Title: Immiscible Diffusion: Accelerating Diffusion Training with Noise Assignment
Venue: NeurIPS 2024
Accepted: yes
Keywords: Diffusion Model

Training Efficiency
Abstract: In this paper, we point out that suboptimal noise-data mapping leads to slow training of diffusion models. During diffusion training, current methods diffuse each image across the entire noise space, resulting in a mixture of all images at every point in the noise layer. We emphasize that this random mixture of noise-data mapping complicates the optimization of the denoising function in diffusion models. Drawing inspiration from the immiscibility phenomenon in physics, we propose *Immiscible Diffusion*, a simple and effective method to improve the random mixture of noise-data mapping. In physics, miscibility can vary according to various intermolecular forces. Thus, immiscibility means that the mixing of molecular sources is distinguishable. Inspired by this concept, we propose an assignment-then-diffusion training strategy to achieve *Immiscible Diffusion*. As one example, prior to diffusing the image data into noise, we assign diffusion target noise for the image data by minimizing the total image-noise pair distance in a mini-batch. The assignment functions analogously to external forces to expel the diffuse-able areas of images, thus mitigating the inherent difficulties in diffusion training. Our approach is remarkably simple, requiring only *one line of code* to restrict the diffuse-able area for each image while preserving the Gaussian distribution of noise. In this way, each image is preferably projected to nearby noise. To address the high complexity of the assignment algorithm, we employ a quantized assignment strategy, which significantly reduces the computational overhead to a negligible level (e.g. 22.8ms for a large batch size of 1024 on an A6000). Experiments demonstrate that our method can achieve up to 3x faster training for unconditional Consistency Models on the CIFAR dataset, as well as for DDIM and Stable Diffusion on CelebA and ImageNet dataset, and in class-conditional training and fine-tuning. In addition, we conducted a thorough analysis that sheds light on how it improves diffusion training speed while improving fidelity. The code is available at https://yhli123.github.io/immiscible-diffusion

Immiscible Diffusion: Accelerating Diffusion Training with Noise Assignment

Abstract

In this paper, we point out that suboptimal noise-data mapping leads to slow training of diffusion models. During diffusion training, current methods diffuse each image across the entire noise space, resulting in a mixture of all images at every point in the noise layer. We emphasize that this random mixture of noisedata mapping complicates the optimization of the denoising function in diffusion models. Drawing inspiration from the immiscibility phenomenon in physics, we propose Immiscible Diffusion , a simple and effective method to improve the random mixture of noise-data mapping. In physics, miscibility can vary according to various intermolecular forces. Thus, immiscibility means that the mixing of molecular sources is distinguishable. Inspired by this concept, we propose an assignment-then-diffusion training strategy to achieve Immiscible Diffusion . As one example, prior to diffusing the image data into noise, we assign diffusion target

noise for the image data by minimizing the total image-noise pair distance in a minibatch. The assignment functions analogously to external forces to expel the diffuseable areas of images, thus mitigating the inherent difficulties in diffusion training. Our approach is remarkably simple, requiring only one line of code to restrict the diffuse-able area for each image while preserving the Gaussian distribution of noise. In this way, each image is preferably projected to nearby noise. To address the high complexity of the assignment algorithm, we employ a quantized assignment strategy, which significantly reduces the computational overhead to a negligible level ( e.g. 22.8ms for a large batch size of 1024 on an A6000). Experiments demonstrate that our method can achieve up to 3x faster training for unconditional Consistency Models on the CIFAR dataset, as well as for DDIM and Stable Diffusion on CelebA and ImageNet dataset, and in class-conditional training and fine-tuning. In addition, we conducted a thorough analysis that sheds light on how it improves diffusion training speed while improving fidelity. The code is available at https://yhli123.github.io/immiscible-diffusion

1 Introduction

Diffusion models have made impressive progress in image generation by framing the process as a phase of denoising random Gaussian noise into the final image. Despite the advancements, training a diffusion model is resource intensive. For example, even in the primary image dataset CIFAR-10, the representative few-step diffusion model, Consistency Model [47], requires training for 10 days on 4 A6000 GPUs to reach a desired FID score of around 10. Similarly, with fewer model parameters, multiple-step diffusion model DDIM [44] still requires 24 hours on an A5000 GPU on the CIFAR-10 dataset. Although recent remarkable achievements in accelerating the inference of diffusion models [19, 33, 47, 28, 30, 31] have been accomplished, the inefficiency of diffusion training remains a significant bottleneck, hindering the iterative development of vision generative AI.

Previous methods for improving diffusion training have focused on various strategies, such as balancing the impact of activation layers and neural weights [16], modifying hyperparameters and design choices [46], and leveraging patchifying strategies [53] etc. Specifically, Karras et al. [16] modifies the activation magnitude, neural weight standardization, and group normalization, achieving significant acceleration in diffusion training. Besides, previous work [46] proposes a customized method for the Consistency Model to improve the performance and diffusion training. Our method is orthogonal to these previous methods. We got inspired by the Immiscible Diffusion in physics. As illustrated in Fig. 2 (a) left, miscible particles tightly jumble together after the diffusion process, making it difficult to separate them individually during the denoising phase. However, when the particles are rendered immiscible, they can still achieve a similar overall distribution while remaining clearly distinguishable (see Fig. 2 (a) right). This insight inspires our strategy for improving the disentanglement of diffused data.

We draw an analogy from the phenomenon of Immiscible Diffusion and relate the distribution of image data to the behavior of particles discussed above. In traditional diffusion processes, each image can be diffused to any point in the noise space, and conversely, each point in the noise space can be denoised to any source image, as illustrated in the left image of Fig. 2 (b). We hypothesize that the jumbled image-noise mapping creates a miscible diffusion effect and makes the optimization of the diffusion model difficult. Inspired by the Immiscible Diffusion, we are motivated to make the mixed diffusion phase distinguishable.

