Title: Adaptive Discretization for Consistency Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion models

Consistency models

One-step generation
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

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 https://github.com/ rainstonee/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, 19] and 3D contents [40, 27, 22]. However, DMs require numerous iterations to achieve high-quality generation, significantly slowing sampling speed and making it resource-intensive. Recently, many fast-sampling methods for DMs have been proposed, including training-free methods [35, 14, 24, 48] and distillationbased approaches [25, 31, 46, 6, 26, 42, 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 to 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, 21] have shown that discretization is 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 a specific time interval, leading to a loss of consistency [17]. To mitigate these challenges and ensure a balanced training process, most existing works adopt empirical discretization strategies, which require manual adjustments based on different noise schedules and datasets [7].

∗

0

100

200

300

400

Training Images (Mimgs)

2.5

3

4

6

10

Fréchet Inception Distance (FID)

ADCM (Ours)

ADCM (Ours)

ECM

ECM

sCD

sCT

CT

CD

iCT

(a) CIFAR-10

100

800

1700

Training Images (Mimgs)

2

4

8

13

Fréchet Inception Distance (FID)

ADCM (Ours)

ADCM (Ours)

ECM

sCD

sCT

CT*

CD*

iCT-deep

20

40

60

3.0

3.5

4.0

(b) ImageNet

64

×

64

Our fundamental goal is to adaptively determine the discretization strategy for CMs' training 2 , considering 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 denoising error can lead to instability in CMs' training, thereby degrading the efficiency of CMs. Hence, we introduce global consistency as a 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 a Lagrange multiplier to express the trade-off between them. To achieve adaptive discretization, we propose using the Gauss-Newton method to obtain an analytical solution to the optimization problem. We refer to our method as Adaptive Discretization for Consistency Models (ADCMs). Our analysis reveals that ADCMs adaptively adjust discretization steps by jointly considering local and global consistency, thus achieving a 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 1a, ADCMs exhibit outstanding training efficiency. We achieve a 1-step FID of 3 . 16 with only a training budget of 12 . 8 Mimages. In contrast, ECM [7], the most efficient CM from previous work, requires 51 . 2 Mimages to reach a FID of 3 . 60 . Moreover, we attain a 1-step FID of 2 . 80 using only 76 . 8 Mimages, outperforming iCT [36], which requires 409 . 6 Mimages to achieve comparable performance. On class-conditional ImageNet 64 × 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 . 8 Mimages. When the training budget increases to 51 . 2 Mimages, ADCMs achieve a 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 consistency and global consistency, we investigate the impact of discretization steps on the trainability and stability of CMs. Guided by these two principles, we formulate a 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, we employ the Gauss-Newton method to derive an analytical solution to the optimization problem, enabling adaptive discretization steps that effectively balance

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, 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. Consistency Models were first proposed by [37], achieving the distillation of Diffusion Models by mapping any point on the PF-ODE trajectory to 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 near the data end. It divided CMs into two categories: Consistency Distillation (CD) and Consistency Training (CT), corresponding to sampling trajectory points using a 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 [17] discovered that CMs struggle to map the entire trajectory using a single model. Therefore, it 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.

Discretization for CMs. The training of CMs fundamentally relies on the selection of adjacent trajectory points, a 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 [11]. ECM [7] introduced a decoupled approach, employing two functions: one to determine the overall magnitude of discretization steps and another to regulate their distribution over time. Both iCT and ECM adopt exponentially decreasing time steps to enhance the stability of CMs' training. Alternatively, CCM [21] introduced an adaptive discretization scheme by iteratively solving for the discretization step based on a Peak Signal-to-Noise Ratio (PSNR) threshold, ensuring a more balanced training across different times. sCM [23] 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 a dataset with an underlying distribution p data , DMs generate samples by learning to reverse a noising process that progressively adds random Gaussian noise to the data, eventually transforming it into pure noise. Specifically, for a data sample x 0 ∼ p data and a noise sample z ∼ N (0 , I ) , the diffusion process is defined as:

where t ∈ [ ϵ, T ] , and ϵ 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) to predict the unknown x 0 in the PF-ODE. The optimization objective of DMs is given by:

where f θ ( x t ) = c skip ( t ) x t + c out ( t ) F θ ( c in ( t ) x t , c noise ( t ) ) , w ( t ) is a weighting function and F θ is an NN with parameters θ . We write f ( x t , t ) as f ( x t ) for simplicity. Most DMs, including DDPM [9], EDM [11], and Flow Matching [18], have training objectives that can be equivalently expressed as Eq. (1) through the design of precondition c skip ( t ) and c out ( t ) .

