID | \n|:---:|:---:|\n|L2          |3.54 |\n|Pseudo-Huber(c=0.0003)         | 3.44|\n|Pseudo-Huber(c=0.003)          | $\\underline{3.42}$ |\n|Pseudo-Huber(c=0.03)       |**3.16** |\n|Pseudo-Huber(c=0.3)        |4.42 |\n|Pseudo-Huber(c=3)        |5.23 |\n|Squared L2        |5.33 |\n\n\n**Some Empirical Results deferred to appendix:** We will bring key ablation results and empirical settings from the appendix back into the main text of the revised manuscript. Thank you for your valuable suggestion.\n\n[^1]:Geng Z, Pokle A, Luo W, et al. Consistency models made easy. ICLR 2025.\n[^2]:Song Y, et al. Improved Techniques for Training Consistency Models. ICLR 2024.\n[^3]:Lee S, Xu Y, Geffner T, et al. Truncated consistency models. ICLR 2025.\n[^4]:Lu C et al. Simplifying, stabilizing and scaling continuous-time consistency models. ICLR 2025.\n[^5]:Karras T, Aittala M, Lehtinen J, et al. Analyzing and improving the training dynamics of diffusion models. CVPR 2024."}, {"review_id": "7N6trCKTkV", "reviewer": "Reviewer_gb53", "summary": "Existing works on consistency models often rely on manually designed or ad-hoc discretization schemes, which can be inefficient and lead to poor training outcomes. This paper introduces ADCMs, a unified framework that addresses this by automatically and adaptively determining the discretization strategy. The method formulates discretization as a constrained optimization problem, using local consistency—minimizing the error between adjacent trajectory points—as the primary objective to ensure the model is trainable. Simultaneously, it imposes a constraint on global consistency—controlling the accumulated error relative to the ground-truth data—to maintain training stability and prevent divergence. To effectively balance this trade-off between trainability and stability, the authors employ a Lagrange multiplier and leverage the Gauss-Newton method to derive an efficient, analytical solution for the optimal discretization step. Several experiments are conducted to demonstrate that the proposed method, ADCMs, improve the efficiency of CMs.", "questions": "- In eqn 2, why we optimize over f_\\theta(x_t) rather than f_\\theta- (x_t) \n- In eqn 6, why the bound \\delta is the same over all t, i.e., why don't you use \\delta_t ?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is well-written and easy to follow. All math notations are defined and explained before being used.\n- The proposed optimization objective has a clear motivation and is easy to implement.\n- Several experiments and ablation studies are conducted to demonstrate the performance of the proposed method.\n\n\nWeaknesses:\n- line 173: please explain further on why you should optimize over f_\\theta- (x_t-\\delta t) rather than  f_\\theta- (x_t) \n- The function stopgrad() is not defined before usage\n- The computational complexity of the algorithm 1 is not analyzed. The paper also lacks the analysis of the convergence rate.\n- The approximation error of the methods in Sec 3.2 is not discussed\n- The experiments only adopt CIFAR-10 and ImageNet datasets, which are very limited. More advanced datasets should be adopted.\n- The ablation studies should cover more values of \\lambda (\\lambda = \\infty)", "was_revised": true, "final_justification": "I raise my score from 3 to 4 as the author addressed most of my concerns in the rebuttal.", "rebuttal": "We thank the reviewer for your careful consideration. We greatly appreciate the positive comments and address your concerns below.\n\n**Optimization over $f_{\\theta^-}(x_t-\\Delta t)$:** We kindly note that our objective is to enforce global consistency by optimizing $\\Delta t$. Global consistency is reflected in **the cumulative error between the practical training target $f_{\\theta^-}(x_t - \\Delta t)$ in Eq.(2) and the actual target $x_0$ (i.e., the data)**. This implies that, when $\\Delta t = 0$, the practical target becomes $f_{\\theta^-}(x_t)$, which often leads to a larger cumulative error. In this case, global consistency is at its weakest, resulting in unstable training. Therefore, we choose to optimize $f_{\\theta^-}(x_t-\\Delta t)$ instead of $f_{\\theta^-}(x_t)$.\n\n**Stopgrad Definition:** $\\operatorname{stopgrad}(\\theta)$ function is is a standard component in  previous Consistency Model works[^1][^2][^3][^4]. To address your concern, we will clarify its definition and explain why $\\theta^-$ can't be optimized after applying $\\operatorname{stopgrad}$. It treats the given parameter $\\theta$ as a constant during gradient computation. This means that while the forward pass still uses the actual value, the backward pass cuts off the gradient flow.   