We propose one simple Immiscible Diffusion method. Note that we still sample Gaussian noise but perform a batch-wise assignment of noise to each image based on the distance between them during training. This approach ensures that each image is only diffused to surrounding areas while maintaining the overall Gaussian distribution of all noises. This technique was also used in flow matching-based methods [36, 50] for optimizing the image-noise flow. Nevertheless, we find that the image-noise distance is reduced by only ~2% after assignment, as provided in Part 4.3. This motivates us to ask which factors dominate the performance improvement? To investigate it, we propose another method that qualifies Immiscible Diffusion in Part 4.3. Experiments show that the training efficiency improvement is comparable to the function of flow optimization, demonstrating that immiscibility is the dominant factor.

Image

Distribution

Noise

Distribution

Miscible (Vanilla)

Immiscible

(a)

(b)

Immiscible Diffusion in Physics

Immiscible Diffusion in Image Generation

Immiscible

Miscible

However, technically, to achieve immiscibility with the image data-noise assignment has an O( N 2 logN )-O( N 3 ) complexity. This introduces significant overhead during training, especially for large-scale training with huge batch sizes and high-resolution images. To address this, we employ a novel quantization method during assignment. We quantize the noise and image data into low-precision formats (e.g., 16-bit) during conducting the assignment algorithm. We highlight that this assignment operation only involves one line of code , and is performed only during the training phase without modifying the model architecture, the noise scheduler, the sampler, or the method of inference.

We conduct extensive experiments on three common modes: unconditional, conditional, and finetuning on three diffusion baselines: Consistency Models, DDIM and Stable Diffusion and three datasets: CIFAR-10, CelebA and ImageNet datasets. Results show that our proposed method significantly improves the training efficiency in all experiments. Specifically, we achieve 3x training efficiency for the CIFAR-10 dataset with immiscible unconditional Consistency Model compared to the original Consistency Model. Furthermore, we show that the FID is even lower with our method used, confirming the fidelity of our generated images. We also provide images generated from models trained with vanilla and immiscible models experiencing the same training steps, where we see that those from immiscible models are much more complete and clearer, further proving the training efficiency enhancement resulted from the Immiscible Diffusion. Examples are shown on the right of Fig. 1. Deeper analysis shows that our method, although with only one line of code and involving ~2% image-noise datapoint distance changes, achieves all the benefits above in negligible running time.

To sum up, our contributions are as follows:

We clearly and specifically identify the miscibility issue in noisy diffusion steps, which leads to slow convergence of diffusion training.

To tackle the miscibility issue, we propose a simple and effective method, Immiscible Diffusion, a strategy that can only requires one-line of code, to improve training efficiency for diffusion training.

Experiments demonstrate the effectiveness of our proposed method on several popular diffusion models across multiple datasets, and across unconditional, conditional, and fine-tuning tasks. In addition, we conducted thorough analyses and ablation studies to elucidate how our method works and dominates the effectiveness.

2 Related Work

2.1 Diffusion Model with Efficient Inference

Diffusion models [48, 12, 39, 35] have been attracting huge attentions because of their high-fidelity image and video generation [11, 13, 34], data-efficient perception [49, 32, 55], and even representation abilities for robotics [3, 37, 1]. However, slow inference is one of the key bottlenecks for diffusion models. To address this issue, various approaches have been proposed. For instance, techniques such as DDIM [44] have reduced the number of denoising steps from 1000 to 10, significantly speeding up the process. Furthermore, the introduction of Consistency Models [47] and LCM [33],

which utilize the properties of self-consistency, enables denoising in as few as 1-4 steps, further enhancing the generation speed of diffusion models. Subsequently, the development of SD-turbo [42], which leverages GAN [8] loss for high-definition image generation in a single step, has occurred. The Consistency Trajectory Models [18, 38] improve the generation quality of Consistency Models and accelerate research on efficient inference for diffusion models. Additionally, beyond reducing denoising steps, efforts to improve the inference efficiency of single function evaluation are being explored in various ways, including model quantization [26] and partitioning the generative components [25]. Moreover, StreamDiffusion [19] streamlines denoising steps to achieve real-time inference at the pipeline level optimization. The improvement of the inference efficiency significantly pushes forward real applications based on diffusion models. Yet accelerating diffusion training is still under-explored.

2.2 Diffusion Model with Efficient Training

Improving the training efficiency of diffusion models is crucial. Various strategies have been proposed, including architectural modifications [16], approximating the diffusion phase with flow [28], and designing parameter choices [46] etc . Specifically, in [16], the authors discover that the magnitude of activation and the magnitude of neural weights significantly impact the training dynamics of diffusion models. They propose adjusting the activation magnitude and standardizing neural weights, as well as modifying the normalization layers to make diffusion training in a more smooth dynamic. Besides, Song et al. [46] aims to enhance the training efficiency of Consistency Models through customized design choices, significantly improving both training speed and fidelity. Furthermore, leveraging approximation strategies based on ODE assumptions [28] improves not only inference efficiency but also training efficiency since diffusion trajectories are prone to deterministic. Beyond improving diffusion training with either architecture adjustment or selection parameters, Wang et al. [53] introduce a novel patch strategy to control the ease of diffusion training, achieving both training and data efficiency. Gleichzeitig, Wang et al. [52] notices that the denoising of some noisy diffusion steps contains little information and is too easy to learn, so focusing more on other steps would significantly improve training efficiency. Our method differs from previous works by clearly highlighting an under-explored problem: the miscibility problem of image data in the noise space, which plays a crucial role in training diffusion models. Our proposed Immiscible Diffusion is extremely simple yet significantly improves training efficiency.

2.3 Image-noise Optimal Transport in Generative Models

In ODE-based methods such as flow matching [27], straightening the flow with optimal transport (OT) has been used as a tool to improve the generation performance. Specifically, optimizing image-noise transport in a batch [36, 50] has been found to be an effective way to improve performance. Training efficiency improvement was also observed [36, 50], and explained or posited with reduction of the variance of the training goal. However, the standard deviation reduction is only ~4% in [36]. Moving forward, [24] pointed out the curvature problem in the ODE paths caused by the collapse of the reverse trajectories in the average direction. However, they replaced Gaussian noise sampling with a VAE encoder-style structure to eliminate such curvatures, which destroys the strict Gaussian distribution in the noise space. Concurrent to our work, [17] applies batch-wise OT to diffusion models to achieve better FIDs, making posits in curvature reduction for the enhancement. Several methods were proposed to improve the speeds and effectiveness of OT, such as pre-training with PF-ODE [54], using Schrodinger bridging [5], utilizing conditional Wasserstein distance [2] and generator-induced coupling in Consistency Models [14]. However, we are the first to emphasize that the dominant reason for training efficiency enhancement is the miscibility problem in noisy layers, which we prove in Part 4.3. Furthermore, most of previous methods are for unconditional flow matching methods only. Our work demonstrates the effectiveness in multiple diffusion models, datasets, conditional generation, and fine-tuning experiments.