2.2 Consistency Models

CMs [37, 23] aim to map any point x t on the PF-ODE trajectory to the corresponding data x 0 , i.e., f θ ( x t , t ) = x 0 . To achieve this, (1) at t = 0 , CMs require that f θ satisfy the boundary condition f θ ( x ϵ , ϵ ) ≡ x 0 , which implies that c skip ( ϵ ) = 1 and c out ( ϵ ) = 0 ; (2) for t > 0 , CMs are trained to produce consistent outputs for any two adjacent points on the PF-ODE trajectory. Specifically, the optimization objective of CMs is given by:

where θ -stands for stopgrad( θ ) and ∆ t is the time interval that defines the adjacent time step corresponding to a given time t , which in turn determines the training target on the PF-ODE trajectory. When retrieving the adjacent point x t -∆ t , we adopt Consistency Training (CT) paradigm [37] which enables unbiased estimation of the score function, expressed as x t -α t x 0 β 2 t . This approach eliminates the numerical errors introduced by solvers required for Consistency Distillation (CD).

The choice of ∆ t , referred to as the discretization of CMs, plays a crucial role in their training [37, 21, 23]. Previous works have proposed various discretization strategies, which can be categorized into two types, as outlined below.

Discrete-CMs. When ∆ t > 0 is not infinitesimally small, CMs fall to discrete-CMs. Discrete-CMs require careful planning of the discretization schedule. iCT [36] divides the time interval [ ϵ, T ] into multiple segments using the sampling time steps in DMs [11], i.e., T = { t 0 , . . . , t N } where t 0 = ϵ and t N = T , and samples adjacent time points within T as t and t -∆ t . 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 t in continuous time and then maps the corresponding time step through a manually designed function. CCM [21] points out that the discretization step size affects the training difficulty of CMs at different times. CCM proposes setting a PSNR threshold for CMs and solving the PF-ODE iteratively until the selected time steps satisfy this threshold.

Continuous-CMs. Taking the limit ∆ t → 0 , CMs fall to continuous-CMs, which can be considered equivalent to 'infinite' discretization. [37] proves that continuous-CMs can be optimized with:

which is a continuous version of Eq. (2). Continuous-CMs effectively avoid the discretization schedule of discrete-CMs. However, continuous-CMs often face significant instability challenges. sCM [23] 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 a more general setting.

Overall, the choice of discretization step ∆ t is still challenging. If ∆ t is too large, CMs struggle to learn meaningful information, while if it is too small, instability issues arise [23]. Additionally, how ∆ t varies w.r.t. t determines the training emphasis at different times [36, 7], and suboptimal strategies can lead the model to focus on specific time intervals, negatively impacting overall performance. Although various discretization strategies have been proposed, they often fail to identify the optimal ∆ t 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-CMs 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.

3 Methodology

3.1 ADCMs: Adaptive Discretization for Consistency Models

In Section 2.2, 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, i.e., the discretization step ∆ t for a given time t . When we fix the NN's parameters θ -= stopgrad( θ ) and time t , we aim to find an optimal ∆ t that contributes the most to the following training objective of CMs:

We define Eq. (4) as local consistency as it reflects the properties of CMs in a local interval. First, we need that the objective in Eq. (4) is trainable. To achieve this, we need to choose an appropriate ∆ t such that the objective is as small as possible, thereby satisfying local consistency, namely:

It can be observed that when ∆ t = 0 , the local consistency in Eq. (5) is minimized. This implies that we need to choose ∆ t as small as possible. However, previous works [37, 7, 23] have shown that when ∆ t 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.e., f θ -( x t -∆ t ) , fails to precisely denoise x t -∆ t to the ground-truth x 0 , leading to the global denoising error quantified as:

Remark 3.1 . This denoising error is also an upper bound on the squared Wasserstein-2 distance between p data and the data distribution generated by f θ -at time t -∆ t . Moreover, this error can be regarded as a lower bound on the accumulated error from previous time steps, namely:

where ∆ t i is the discretization step corresponding to time t i , satisfying t i -∆ t i = t i -1 .