As a result, we cannot optimize over $f_{\\theta^-}(x_t)$ since it does not participate in gradient computation.\n\n\n**Computational Complexity of Algorithm 1:** The computational overhead in Algorithm 1 compared to other methods comes from the computation of $\\mathbb{T}$. For each value of $\\Delta t$, we use a mini-batch to estimate the expectation and compute the Jacobian. The computational overhead introduced by the Jacobian calculations is negligible. As noted in line 224, the Jacobian can be efficiently computed using **PyTorch’s built-in Jacobian-vector product (JVP)**.\n\nMoreover, since we adopt the Gauss-Newton method, which incorporates second-order approximation information, the optimization typically converges faster than first-order methods[^5], further reducing the overall computational cost. Additionally, due to the infrequent updates of $\\mathbb{T}$, the overhead of computing $\\mathbb{T}$ is minimal relative to the total training time.\n\nAs shown in Figure 3a, the additional computational cost is typically less than 4% under the same number of training epochs, while significantly improving performance. As shown in Figure 3b, ADCM significantly accelerates the convergence of Consistency Models (CM), allowing it to achieve better generative performance with fewer training steps compared to other CMs.\n\n**Approximation Error**: To solve the optimization problem in Eq.(8), we adopt the Gauss-Newton method, as it is well-suited for problems involving complex neural networks. Specifically, we approximate $f_{\\theta^-}(x_t - \\Delta t)$ using the first-order Taylor expansion of the neural network. It is important to note that this approximation error is acceptable, since, as stated in line 265, we do not require precise step sizes but only need to capture the trend of step size changes over time $t$. To further address your concern, we calculated the average approximation error incurred during the computation of $\\mathbb{T}$ with different $\\lambda$. The error is measured by $\\frac{\\|f_{\\theta^-}(x_{t-\\Delta t}) - (f_{\\theta^-}(x_{t}) - v \\Delta t) \\|}{\\| f_{\\theta^-}(x_{t-\\Delta t}) \\|}$. The detailed results are presented in the following table. It can be observed that the approximation error is very small even with a large $\\lambda$, demonstrating that the Gauss-Newton method provides a sufficiently accurate analytical solution in our setting.\n\n| $\\lambda$ | Approximation Error |\n|:---:|:---:|\n|0.64           |0.0357 |\n|0.16           |0.0050 |\n|0.04           |0.0004 |\n|0.01          |0.00002|\n\n\n**Generation Results on CIFAR-10 and ImageNet:** We kindly note that all baseline works conduct experiments exclusively on CIFAR-10 and ImageNet, which is consistent with our experimental setup. Therefore, our choice of datasets and resolutions aligns with established prior work and should not be considered a limitation.\n\n**Discussion on $\\lambda \\to \\infty$:** As noted in Remark 3.1, when $\\lambda \\to \\infty$, this is equivalent to choosing the maximum optimization step $\\Delta t = t - \\epsilon$. In this case, the optimization objective of the Consistency Model in Eq.(2) degenerates into that of the Diffusion Model in Eq.(1), thereby losing the ability to perform single-step generation.\n\n\n**Discussion on bound $\\delta$:** In Eq.(6), we use a constant $\\delta$ to bound the global consistency, rather than designing it as a time-dependent function $\\delta(t)$. This is due to the complexity of neural networks, which makes it difficult to determine an appropriate $\\delta(t)$ for every $t$. Therefore, we adopt a uniform constraint using a constant $\\delta$ that satisfies $\\delta \\geq \\max_t \\delta(t)$. It is important to note that the choice of $\\delta$ does not affect our subsequent conclusions, as we will relax the original optimization problem from Eq.(7) to Eq.(8).\n\n\n\n[^1]:Song Y, et al. Consistency Models. ICML 2023.\n[^2]:Geng Z, Pokle A, Luo W, et al. Consistency models made easy. ICLR 2025.\n[^3]:Lu C et al. Simplifying, stabilizing and scaling continuous-time consistency models. ICLR 2025.\n[^4]:Frans K, Hafner D, Levine S, et al. One step diffusion via shortcut models. ICLR 2025.\n[^5]:Gratton S, Lawless A S, Nichols N K. Approximate Gauss–Newton methods for nonlinear least squares problems.SIAM Journal on Optimization."