3 Method

3.1 Physics Intuition

Diffusion models mimic the reverse thermodynamic diffusion phenomenon [43] to ease the denoising process. However, when the sources are miscible , as shown in the left of Fig. 2 (a), they end up

messily mixed. Predicting the reversal process from such a random mixture encounters significant difficulties, and unfortunately, this is a problem diffusion model always facing during denoising.

However, we notice that mixing can also be organized when sources are immiscible . Under that circumstance, the sources would take different continuous areas after diffusion, while the whole diffused area remains the same, as shown in the right image of Fig. 2 (a). Thereafter, the reversal process becomes smooth. Inspired by Immiscible Diffusion, we then introduce it to the diffusion models, with the aim of making the optimization easier and to achieve a higher training efficiency.

3.2 Immmiscible Diffusion Model

Similar to the physics phenomenon, we find that for diffusion models, any images are diffused to every corner of the noise space, which also means that each noise point can go back to any image. This would cause the denoising model to be confused on which image to go to, as shown in the left of Fig. 2 (b).

Mimicing the immiscible phenomenon in physics, we hope to design similar processes where each noise point is only matched to limited images, so as to avoid the confusion for the denoising model. However, the noise space must remain Gaussian to help the sampling process. Therefore, we propose our first implementation way of Immiscible Diffusion , which assigns the batch of noises to the batch of images during training according to the image-noise distance in their shared space. We minimize the total distance of the image-noise pairs in a batch during assignment. Here we use the L2 distance for assignment, which is ablated in Part A.1.3 in the Supplemental Materials. After assignment, the noise is still Gaussian, while each noise is assigned to nearer images like what happens in the immiscibility phenomenon, which significantly eases the difficulties for the denoising. Fig. 2 (b) right illustrates an extreme example of the Immiscible Diffusion, where the noise corresponding to each image is relatively separated.

For implementation, all we need to do is to perform a linear assignment [21] between the batch of images and noises according to their distances. This can be achieved in only one line of code using Scipy [51]. The algorithm is shown below:

Algorithm 1 Batch-wise Image-Noise Assignment

1: Input: Image batch x b , random noise batch n rand,b , sampled diffusion steps t b and diffusion schedule α

2: assign_mat ← scipy.optimize.linear_sum_assignment (dist( x b , n rand,b )) √

3: x t,b ← √ α t b x b + 1 -α t b · n rand,b [assign_mat]

4: Output: Diffused image batch x t,b

While linear assignment qualifies Immiscible Diffusion, Immiscible Diffusion can be achieved with multiple paths. In part 4.3, we ablate the linear assignment by letting it remain immiscible while disqualifying optimal transport, proving that immiscibility plays the dominant role in performance improvement.

3.3 Mathematical Illustration

In this section, we mathematically elucidate the denoising difficulty for traditional diffusion models based on DDIM [44, 12] and how our proposed Immiscible Diffusion reduces such difficulties.

In DDIM, we know for any image data-point x 0 , when it is diffused to the last diffusion step T, i.e. t = T , the image is sufficiently wiped out and nearly only gaussian noise is remaining, therefore,

Where q refers to the Utilizing Bayes' Rules and Equation 1, we can find that for a specific x T :

which indicates that the distributions of the corresponding images for any noise data-point are the same as the distribution of all images.

Immiscible DDIM

Image Output

Denoised Image

Sampled Noise

(b)

DDIM

Image Output

Denoised Image

Sampled Noise

(a)

τ = S-2

τ = S-3

τ = S-4

τ = S

τ = S-1

τ = S-2

τ = S-3

τ = S-4

τ = S

τ = S-1

τ = S-2

τ = S-3

τ = S-4

τ = S

τ = S-1

τ = S-2

τ = S-3

τ = S-4

τ = S

τ = S-1

The simplified training objective in DDIM [44, 12] is the added noise ϵ ( x t , t ) at each diffusion step t and at the point x t . However, we find that for a specific point x T in the noise space at diffusion step T ,

where a = -α t √ 1 -α t and b = 1 √ 1 -α t are constants, and ¯ x 0 is an average of images in the dataset. When the number of images is large enough, ¯ x 0 contains little meaningful information.

As shown in Fig. 3 (a), we study the predicted noises of different denoising layers in a DDIM model with total inference steps of 20, to illustrate our discovery. Here we refer to [45], specifying the sampling step τ = S as the layer that denoises the pure noise, which is equal to the diffusing step t = T . We can see that the predicted noise for DDIM at τ = S (equal to t = T in diffusing) does not provide much useful information, while denoising with this "noise" does not provide any distinguishable image, which all support our hypothesis for the difficulty in denoising when τ in sampling (or t in diffusing) is large. Similar observations are also shown in the concurrent work [52]. However, in our Immiscible Diffusion, while for each batch, we still have

for each specific data point x T or x 0 , the conditional noise distribution does not follow the Gaussian distribution because of the batch-wise noise assignment

̸

Instead of the Gaussian distribution, we assume that the predicted noise with noise assignment has a distribution described as follows

where f is a function denoting the influence of assignment on the conditional distribution of x T , and bs is the training batch size. Apparently, according to the definition of linear assignment problem [21], f decreases when x T -x 0 increases its norm, specifically the L2 norm as in our default setting.

Therefore, from Equation 2 and 6, we have

which means that for a specific noise data-point, the possibility of denoising it to the nearby image data-point would be higher than to a far-away image.