We define Eq. (6) as global consistency because it 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 ∆ t , it is necessary to ensure that the denoising error is constrained, namely:

where δ is an upper bound that needs to be set manually. Clearly, when ∆ t takes the maximum value t -ϵ , due to the boundary condition f θ ( x ϵ , ϵ ) ≡ x 0 , the constraint in Eq. (7) will be satisfied regardless of the value of δ . This implies that we need to choose the largest possible ∆ t .

Notably, the optimization objective in Eq. (5) and the constraint in Eq. (7) respectively impose opposite guidance for ∆ t . When ∆ t 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. (7) to exceed its upper bound. Conversely, when ∆ t 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 a constrained optimization objective to achieve a trade-off between Eq. (5) and Eq. (7), which is given by:

We denote the optimization objective E x 0 , z [ ∥ f θ -( x t ) -f θ -( x t -∆ t ) ∥ 2 2 ] as L local, as it focuses on the local consistency information of CMs and controls the local consistency error for CMs. Therefore, minimizing L local effectively improves the effectiveness of CMs. We denote the constraint E x 0 , z [ ∥ f θ -( x t -∆ t ) -x 0 ∥ 2 2 ] as L global as it focuses on the global consistency and controls the denoising error in the training objective. Consequently, L global 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 simultaneously, enabling an adaptive 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:

Here, the Lagrange multiplier λ acts as a weighting factor balancing the local consistency and the global consistency of CMs. We aim for λ to be a constant independent of t , ensuring that the focus on trainability and stability remains consistent across different time scales. Typically, we set λ ≪ 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 2. We find that previous discretization strategies can be unified in ADCMs. We summarize this as follows.

Global consistency of DMs

ADCMs balance local and

global consistency

Local consistency of CMs

Remark 3.2 . DMs, discrete-CMs and continuous-CMs can be viewed as special cases of Eq. (9). Specifically,

DMs [11, 18]: Choose the maximum optimization step ∆ t = t -ϵ . this corresponds to set λ →∞ in our framework.

Discrete-CMs:

-CM[37], iCT [36], ECM [7]: These methods consider the smoother trajectory changes near the noise end and empirically choose a 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 x 0 , z , L e is always less than a certain threshold δ . Since an analytical solution cannot be obtained directly, CCM requires iterative solving for all x 0 , z , t . This is equivalent to making L local a constant in our framework.

Continuous-CMs [23]: Choose the minimum optimization step ∆ t → 0 . This is equivalent to set λ = 0 in our framework.

Analytical Solution. To achieve an adaptive solution for the discretization step, we propose using the Gauss-Newton method to directly derive an analytical solution to the optimization problem in Eq. (9). Since we assign a higher weight to local consistency in the objective, we approximate f θ -( x t -∆ t ) using its first-order Taylor expansion, which is:

where v can be efficiently computed via the Jacobian-vector product (JVP) for f θ -( · , · ) , evaluated at input vector ( x t , t ) and tangent vector ( d x t d t , 1) , following the method in [23]. Under this approximation, the optimization problem is transformed into a least-squares problem, whose optimal solution is given by:

From the expression of the discretization step, we have the following observations:

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 L local could be very large. Therefore, a smaller step size is required to ensure effectiveness.

The optimal discretization step is proportional to ∥ f θ -( x t ) -x 0 ∥ 2 , which is an effective estimate of L global. This indicates that the denoising error may be very large at this time, and therefore, a larger step size is required to ensure stability.

The optimal discretization step is proportional to the linear correlation between v and f θ -( x t ) -x 0 . This indicates that when v and f θ -( x t ) -x 0 tend toward linearity, the direction of local optimization aligns more closely with the direction of global optimization, allowing for the use of a larger step size.