}, {"review_id": "N900AGHOVq", "reviewer": "Reviewer_nhUt", "summary": "This paper presents a technique for adaptively choosing the discretization steps in consistency models to improve their training stability and efficiency. The idea is to formulate the problem using a combination of local and global consistency losses. Using Gauss-Newton method, the authors derive an analytical solution to the problem. This results in a formulation where \\delta_t can be derived using JVP. Using this, the authors show that superior one step generation results can be obtained with much better training efficiency on Imagenet64 and CIFAR-10 datasets.", "questions": "1. For large-scale runs, how would you recommend picking \\lambda?\n2. From table 1, it looks like ICT or TCM still outperform ADCM. With more training budget, can you match or outperform these methods?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. The paper addresses an important problem of choosing discretization steps in consistency models.\n2. The proposed method is novel. The formulation is intuitive, the objective makes sense. Overall, the results are easy to follow.\n3. Experimental results are very strong. I am impressed with the training efficiency section/\n\nWeakness:\n1. The paper would have been very strong if there were large-scale results on high resolution with LDMs. I would not penalize the authors on this aspect as some labs can have low resources.\n2. Among other consistency models, results are still not SOTA. Other methods can outperform ADCM.", "was_revised": true, "final_justification": "The paper addressed my concerns about large scale experiments. It is promising to see that results on Imagenet with -M model gets good FID. The paper has good contribution, hence I stay with my initial rating of Borderline accept.", "rebuttal": "Thank you for your positive evaluation and kind words regarding our method’s novelty, intuitive formulation, and strong experimental results. We also appreciate your recognition of the training efficiency improvements and address your concerns below.\n\n**Higher Resolution Generation Results:** Following your suggestion, we conduct experiments on ImageNet at $512 \\times 512$ resolution. We choose EDM2[^1] as the base latent diffusion model, which employs SD-VAE for image encoding and decoding. The detailed results are presented in the following table. \nTraining at this scale is time-consuming, which limits the extent of experiments we can perform during the rebuttal period. Moreover, we note that only sCT[^2] reports results at this resolution. Comprehensive comparisons with other baselines are difficult, as training and re-implementing these methods demand significant time. Thus, we choose to compare our method with sCT and ECM, the most efficient prior CM.\nOur results are promising. We outperform ECM[^3],  with a similar training budget. While we do not yet match sCT’s performance, their method involves extremely higher training costs  and specialized architecture. The results demonstrate that ADCM is capable of scaling to high-resolution images while maintaining high efficiency.\n\n| Method | Training Budgets (Mimgs) | 1-step FID |\n|:---:|:---:|:---:|\n|sCT-S                  |204.8|  10.13 |\n|ECM-S             |  6.4 | 25.69 |\n|ADCM-S (Ours)          |  6.4 | 23.12 |\n\n**$\\lambda$ for Large Scale Runs:** Due to the inherent instability of models in large-scale experiments, we recommend using a larger $\\lambda$ to ensure training stability or just employ a larger $\\lambda$ in the early training stage as a warm-up. This recommendation aligns with the design of our theoretical framework: a larger  $\\lambda$  places greater emphasis on global consistency, thereby enhancing stability.\n\n[^1]:Karras T, Aittala M, Lehtinen J, et al. Analyzing and improving the training dynamics of diffusion models. CVPR 2024.\n[^2]:Lu C et al. Simplifying, stabilizing and scaling continuous-time consistency models. ICLR 2025.\n[^3]:Geng Z, Pokle A, Luo W, et al. Consistency models made easy. ICLR 2025."