For the noise prediction task, we see that

where x 0 f ( x T -x 0 , ... ) is the weighted average of x 0 with more weights on image data-points closer to the noisy data-point x T itself. Therefore, the noise predicted would lead to the average of nearby image data-points, which makes more senses than pointing to a constant. Indeed, in Fig. 3, we see that even for the pure noise layer, immiscible DDIM can predict the noise effectively pointing to the shape of the horse image, and the prediction in one step by subtracting the predicted noise shows the outline of the horse correctly.

3.4 Accelerating Assignment in Immiscible Diffusion

The assignment problem has been studied extensively for decades [6, 22, 7]. In this paper, we use the Hungarian algorithm [22] as our main assignment method. However, Hungarian matching has high complexity with O ( N 3 ) , which drastically slow down the training especially when we have high-dimensional image data ( e.g., even using the mini image data 32 × 32 × 3 = 3072 ). To mitigate this issue, we make a novel use of quantization for image data and noise, that is, we quantize the fp 32 image and noise data to fp 16 to carry out the assignment, while maintaining the same precision input to diffusion models. This trick significantly reduces the overhead to a negligible level.

To efficiently perform Immiscible Diffusion when running on multiple GPUs, we assign the imagenoise distance matrix computation to each process, and then gather them to execute the assignment. This is particularly important as high resolutions and large batch sizes are frequently required in applications.

4 Experiments

4.1 Experiment Settings

To elaborate the performance of Immiscible Diffusion, we utilize the proposed method on Consistency Models [47], DDIM [45] and Stable Diffusion [41], and using CIFAR-10 [20], CelebA [29], tiny, random picked 10% and the full ImageNet [4] datasets due to the limitation of computation resource. The training hyperparameters are shown in Tab. 1. Unspecified hyperparameters are taken the same as those in their baseline methods' original papers. For evaluations, we compare the results generated by our Immiscible Diffusion method and the baseline using both the quantitative evaluation metric FID [10] and qualitative assessments.

Note that for Consistency Models, we use the single-step generation consistency training. For DDIM, we add no noise during the sampling and use linear scheduling for picking sampling steps. For Stable Diffusion, we directly use the implementation from Diffusers of Huggingface team [40]. For fine-tuning, we use Stable Diffusion v1.4 [40] as the pre-trained model.

4.2 Training Efficiency Improvement with Linear Assignment

Unconditional generation with Consistency Models: In Fig. 4, we show the FIDs of images generated with baseline and immiscible Consistency Models trained with different training steps on

3X

(a)

(b)

(c)

CIFAR

CelebA

tiny-lmageNet

40

80

40

Immiscible Consistency Model

Immiscible Consistency Model

Immiscible Consistency Model

58

24

020

FID

FID

E

16

42

10

6

12

34

8

30

0

100

200

300

400

500

600

700

800

10

20

40

60

80

100

28

40

60

80

100

120

Training Steps (k)

Training Steps (k)

the CIFAR-10 dataset, the CelebA dataset and the tiny ImageNet dataset, respectively. We observe that the immiscible Consistency Model trains much faster than the baseline Consistency Model, and converges to a significantly lower FID on all these datasets. We also show the images generated by immiscible and baseline Consistency Models trained for 100k steps in Fig. 9 in the Supplemental Materials, where we find that the images generated by the Immiscible Consistency Model are much more complete and realistic. Tab. 3 in the Supplemental Materials further presents the training steps necessary to achieve specific reasonable FID thresholds. We find that the immiscible Consistency Model significantly improves the training efficiency by around 3x, proving the effectiveness of Immiscible Diffusion in training accelerations.

In the main experiment, we observe that our method on top of the Consistency Model is effective across the datasets varying from different data sizes and resolutions. Indeed, the Consistency Model is a few-step diffusion model, and our proposed Immiscible Diffusion especially works on improving the denoising effect when the noise level is high, as shown in Fig. 3. The improvement of the training efficiency on such a few-step diffusion model further validates our findings.

One characteristic of the Consistency Model is that it approximates the SDE-diffusion model with the ODE approximation. Thus, the original image-noise mapping is highly jumbled together since it is highly possible that closed image data points are diffused to distant noise points. Our Immiscible Diffusion improves this issue by adjusting the trajectories of image-noise mapping and making them more distinguishable.

S = 20 Steps

S = 50 Steps

S = 100 Steps

(a)

(b)

(c)

13

DDIM

10

DDIM

6.5

DDIM

12

Immiscible DDIM

9

Immiscible DDIM

6

Immiscible DDIM

11

8

5.5

10

FID

FID

7

5

6

8

6

4.5

100

150

200

250

100

150

200

250

100

150

200

250

Training Steps (k)

Training Steps (k)

Training Steps (k)

More baselines: To show the generalization of Immiscible Diffusion for more baselines, we further conduct experiments on two baselines: DDIM [45] and Stable Diffusion [39] on the CIFAR-10 and the randomly picked 10% ImageNet dataset, respectively. As shown in Fig. 5, and detailed in Tab. 4 in Supplemental Materials, we find that our immiscible DDIM significantly improves the training speed and the FID compared to those of the baseline DDIM on the CIFAR-10 dataset, and the improvement

is more significant when the sampling step is lower. This demonstrates the effectiveness of our proposed method works beyond the Consistency Model and can be generalized to more few-step denoising models. We also provide a discussion in Part A.1.4, showing that the effectiveness of Immiscible Diffusion can persist in a wide range of batch sizes. To further evaluate generalizability on the popular baseline, Stable Diffusion [39], we also conduct unconditional generation experiments on the ImageNet dataset. We observe that immiscible Stable Diffusion and baseline Stable Diffusion achieve similar FID without significant gap, yet our immiscible Stable Diffusion is able to generate much higher quality images from a subjective human judgement. For example, Fig. 14 in the Supplemental Materials shows that our proposed method generates significantly clearer images compared to the baseline. More visualization without any cherry-picking can be seen in Fig. 15 in the supplementary materials. We indicate that even though FID is the primary metric and is remarkably successful, the metric is known to sometimes disagree with human judgement [23].