Algorithm 1 Adaptive Discretization for Consistency Models

Input: dataset D , diffusion parameter α t and β t , time range [ ϵ, T ] , network parameter θ , weighting factor λ , update frequency m , batch size b θ -← θ

repeat

Initialize an empty set T and t ← T

The above analysis demonstrates that the proposed discretization step can be adaptively adjusted based on the current state of the NN, considering both L global and L local. As a result, we achieve an adaptive balance between the stability and trainability of CMs at different times through Eq. (10). Starting from t = T , we iteratively solve the optimization problem to derive the adaptively optimal time segmentation T = { t ∗ 1 , . . . , t ∗ N } .

3.2 Putting ADCMs into Practice: Further Training Strategies

Adaptive Weighting Function. Through the analysis in Section 3.1, we know that L global fundamentally determines the training stability of CMs at the current time t . However, during the training of CMs, the NN only optimizes for L local. Therefore, to further balance the impact of L global over time, we propose the following adaptive weighting function:

When L global is very large, the training of CMs will suffer from instability. Therefore, a smaller weighting is needed. On the other hand, when L global is small, the CMs' training objective aligns closely with the true target, and thus a larger weighting is required.

Adaptive Distance Metric. Previous works [7, 36] have shown that compared to the squared L 2 metric, Pseudo-Huber metric can effectively reduce training variance, which is given by:

where c is a constant. ADCMs also use 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 as:

where ∆ t ∗ is obtained with Eq. (10). See Appendix B for more discussion on loss function design.

Putting It Together. We alternately optimize the time segmentation T and the NN's parameters θ during training. We typically update T after updating θ for m = 25000 times, as the changes in the

NN are relatively slow. Before training the network, we fix its parameters and perform simulationbased optimization starting from t = T . We iteratively update t using Eq. (10) until t = ϵ , recording the optimization process as T = { t ∗ 1 , . . . , t ∗ N } . We observe that during the optimization process, the expectation in Eq. (10) is well approximated using a 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 1.

4 Experiments

To validate the effectiveness of ADCMs, we perform unconditional and class-conditional generation experiments on CIFAR-10 [16] and ImageNet 64 × 64 [3], respectively. For CIFAR-10, we initialize CMs with pretrained DM from [11]. For ImageNet 64 × 64 , we adopt the pretrained DM from [12]. If not 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 1. FIDs with ∗ indicate that they have additional training costs compared to other CMs, such as a larger model or an auxiliary model used during training. Experiments show that ADCMs achieve high-quality singlestep generation without additional training costs. See Appendix C for multi-step generation results.

4.1 Efficiency of ADCMs

We evaluate the training efficiency of ADCMs on both unconditional CIFAR-10 and class-conditional ImageNet 64 × 64 . For CIFAR-10, we use a unified model size and measure computational cost by the total number of training images. For ImageNet 64 × 64 , both model size and training budgets 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.

0

2

4

6

1-Step FID (

)

4.54

30.2h

3.16

31.5h

CIFAR-10

5.51

17.2h

5.12

17.8h

ImageNet 64×64

w/o Adaptvie Discretization

w Adaptvie Discretization

(a)

0

20

40

60

80

100

Training steps (×10 3 )

3

4

6

10

50

200

400

FID

CD

CT

iCT

sCT

ECM

ADCM (ours)

(b)

0

20

40

60

80

100

Training steps (×10 3 )

3

4

6

10

50

200

400

FID

=0.1

=0.05

=0.02

=0.01

=0.005

0 (sCT)

(c)

10

2

10

1

10 0

10 1

10 2

t

10

5

10

4

10

3

10

2

10

1

10 0

t

iCT

ECM

ADCM (ours)

(a) EDM (VE)

10

2

10

1

10 0

t

10

5

10

4

10

3

t

iCT

ECM

ADCM (ours)

(b) Flow Matching

Data Efficiency. On unconditional CIFAR-10, as shown in Table 2, ADCMs achieve high-quality one-step generation results with a training budget of only 12 . 8 Mimages. Compared with ECM [7], 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 . 8 M training images. On class-conditional ImageNet 64 × 64 , as shown in Table 3, ADCMs significantly reduce the training budgets of CMs. Compared to the most efficient ECM [7], ADCMs can achieve a 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 2 × model size, ADCMs achieve a 1-step FID of 3 . 49 with a training budget of only 12 . 8 Mimages. 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 with different CM approaches, as shown in Figure 3b. It 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 ∆ t of ADCMs and time t 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. In addition, compared to other discretization schemes, ADCMs adopt finer discretization at smaller t and coarser discretization at larger t . As a result, ADCMs place greater emphasis on time intervals closer to the data during training, which aligns with empirical practices in both DMs and CMs [11, 36, 7].