}, {"review_id": "xYY3GJpjqD", "reviewer": "Reviewer_TPUd", "summary": "The paper introduces a form of adaptive time discretization of consistency training designed to replace the fixed discretization schedules that are commonly adopted in the literature. The choice of the time step Dt is obtained as the result of an optimization of a linear mixture the standard consistency loss and a denoising autoencoder loss. This choice is interpreted in terms of a trade-off between local error (consistency) and global error. The proposed method seems to work well in practice, especially on CIFAR10 where it achieves SOTA performance. The method oslo seems to have an advantage for smaller dataset sizes, although it is unclear if this advantage would scale in the large data regime.", "questions": "Can you explain to me why you are interpreting Eq.6 as the global accumulated error of the consistency model? (see above). \n\nThe correct expression for the true global error should be:\n\nglobal_error = E_{x_0}||  f(g_t(x_0)) - x_0 ||^2     \n\nwhere g_t(x_0) is the ODE flow. This expression is zero when f = g^{-1}, which is the optimum of a consistency model. On the other hand, the expression \n\ndenoising_error = E_{x_0, x_t}||  x_t - x_0 ||^2 \n\nis only exactly equal to zero when p(x_0 | x_t) is a delta distribution.", "limitations": "There is no \"Limitation' section in the main text and the section in the Supps. is very limited. I would have appreciated a more extensive discussion, especially since it is not clear if the proposed approach would scale well to the large data regime.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "I think that this is a very important problem area that might drive consistency training to generative SOTA. The standard discretization approaches are largely handcrafted and often ignore the rich mathematical structure that is implicit in these models. I do think that  the algorithm proposed by the authors is an important step in the right direction. I find the results to be very exciting since it is notoriously hard to improve on the very optimized iCT baseline.\nI really appreciate the use of principled closed-form and Taylor methods to obtain the ‘optimal’ schedule, which is a neat example of interplay between theory and practical machine learning. \n\nMy main issue with this work is on the interpretation of the main constraint is Eq.6. As far as I understood it, the authors characterize the expression as the global error due to the accumulation of training errors along the ODE flow trajectory. However, this interpretation is just wrong since, even when the network is perfectly trained, the denoising error given in the constraint will be generally non-zero.\nThe reason is that, if the noise is sampled independently under the forward process, the state x_t will generally not lie on the same ODE trajectory as x_0. In fact, the optimum of the denoising loss can be expressed in terms of the score function instead of the consistency map.\nSo said, the denoising autoencoder loss is a central quantity in both diffusion and consistency, so it can well be very meaningful to have it as a constraint.", "was_revised": true, "final_justification": "The methodological approach is solid and the results are promising. Some parts of the theoretical explanation need to be revised based on our discussion.", "rebuttal": "Thank you very much for your encouraging feedback, your appreciation of our theoretical and practical contributions, and your deep understanding of this important problem. Below, we provide point-by-point responses to your concerns and questions.\n\n**Global Error:** We first emphasize that ADCM focuses on the Consistency Training (CT) paradigm[^1] due to its superior empirical performance[^2][^3][^4][^5], as noted in line 38. Under CT paradigm, ADCM has access to the data point $x_0$, which allows it to obtain **an unbiased score estimation $s(x, t) = \\frac{x - \\alpha_t x_0}{\\beta_t^2}$**, as described in the original CM paper[^1]. Under this assumption，$x_t = \\alpha_t x_0 + \\beta_t z$ and $x_0$ lie on the same Probability Flow ODE (PF-ODE) trajectory for $\\forall t \\geq0$. For simplicity, we take the VE setting as an example. At this point, $x_t$ is given by $x_t = x_0 + tz$, and PF-ODE is given by\n\n$\\frac{dx(r)}{dr} = -r \\cdot s(x(r),r) \\quad \\to \\quad \\frac{dx(r)}{dr} = \\frac{x_0 - x(r)}{r}$.\n\nWe will prove $x_t = x_0 + tz$ and $x_0$ lie on the same PF-ODE trajectory from two perspectives.\n\n1. We first consider the standard PF-ODE sampling process, i.e., taking $x(t) = x_t = x_0 + t z$ as the initial condition and solving the PF-ODE starting from time $t$. \n\nIn this case, the solution to the PF-ODE is given by $x(r) = x_0 + r z$. Substituting $r = 0$ yields the original data point $x_0$, confirming that $x_t$ and $x_0$ lie on the same PF-ODE trajectory.\n\n2. To further address your concern, we consider the global error proposed by the reviewer. Under the CT paradigm, the ODE flow $g_t(x_0)$ mentioned by the reviewer corresponds to the solution $x(t)$ of the PF-ODE with initial condition $x(0) = x_0$, solved forward from $r = 0$ to time $t$. \n   \nIn this case, the PF-ODE can only be solved under the assumption that $\\lim_{r \\to 0} \\frac{x_0 - x(r)}{r} = z$, where $z$ is an arbitrary vector. Under this condition, the solution to the PF-ODE is $x(r) = x_0 + r z$. By taking $r = t$, we obtain $g_t(x_0) = x(t) = x_0 + t z$. Therefore, under the CT assumption, our definition of the global error is equivalent to that proposed by the reviewer.