Class-conditional Generation: We extend Immiscible Diffusion to class-conditional generations on ImageNet dataset with Stable Diffusion [41], to explore the performance of Immiscible Diffusion in conditional generations. Results are shown in Fig. 6 (a), where we observe that in 20k training steps, the FID for immiscible class-conditional Stable Diffusion is 16.43, which is 1.49 lower than our Stable Diffusion baseline. We further confirm such improvements on CMMD [15], where the immiscible and vanilla models get 1.385 and 1.436 respectively. Additional evaluation on CLIPScore [9] shows that both the immiscible and the baseline models generate images with CLIPScores of 28.55, with a standard deviation of 0.01 and 0.02 respectively, indicating that Immiscible Diffusion does not hurt the image-prompt correspondence in complicated ImageNet dataset. Qualitative comparisons in Fig. 16 in Supplemental Materials further prove such performance enhancements, which augment the effectiveness of Immiscible Diffusions into more commonly-used conditional generations.

Fine-tuning: Our Immiscible Diffusion can also be applied to enhance the fine-tuning process where numerous applications fall in. We fine-tune the stable diffusion v1.4 model [41] on ImageNet dataset, finding that immiscible fine-tuning achieves an FID of 10.28 compared to 11.45 for vanilla fine-tuning with 5k training steps. Detailed results are shown in Fig. 6. This further broadens the application of Immiscible Diffusion.

(a)

(b)

4.3 Discussion

To further understand the proposed Immiscible Diffusion method, we delve into several key questions to ablate our approach:

How much does image-noise distance reduce in the assignment? Tab. 2 shows the reduction in distance after the image-noise assignment. We find that the L2 distance only reduces by about 2%, with a slight increase observed at higher batch sizes. However, as shown in Fig. 3, the assignment is sufficient to effectively activate denoising at high noise levels, significantly boosting training efficiency, even though the distance change is low. We attribute the low distance reduction rate after the assignment to the extremely high dimensionality (3072 for each image of the CIFAR-10 dataset) of the image and noise space.

How much time does image-noise assignment cost? In Tab. 2, we indicate that our assignment method does not introduce significant extra overhead due to our utilization of quantized assignment in our practical implementation. Even for a large batch size per GPU of 1024, our algorithm only brings in an additional 22.8 ms, demonstrating the potential of utilization for future applications.

Immiscible and OT: who dominates the training efficiency enhancement? Our Immiscible Diffusion claims to enhance training efficiency by improving miscibility in noisy diffusion steps. However, the method we take towards Immiscible Diffusion, i.e. linear assignment between image and noise, also serve as a roughly approximate OT between image and noise, which might intuitively benefit the diffusion through straightening the diffusion paths [36, 50]. However, the previous section shows that the image-noise distance is only reduced by ~2%, motivating us to ask if OT is really the dominant factor?

To answer this question, we ablate these two factors: OT and immiscibility. We design a non-OT Immiscible Diffusion experiment which keeps the immiscible property while not involving the OT. This is achieved by assigning images to the flipped noise whose all dimensions are reversed, while using the original noise during diffusion. In such a way, the image-noise pair no longer follows OT, but still qualifies the Immiscible Diffusion - i.e. images are still assigned to a limited area. Interestingly, we observe that the non-OT Immiscible Diffusion can still accelerate and enhance the diffusion training, which is nearly comparable to the OT Immiscible Diffusion in final stages, as shown in in Fig. 7. Considering that the non-OT version introduce miscibility in middle diffusion layers, which we posit for its difference to OT version, we conclude that Immiscible Diffusion is dominant in enhancing the diffusion model's performance, compared to the benefits from OT.

5 Conclusion, Limitations, and Future Work

Inspired by the immiscibility phenomenon in physics, we introduce Immiscible Diffusion, a method to improve image-noise mapping to accelerate diffusion training. Specifically, Immiscible Diffusion is an assignment-then-diffusion strategy. One way of it is to minimize the image-noise pair distance within a mini-batch so that each image is diffused to nearby noise areas. This simple approach requires only one line of code and includes a quantized-assignment strategy to reduce computational overhead.

Experiments show our Immiscible Diffusion approach speeds up Consistency Model's training by approximately 3x on the CIFAR-10 dataset, 1.3x on the CelebA dataset, and 1.2x on the tinyImageNet dataset, as well as in conditional generation and fine-tuning on Stable Diffusion. Thus, we show that Immiscible Diffusion can generalize to across datasets, baselines and tasks. Further analysis is provided to explain how this works.

Limitation. The assignment strategy is one straightforward way for Immiscible Diffusion, but not necessarily optimal. Due to the limited computational resources, our experiments are mainly conducted on small-scale datasets, so we lack the validation on larger-scale datasets such as LAION. In future work, we will improve the assignment strategy to cater to practical utilization of conditional generation such as accelerating the general text-to-image or text-to-video diffusion training.

Broader impact. With the increased use of diffusion models for image and video generation, the training of diffusion models is certain to become an increasing portion of data center workloads. Moreover, training time is a significant bottleneck in model development. Our proposed method significantly improves the efficiency of diffusion model training. We believe that our method has the potential to accelerate progress and reduce the cost of development in this field.

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A Supplemental Materials

A.1 Additional Experiment Results

A.1.1 Quantitative Training Efficiency Improvements for Immiscible Consistency Model

A.1.2 Quantitative FID Improvements for Immiscible DDIM with Different Inference Steps.

A.1.3 Ablation on the Distance Measurement Methods in Noise Assignment.

We use the L2 norm for our experiments. However, we note that the L2 norm may face more challenges in distance evaluation in high-dimensional spaces compared to the L1 norm. Therefore, we compare the performance of immiscible DDIMs using assignments based on the L1 and L2 norms. The results, as illustrated in Tab. 5, show that using the L2 norm provides better performance than the L1 norm.

A.1.4 Ablation on the Batch Size on Immiscible DDIM.

The effectiveness of image-noise assignment can intuitively rely on the batch size. Therefore, we perform a comparison to see the effectiveness of Immiscible Diffusion on DDIM across a selected range of batch sizes, whose result is shown in Fig. 8. We observe that while larger batch sizes consistently accelerate the training as expected, its training efficiency enhancement is not as large as that from Immiscible Diffusion. Immiscible Diffusions continously improve the training efficiency and the performance in the whole selected range of batch sizes.