λ as a Trade-off between Stability and Effectiveness. We control the trade-off between the trainability and stability of ADCMs through the Lagrange multiplier λ according to Eq. (9). We perform an ablation study on λ by examining the training dynamics of ADCMs on unconditional CIFAR-10, as shown in Figure 3c. We find that when λ is small, i.e., more emphasis is placed on L global, CMs converge quickly, but the final generation quality is relatively poor. When λ is large, i.e., more emphasis is placed on L local, CMs become more unstable, making them difficult to converge and reach the optimal solution. Ablation study on loss function are deferred to Appendix B.

ADCMsAdapttoDifferent Variants of DMs. Weconduct experiments on Flow Matching [18, 20], an advanced variant of DMs. We initialize CMs with pretrained DMs from [20] and compare ADCMs with other Flow Matching-based distillation methods. Additionally, we conduct experiments on ECM and sCT, two state-of-the-art CMs. All CMs are trained under a training budget of 12 . 8 M images. As shown in Table 4, ADCMs achieve superior performance over other CMs without manual adjustments, which demonstrates the strong adaptability.

Scalability to High-Resolution Images. To further assess the scalability of ADCM, we conduct experiments on ImageNet 512 × 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 [7], the two most efficient prior CMs. The detailed results are reported in Table 5. 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 an optimization problem, we introduce local consistency as the optimization objective and global consistency as a 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 (2023B1212010001). Z. Deng is partially supported by the National Natural Science Foundation of China (via NSFC-92470118 and NSFC-62306176) and the Natural Science Foundation of Shanghai (23ZR1428700). R. C. Qiu is partially supported in part by the National Natural Science Foundation of China (via NSFC-12141107), the Key Research and Development Program of Wuhan (2024050702030100), and the Key Research and Development Program of Guangxi (GuiKe-AB21196034).

References

David Berthelot, Arnaud Autef, Jierui Lin, Dian Ang Yap, Shuangfei Zhai, Siyuan Hu, Daniel Zheng, Walter Talbott, and Eric Gu. Tract: Denoising diffusion models with transitive closure time-distillation. arXiv preprint arXiv:2303.04248 , 2023.

Andreas Blattmann, Tim Dockhorn, Sumith Kulal, Daniel Mendelevitch, Maciej Kilian, Dominik Lorenz, Yam Levi, Zion English, Vikram Voleti, Adam Letts, et al. Stable video diffusion: Scaling latent video diffusion models to large datasets. arXiv preprint arXiv:2311.15127 , 2023.

Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition , pages 248-255. Ieee, 2009.

Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems , 34:8780-8794, 2021.

Kevin Frans, Danijar Hafner, Sergey Levine, and Pieter Abbeel. One step diffusion via shortcut models. arXiv preprint arXiv:2410.12557 , 2024.

Zhengyang Geng, Ashwini Pokle, and J Zico Kolter. One-step diffusion distillation via deep equilibrium models. Advances in Neural Information Processing Systems , 36, 2024.

Zhengyang Geng, Ashwini Pokle, William Luo, Justin Lin, and J Zico Kolter. Consistency models made easy. arXiv preprint arXiv:2406.14548 , 2024.

Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems , 30, 2017.

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems , 33:6840-6851, 2020.

Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video diffusion models. Advances in Neural Information Processing Systems , 35:8633-8646, 2022.

Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-based generative models. Advances in neural information processing systems , 35:26565-26577, 2022.

Tero Karras, Miika Aittala, Jaakko Lehtinen, Janne Hellsten, Timo Aila, and Samuli Laine. Analyzing and improving the training dynamics of diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 24174-24184, 2024.

Dongjun Kim, Chieh-Hsin Lai, Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: Learning probability flow ODE trajectory of diffusion. In The Twelfth International Conference on Learning Representations , 2024.