\n\nThe denoising error proposed by the reviewer does not involve any NN as it directly measures the distance between the noised data $x_t$ and the original data $x_0$. As such, its meaning is not entirely clear to us. We would greatly appreciate it if the reviewer could provide further clarification on the denoising error. We would be pleased to address any questions or concerns the reviewer may have.\n\n**Large Scale Results:** To address your concern, we conduct experiments on ImageNet at $512 \\times 512$ resolution. We choose EDM2[^6] as the base latent diffusion model which employ SD-VAE for image encoding and decoding. The detailed results are presented in the following table. We choose to compare our method with ECM[^2] and sCT[^3], the most efficient prior CM, and our results are promising. We outperform ECM,  with a similar training budget. While we do not yet match sCT’s performance, their method involves extremely higher training costs  and specialized architecture. The results demonstrate that ADCM is capable of scaling to high-resolution images while maintaining high efficiency.\n\n| Method | Training Budgets (Mimgs) | 1-step FID |\n|:---:|:---:|:---:|\n|sCT-S                  |204.8|  10.13 |\n|ECM-S             |  6.4 | 25.69 |\n|ADCM-S (Ours)          |  6.4 | 23.12 |\n\n[^1]:Song Y, et al. Consistency Models. ICML 2023.\n[^2]:Geng Z, Pokle A, Luo W, et al. Consistency models made easy. ICLR 2025.\n[^3]:Song Y, et al. Improved Techniques for Training Consistency Models. ICLR 2024.\n[^4]:Lu C et al. Simplifying, stabilizing and scaling continuous-time consistency models. ICLR 2025.\n[^5]:Lee S, Xu Y, Geffner T, et al. Truncated consistency models. ICLR 2025.\n[^6]:Karras T, Aittala M, Lehtinen J, et al. Analyzing and improving the training dynamics of diffusion models. CVPR 2024."}]

Author rebuttal:

Meta-review: This paper introduces a new method for adaptively learning the discretization schedule for Consistency Models, moving away from the fixed, handcrafted schedules used in prior work. The approach formulates the selection of time steps as an optimization problem that balances local consistency (for trainability) and a global consistency constraint (for stability), deriving an efficient analytical solution. The paper's main strength, as identified by all reviewers, is its strong empirical performance. The method shows improvements in training efficiency and sample quality on benchmark datasets like CIFAR-10 and ImageNet, and demonstrates robustness by adapting to different diffusion model variants without manual retuning. This is a valuable practical contribution to a problem of interest in generative modeling.

The primary weakness of the submission, identified in a critical discussion with Reviewer TPUd, was a flaw in the original theoretical interpretation of the "global consistency" constraint. The reviewer correctly pointed out that this term is a denoising autoencoder loss, not a measure of accumulated error along a single ODE trajectory as initially claimed. The discussion period was highly constructive in addressing this and other points. In response to reviewer feedback, the authors conducted new experiments on higher-resolution images, strengthening the empirical claims of the paper. Crucially, after discussion, they acknowledged the theoretical misinterpretation and agreed to revise the manuscript to reflect a more accurate framing of their method's motivation.

The decision to accept is based on the paper's significant and well-supported practical contributions. While the theoretical narrative required a substantial correction, the final methodology is sound and its effectiveness is convincingly demonstrated. The authors' transparent engagement during the rebuttal, particularly their willingness to correct the theoretical framing and provide additional experiments, was a key factor. Although the reviewer scores are borderline, the paper presents a useful and efficient method that will be of interest to the community. The authors are strongly encouraged to incorporate all reviewer feedback into the final version, with special attention to carefully revising the theoretical motivation as discussed with Reviewer TPUd.
Decision: Accept (poster)