Vanilladdim-128

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Immiscibleddim-256

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A.2 Qualitative Evaluations of Immiscible Diffusion

A.2.1 Generated images from immiscible and baseline Consistency Models trained on CIFAR-10 (Top) and CelebA (Down) with the same training steps.

Immiscible Consistency model

Consistency modell

Immiscible Consistency model

Consistency model

A.2.2 Generated images from immiscible and baseline Consistency Models trained on CIFAR-10 Dataset for 100k steps without cherry-picking.

A.2.3 Generated images from immiscible and baseline Consistency Models trained on CelebA Dataset for 100k steps without cherry-picking.

A.2.4 Generated images from immiscible and baseline stable diffusion models trained unconditionally on 10% ImageNet for 70k steps.

Immiscible StableDiffusion

Stable Diffusion

Immiscible Stable Diffusion

StableDiffusion

A.2.5 Generated images from immiscible and baseline stable diffusion models trained unconditionally on 10% ImageNet Dataset for 70k steps without cherry-picking.

Immiscible Stable Diffusion

Stable Diffusion

Immiscible Stable Diffusion

StableDiffusion

Immiscible StableDiffusion

Stable Diffusion

A.2.6 Generated images from immiscible and baseline stable diffusion models trained conditionally on ImageNet Dataset for 20k steps.