Zhifeng Kong and Wei Ping. On fast sampling of diffusion probabilistic models. In ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models , 2021.

Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. Diffwave: A versatile diffusion model for audio synthesis. In International Conference on Learning Representations , 2021.

Alex Krizhevsky et al. Learning multiple layers of features from tiny images. 2009.

Sangyun Lee, Yilun Xu, Tomas Geffner, Giulia Fanti, Karsten Kreis, Arash Vahdat, and Weili Nie. Truncated consistency models. arXiv preprint arXiv:2410.14895 , 2024.

Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations , 2023.

Haohe Liu, Zehua Chen, Yi Yuan, Xinhao Mei, Xubo Liu, Danilo Mandic, Wenwu Wang, and Mark D Plumbley. Audioldm: Text-to-audio generation with latent diffusion models. In International Conference on Machine Learning , pages 21450-21474. PMLR, 2023.

Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In The Eleventh International Conference on Learning Representations , 2023.

Yunpeng Liu, Boxiao Liu, Yi Zhang, Xingzhong Hou, Guanglu Song, Yu Liu, and Haihang You. See further when clear: Curriculum consistency model. arXiv preprint arXiv:2412.06295 , 2024.

Xiaoxiao Long, Yuan-Chen Guo, Cheng Lin, Yuan Liu, Zhiyang Dou, Lingjie Liu, Yuexin Ma, Song-Hai Zhang, Marc Habermann, Christian Theobalt, et al. Wonder3d: Single image to 3d using cross-domain diffusion. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 9970-9980, 2024.

Cheng Lu and Yang Song. Simplifying, stabilizing and scaling continuous-time consistency models. arXiv preprint arXiv:2410.11081 , 2024.

Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems , 35:5775-5787, 2022.

Eric Luhman and Troy Luhman. Knowledge distillation in iterative generative models for improved sampling speed. arXiv preprint arXiv:2101.02388 , 2021.

Thuan Hoang Nguyen and Anh Tran. Swiftbrush: One-step text-to-image diffusion model with variational score distillation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 7807-7816, 2024.

Ben Poole, Ajay Jain, Jonathan T. Barron, and Ben Mildenhall. Dreamfusion: Text-to-3d using 2d diffusion. In The Eleventh International Conference on Learning Representations , 2023.

Vadim Popov, Ivan Vovk, Vladimir Gogoryan, Tasnima Sadekova, and Mikhail Kudinov. Gradtts: A diffusion probabilistic model for text-to-speech. In International Conference on Machine Learning , pages 8599-8608. PMLR, 2021.

Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In International conference on machine learning , pages 8821-8831. Pmlr, 2021.

Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. Highresolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 10684-10695, 2022.

Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. In International Conference on Learning Representations , 2022.

Axel Sauer, Dominik Lorenz, Andreas Blattmann, and Robin Rombach. Adversarial diffusion distillation. In European Conference on Computer Vision , pages 87-103. Springer, 2025.

Axel Sauer, Katja Schwarz, and Andreas Geiger. Stylegan-xl: Scaling stylegan to large diverse datasets. In ACM SIGGRAPH 2022 conference proceedings , pages 1-10, 2022.

Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learning , pages 2256-2265. PMLR, 2015.

Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In International Conference on Learning Representations , 2021.

Yang Song and Prafulla Dhariwal. Improved techniques for training consistency models. In The Twelfth International Conference on Learning Representations , 2024.

Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. In International Conference on Machine Learning , pages 32211-32252. PMLR, 2023.

Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations , 2021.

Joshua Tian Jin Tee, Kang Zhang, Hee Suk Yoon, Dhananjaya Nagaraja Gowda, Chanwoo Kim, and Chang D Yoo. Physics informed distillation for diffusion models. arXiv preprint arXiv:2411.08378 , 2024.

Arash Vahdat, Francis Williams, Zan Gojcic, Or Litany, Sanja Fidler, Karsten Kreis, et al. Lion: Latent point diffusion models for 3d shape generation. Advances in Neural Information Processing Systems , 35:10021-10039, 2022.

Yaohui Wang, Xinyuan Chen, Xin Ma, Shangchen Zhou, Ziqi Huang, Yi Wang, Ceyuan Yang, Yinan He, Jiashuo Yu, Peiqing Yang, et al. Lavie: High-quality video generation with cascaded latent diffusion models. International Journal of Computer Vision , pages 1-20, 2024.

Zhengyi Wang, Cheng Lu, Yikai Wang, Fan Bao, Chongxuan Li, Hang Su, and Jun Zhu. Prolificdreamer: High-fidelity and diverse text-to-3d generation with variational score distillation. Advances in Neural Information Processing Systems , 36, 2024.

Ling Yang, Zixiang Zhang, Zhilong Zhang, Xingchao Liu, Minkai Xu, Wentao Zhang, Chenlin Meng, Stefano Ermon, and Bin Cui. Consistency flow matching: Defining straight flows with velocity consistency. arXiv preprint arXiv:2407.02398 , 2024.

Tianwei Yin, Michaël Gharbi, Taesung Park, Richard Zhang, Eli Shechtman, Fredo Durand, and William T Freeman. Improved distribution matching distillation for fast image synthesis. arXiv preprint arXiv:2405.14867 , 2024.

Tianwei Yin, Michaël Gharbi, Richard Zhang, Eli Shechtman, Fredo Durand, William T Freeman, and Taesung Park. One-step diffusion with distribution matching distillation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 6613-6623, 2024.

Hongkai Zheng, Weili Nie, Arash Vahdat, Kamyar Azizzadenesheli, and Anima Anandkumar. Fast sampling of diffusion models via operator learning. In International conference on machine learning , pages 42390-42402. PMLR, 2023.

Mingyuan Zhou, Huangjie Zheng, Zhendong Wang, Mingzhang Yin, and Hai Huang. Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation. In Forty-first International Conference on Machine Learning , 2024.

Zhenyu Zhou, Defang Chen, Can Wang, and Chun Chen. Fast ode-based sampling for diffusion models in around 5 steps. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 7777-7786, 2024.

A Experiments Details

A.1 Precondition

For VE-based ADCMs, we follow the parameterization of EDM. Specifically, we set:

For ADCMs on the Flow Matching setting, using EDM's precondition causes the model output to deviate from x 0 , which contradicts the objective of CMs to estimate x 0 . We use a pretrained model from rectified flow [20] whose output is:

which implies c in ( t ) = 1 and c noise ( t ) = t . To ensure the model's final output matches x 0 , we accordingly modify the preconditioning to:

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 . 8 Mimages. We use a batch size of 1024 with an EMA decay rate of 0 . 99993 for training budget of 76 . 8 Mimages. For class-conditional ImageNet 64 × 64 , we set the batch size to 128 , 256 , and 512 , corresponding to training budgets of 12 . 8 M, 25 . 6 M, and 51 . 2 Mimages, respectively. We use Power function EMA for class-conditional ImageNet 64 × 64 following [12].

Time Sampling. For unconditional CIFAR-10, we follow previous works [36, 7, 23] and use a log-normal SNR distribution for time sampling, which can be expressed as:

where SNR( t ) = β t α t , P mean = -1 . 1 , P std = 2 . 0 . Since the time segment T is discrete, we also apply discretization to the sampling results following [36]. For class-conditional ImageNet 64 × 64 , we sample uniformly within the time segment T .

Lagrange Multiplier λ . For unconditional CIFAR-10, we set λ = 0 . 01 . For class-conditional ImageNet 64 × 64 , we find that starting with a small λ led to training instability on ImageNet 64 × 64 . Therefore, we follow previous work [36, 7] and select λ = 0 . 64 for warm-up, gradually decreasing it to λ = 0 . 01 . We summarize the hyperparameter settings in Table 6.

A.3 Baseline Reimplementation.

Some of the baseline results in this paper, including those in Figure 3b, Table 4 and sCM [23] under VE settings, are obtained from our own reproductions. Under the VE SDE setting, for a fair comparison, we initialize all neural networks using the pretrained DM provided by EDM [11]. 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 0 , as dropout can lead to a 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 of time t to being functions of SNR . It is important to note that without manual adjustments, the performance of these baselines degrades significantly (e.g., the 1-step FID of ECM drops from 5 . 82 to 15 . 55 ). The implementation code is available in the supplementary material.