Immiscible Stable Diffusion

Stable Diffusion

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nDuring training of Diffusion Models, images are corrupted with standard Gaussian noise and the Diffusion Model is tasked to denoise the corrupted images. The Gaussian noise used during training is often sampled independently of the clean image.\n\nThis submission presents the following \"Immiscible Diffusion\" method: within each mini-batch, we should reassign the pairs of images and noise samples, so that each image is corrupted with a ‘closer’ noise sample.\n\nThe submission claims \"Immiscible Diffusion\" is faster to train to reach the same FID threshold, with quantitative evaluation, and \"can catch more details and more feature of objects\" with some cherry-picked qualitative results.\n\nSTRENGTHS:\n**S1/** The metaphor of “immiscible fluids” is creative.\n\n**S2/** The described method is simple.\n\n**S3/** L489: The authors say they will publish their code (although the paper claims it is only one line)\n\nWEAKNESSES:\nThe main weaknesses and reasons to reject the paper are the following:\n\n**W1/** The notations and mathematical derivations are imprecise, unnecessarily complex, and potentially incorrect.\n\n**W2/** Similar methods have already been presented in previous literature. The submission does not discuss these existing works, nor compare with them.\n\n**W3/** The presentation, including the figures quality and writing, needs to be improved.\n\n**W4/** The presented results are not convincing.\n\n**More details:**\n\n**W1/ The notations and mathematical derivations are imprecise, unnecessarily complex, and potentially incorrect.**\n\n* The Equations in the Section 3.3 are very hard to understand, and I am not sure they are necessary as they make understanding more difficult. The section could simply be rephrased as follows without the need to introduce complex and potentially incorrect equations: When the noise samples and data points are sampled independently, the optimal (in MSE sense) denoising prediction from a completely noisy observation is the mean of the data distribution. If we additionally know that data points are only corrupted with \"close\" noise, then the optimal denoising prediction from a completely noisy observation is, intuitively, the average of only \"close\" data points, which can allow to converge faster to a data point at generation time.\n* The notation \"When $t → \\infty $, $p(x_t|x_0) = N(0, I)$\" in Equation 1 is not precise (equality? convergence?).\n* ​​L155 states that Equation 2 indicates \"the distributions of the denoised images for any noise data-point are the same, which is equal to the distribution of the image data\", but Equation 2 does not contain any \"denoised images\", only clean images $x_0$ and their corrupted versions $x_t$.\n* In Equation 3, the formulas for $a$ and for $b$ are inverted. Also the \"minus\" sign should be in front of $a$ and not $b$. The equation further does not really make sense according to the paper’s notations, since \"p(x_0|x_t) = p(x_0)\" only “when $t → \\infty$”. It is not mathematically correct to replace only a few terms of an equation by their limit, but keeping dependencies in $t$ for other terms. $a$ and $b$ are said to be constant, but their formula clearly shows they depend on $t$ (or are $a$ and $b$ the limits “when $t → \\infty$”). \n* L158 states that the average of a large number of images is a solid color. This is incorrect, see for instance Figure 1 in **[1]**.\n* In Equation 5, the notation ${\\{ \\}}_{\\textnormal{batch}}$ is not introduced and it is not clear. It seems to contradict equation 6.\n* Equation 7 is not justified. Why would $f$ only depend on the difference between $x_t$ and $x_0$?\n\n**References:**\n\n**[1]** Torralba, Antonio, and Aude Oliva. \"Statistics of natural image categories.\" Network: computation in neural systems 14.3 (2003): 391.\n\n**W2/ Similar methods have already been presented in previous literature. The submission does not discuss these existing works, nor compare with them.**\n\n* The notions of “coupling” and “optimal transport” between data points and noise samples have already been presented in the previous literature. See for instance **[2]** and **[3]**.\n* To understand the significance of this submission, it would be necessary to discuss these similar works and highlight the differences if any, and compare with such methods if possible. \n* As such, I do not see any novelty in this work (If I miss any, the authors fail to highlight the difference by discussing related work).\n\n**References:**\n\n**[2]** Tong, Alexander, et al. \"Improving and generalizing flow-based generative models with minibatch optimal transport.\" arXiv Feb 2023 + TMLR + ICML Workshop \n\n**[3]** Pooladian, Aram-Alexandre, et al. \"Multisample flow matching: Straightening flows with minibatch couplings.\" arXiv April 2023 + ICML \n\n\n**W3/ The presentation, including the figures quality and writing, needs to be improved.**\n\n* The parallel with “Immiscible Fluids” does not bring a better understanding of the method, no physics equations are given. What should we understand from Figure 2a? Even if this is schematic, where would the noise and data distribution be in this image?\n* The quality of Figure 3 is very bad. The images for predicted noise are completely saturated.\n* The authors fail to show any ablation on the “quantization method” they claim. It is not clear if this is really necessary and the choices of fp8 and fp16 appear rather arbitrary.\n* Many sentences do not make sense to me\n    * L163: “T=20 means the layer of pure noise”\n    * L160: what are “rich image classes”?\n    * L162: “the predicted noises of different layers”\n* Many typos:\n    * L89 “its” for diffusion models\n    * text written in math font in equations\n    * L422: “our method accelerate”\n    * L471: “our immsicible diffusion work” (2 typos)\n    * Caption Figure 5: “can catch more details and more feature”\n* Several incoherences:\n    * L93: DDIM is 10 steps. <->  the authors experiment with DDIM but never take less than 20 steps.\n    * linear_sum_assignment from Scipy (algorithm 1) <-> Hungarian matching (text)\n    * Some generated images in Figures 8 and 9 are completely black.\n\n**W4/ The presented results are not convincing.**\n\n* All qualitative results presented in the main submission are cherry picked. The only non cherry picked results are shown in the appendix and the generated images, and the qualitative results seem very similar to me (no significant qualitative improvement).\n* The quantitative results are difficult to trust. There seems to be incoherences between the results:\n    * The “-0.98” in the third row of Table 5 is not coherent with the two first rows.\n    * The plots have different scales and are not clear (why does the plot for imagenet in Figure 4 starts at 20k steps and not 0?)\n    * The quantitative results mix consistency models and diffusion models in an unorganized way (eg, the columns of table 1 alternate between them), making it difficult to understand the results.\n* It is not clear why some datasets use the diffusion approach and some use the consistency approach. Did the authors experiment with both for each dataset and only report the best results?\n* The datasets and evaluation settings are not clearly presented. How many images are there in each dataset? Was the dataset split into training, validation and test sets? Are the reported FID scores computed on training, validation or test set? The number of sampling steps is not reported for all experiments (eg table 2). Only part of the quantitative results are shown (thus potentially cherry picked too, eg not quantitative results for Stable Diffusion)\n\n----\n**Changes following the rebuttal and authors-reviewers discussion:**\n\n**Reduced soundness from 2 to 1**: The scope of the paper has been significantly expanded with the rebuttal (conditional generation, flipped dimension OT, decreased noise levels). Some arguments in the paper do not hold anymore (eg, average of all images). Revised mathematical proof remains imprecise and incorrect.\n\n**Reduced contribution from 2 to 1**: Discussions following the rebuttal made me dive more deeply in the Conditional Flow Matching paper (https://arxiv.org/abs/2302.00482). Most, if not all, findings of the reviewed submission seem to be already known and analyzed more deeply in the Conditional Flow Matching paper, which additionally uses Flow Matching formalism (generalization of diffusion to more prior distributions than Gaussian Noise) and contains sound proofs. I do not see any new meaningful contribution for the Neurips community in the reviewed submission.\n\n**W1** The authors did not address this concern correctly. To expand on the fact math assumption (Equation 7) is incorrect, suppose the target distribution is $P(x_0=0.1) = P(x_0=-0.1) = 0.5$. As $t \\rightarrow +\\infty$, it is clear that, if the batch size is large,  $p(x_t|x_0=0.1)$ is approximately $2 \\times N(x_t; 0, 1)$ if $x_t>0$ and approximately  0 if $x_t <0$. This is clearly not in the form of Equation 7. The ratio also does not decrease with the norm of $x_t – x_0$, as $p(x_t=-0.1|x_0=0.1) \\approx 0$ while $p(x_t=1.1|x_0=0.1) \\approx 0.4$.\n\n**W2** The authors did not address this concern correctly. They included new experiments to claim their method does not fit into “Conditional Flow Matching”, but it seems that it still does (Conditional Flow Matching also seems to allow to choose the coupling via the joint distribution $q(x_0, x_1)$).\n\n**W3** The authors did not address this concern. The quality of Figure 3 is extremely low (all images are either extremely blurry or extremely saturated. It is possible the authors forgot something (dividing by 255?) when saving the images of the predicted noise, and that they upscaled the other images with interpolation or something). The authors did not comment on the incoherences/typos/sentences that need clarifications. The parallel with physics appears decorative without supporting physics equations. \n\n**W4** is not addressed correctly. Qualitative results show marginal improvement on cherry picked results only. Quantitative results are presented weirdly (eg, axes of the plot) with incoherences.