B Ablation Study

We investigate the impact of adaptive loss function in ADCMs, including the choice of weighting function and distance metric. We perform an ablation study on unconditional CIFAR-10 under the same training budget of 12 . 8 Mimages.

Weighting Function. The choice of weighting function is crucial for training CMs. An inappropriate 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 are presented in Table 7. The results show that our designed adaptive weighting function can effectively enhance the generation capability of ADCMs. Notably, even without the loss function improvement, ADCMs still outperform ECM's 1-step FID ( 4 . 54 ). This demonstrates that the improvement 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 [7, 36, 17], we adopt the Pseudo-Huber metric due to its robustness to outliers [36]. The Pseudo-Huber metric is defined as

which provides a smooth interpolation between the L 2 and squared L 2 metrics. Specifically, when c = 0 , it reduces to the standard L 2 distance, while as c →∞ , it approaches the squared L 2 distance. smoothly bridges the gap between the L 2 and squared L 2 metric. When c = 0 , the Pseudo-Huber metric degenerates to the standard L 2 metric. When c →∞ , it becomes equivalent to the squared L 2 metric. To address this phenomenon, we conduct experiments with different values of c , and the

results are presented in Table 8. It can be observed that Pseudo-Huber metric smoothly interpolates between L 2 and squared L 2 through the control of the parameter c , thus achieving performance that surpasses both extremes. These results clearly demonstrate the significance of choosing Pseudo-Huber as 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 Pseudo-Huber with c = 0 . 03 , while applying different distance metrics in the weighting function. As shown in Table 9, a mismatched distance metric leads to degraded performance of ADCMs.

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 to improve generation quality through multi-step sampling. We investigate the two-step generation performance of ADCMs on unconditional CIFAR-10, with results shown in Table 10. We set the intermediate t = 0 . 420 . It can be observed that ADCMs not only maintain optimal single-step generation performance but also demonstrate strong two-step generation capabilities, second only to ECM [7], which is specifically designed for two-step generation.

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

L global 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 to 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.

E License

We list the used datasets, models and their citations, URLs, and licenses in Table 11.

F Additional Samples

We provide additional samples of ADCMs from unconditional CIFAR-10 and class-conditional ImageNet 64 × 64 in Figures 5 - 7.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The main claims made in the abstract and introduction accurately reflect the paper's contributions.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We discuss the limitations of the work. See sec D.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [NA]

Justification: The paper does not include theoretical results.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: The paper fully disclose all the information needed to reproduce the main experimental results of the paper. Detailed algorithm can be found in Algorithm 1. Detailed settings can be found in Sec 4 and Appendix A. Code is available in the supplementary material.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: The paper provide open access to the code. Code is available in the supplementary material.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: The paper specify all the training and test details. Detailed settings can be found in Sec 4 and Appendix A.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [No]

Justification: Since the FID is computed using 50k samples, we find that the standard deviation of ADCMs' FID is very small, and thus it does not affect the conclusions of our experiments.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We provide sufficient information on the computer resources in Table 6.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: The research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We discuss both potential positive societal impacts and negative societal impacts of the work performed. See Appendix D.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: The paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We list the used datasets, models and their citations, URLs and licenses in Appendix E.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: The code is well documented and the documentation is available in the supplementary material.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects. Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nI 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.\n\nQUESTIONS:\nCan 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.\n\nLIMITATIONS:\nThere 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.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-09"}

comments: 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.

STRENGTHS AND WEAKNESSES:
Strength:
1. The paper addresses an important problem of choosing discretization steps in consistency models.
2. The proposed method is novel. The formulation is intuitive, the objective makes sense. Overall, the results are easy to follow.
3. Experimental results are very strong. I am impressed with the training efficiency section/

Weakness:
1. 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.
2. Among other consistency models, results are still not SOTA. Other methods can outperform ADCM.

QUESTIONS:
1. For large-scale runs, how would you recommend picking \lambda?
2. 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
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nExisting 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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)\n\nQUESTIONS:\n- 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 ?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nPros\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).\n\nQUESTIONS:\nSee Cons in the \"Strengths And Weaknesses\".\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-19"}