\n\n**W5** The new experiments introduced in the rebuttal significantly extend beyond the original scope of the paper (e.g., introducing OT with flipped dimensions, decreasing noise levels, and including conditional generation). Arguments made in the paper do not hold in this expanded scope. For instance, the paper's claim that the initial denoising direction points to the average of all images, and is thus not meaningful. This is not true in the conditional generation case, as the initial denoising direction points to the average of images of the desired class. Additionally, the authors seem to say that in immiscible diffusion, any noise sample can still lead to any class (so, is it miscible?), which needs to be discussed/explained/investigated more.\n\n**Reduced overall rating from 3 to 2** Due to the (important) remaining concerns, the extent of the changes proposed in the rebuttal and discussion (would need additional reviews after changes are made), as well as the conflicts of the proposed changes with arguments contained in the original submission.\n\n**Increased confidence from 3 to 5** I participated in the discussion and dived deeper into the CFM paper (which I was initially not very familiar with). I am certain about the need to reject the paper in its current state, with the possibility for the authors to address these issues in a future submission.\n\n ----\n\nQUESTIONS:\nHere are my suggestions for this submission:\n\n**Q1/** Discussion and comparison of related works. By not discussing the related works, it is not possible for me to see the novelty in this submission, given that similar methods have been presented in existing literature. (see also **W2**)\n\n**Q2/** The presentation quality (figures, clarity) and mathematical explanations should be improved (see also  **W1**, **W3** and **W4**).\n\n**Q3/** All qualitative and quantitative results need to be shown clearly, without mistake or cherry picking (see **W4**).\n\nLIMITATIONS:\n**L1/** The authors apply some matching within each mini-batch. So there is now a dependence on batch size, it is not sure the method would work with other batch sizes. For instance on GPU with little memory and batch size of 1, the method actually doesn’t change training at all compared to normal diffusion.\n\n**L2/** Some theoretical results on diffusion may not hold anymore, e.g. that the ideal predicted denoising direction corresponds to the score (gradient of log likelihood) of the distribution of noisy data.\n\n**L3/** The method only works for unconditional generation, which is a severe limitation. (EDIT: the rebuttal contains 1 experiment on conditional generation, but discussion is missing (how come noising is immiscible but denoising is not [noise samples can still lead to any class]?) and arguments like “the average of images does not contain much meaningful information” do not hold anymore.)\n\n**L4/** Like any generative model, this work could be unethically used. It is crucial for researchers to consider designing safeguards to ensure the model can ignore inappropriate requests (eg, training or fine-tuning your Immiscible Diffusion model on inappropriate images). Can you provide a small discussion on existing safeguards and how follow-up work can address potential misuse of your method?\n\n**L5/** The authors did not communicate the details (incl. license and terms of use) of the datasets or the models that they used via structured templates. The checklist incorrectly answered “NA” for “12. Licenses for existing assets”", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 1, "DATE": "2024-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper is motivated by the miscible phenomenon of physics and transfers it into the diffusion training process, which is very interesting. The author proposes a noise-assigned strategy based on distance, which is simple but powerful for faster diffusion model training. The experiment and the theoretical analysis validate the core idea of the paper and show good results.\n\nSTRENGTHS:\n1. The author provides a faster diffusion model training strategy while maintaining the generation quality.\n2. The motivation is novel and interesting with good writing.\n3. The experiments show the robustness and university of the proposed method and the efficiency of the projected module.\n\nWEAKNESSES:\n1. In my view, grouping the proposed noise assignment method is similar to fixing and memorizing the relationship with noise and corresponding image, limiting the generation diversity of the diffusion model. However, the experiment shows that the FID could further be reduced, Hoping that the author can explain this phenomenon.\n2. As shown in the experiment, the noise distance change is only 2%, why such a minimal change could bring remarkable training speed up? The author should do a theory analysis in depth. \n3. To validate my concern, please provide two experiments i) providing the result generated by a noise and its minimal change variant (only 10 pixel value changes, etc). ii) providing the result generated by a certain noise with different seeds.\n4. I’m curious about the scaling law phenomenon. As shown in stable diffusion, training the diffusion model with large image data could bring surprising topological ability. However, the way the author trains the diffusion model, the noise seems to connect with certain images (same class, etc.) which will limit the generation diversity for more data. I know this experiment requires more computational resources, the author could only provide the theory analysis of the scaling law ability of the proposed method.\n\nQUESTIONS:\nSee the weakness.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose an approach to mitigate the random correspondence of noise-data mapping in vanilla diffusion models. They first assign target noise by minimizing the total image-noise pair distance within a mini-batch, and then diffuse the data into noise. Experimental results seem to demonstrate the potential of this approach. However, some of the arguments are mathematically sloppy and not well testified.\n\nSTRENGTHS:\nThe approach sounds novel and interesting. Also the implementation seems simple.\n\nWEAKNESSES:\n1. The way of presentation is not reader friendly. I suggest to review DDIM and define its notations first, explain your motivation from DDIM, and show your algorithm/solution.  \n\n2. No strong evidence showing your claim that vanilla diffusion models are miscible. Any toy examples to support it?\n\n3. The argument from L155-L161 is sloppy and hard to understand the argument. For instance, Eq. (2) holds in what sense (density? certain divergence?). Indeed, I believe it will not hold for most diffusion models.\n\n4. More justification and rationale is needed to the assumption Eq. (7). Do authors suppose that should hold for all $t$? If so, I do not think it is a correct assumption.\n\nQUESTIONS:\n1. Is the proposed method, which does in-batch reassignment, sensitive to batch size?\n\n2. Is the proposed algorithm applied to all $t_b$? In theory, all the arguments that authors make may only hold for large timesteps. Thus, the proposed method should work effectively even if it is applied to large diffusion time. It will be nice to see experimental results related to this.\n\n3. Does the proposed method work effectively with fine-tuning? \n\n4. Even though authors propose to quantize to avoid computation bottleneck, by how much the batch-wise image-noise assignment increases the runtime of training?\n\nLIMITATIONS:\nYes, authors have discussed potential limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper shows that the current diffusion training strategy diffuses each image into the entire noise space, making it difficult to optimize the model and thus slow to converge. Inspired by the fact that miscibility can be changed according to various intermolecular forces in physics, this paper proposes Immiscible Diffusion, a simple and effective method that accelerates diffusion training by pre-assigning noises in each mini-batch before standard training, which can be implemented in just one line code. This assignment operation can be accelerated by fp16/fp8, which greatly improves training efficiency while avoiding algorithm complexity and is applicable to various baselines, including Consistency models, DDIM and Stable Diffusion.\n\nSTRENGTHS:\n1. This paper points out that existing diffusion training will map each image to all points in the Gaussian distribution, which means that each point can be denoised to any source image, leading to inherent difficulties in diffusion training. This is a new perspective on diffusion training acceleration, and it sounds reasonable.\n2. The proposed assignment strategy is simple and effective, and one line of code achieves a significant performance improvement, without much computation complexity.\n3. Immiscible Diffusion improves diffusion training speed while improving the FID metric, and it is orthogonal to existing diffusion training acceleration methods.\n4. Clear and concise presentation, writing and diagrams.\n\nWEAKNESSES:\n1. The noise assignment strategy proposed in this paper essentially introduces a priori assumption into the diffusion training process: different images should correspond to different points in the Gaussian distribution. This prior reduces the difficulty of diffusion training, but also brings concerns about practicality and generalization:\n    - Practicality: The datasets for unconditioned generation used in this paper are all based on image classification (CIFAR/CelebA/ImagenNet), so these data distributions themselves already meet the above prior assumptions (each class is a kind of image distribution). However, for image datasets with more complex distributions, such as LAION, whether the proposed assignment strategy can still be used is an uncertain question, and this paper also lacks relevant experiments. You can demonstrate that this is not a problem by training unconditional generation experiments on a subset of LAION.\n    - Generalization: Since images with the same data distribution are assigned to a specific noise interval during training, this may cause the model to lose the ability that denoising each point in the noise space into any source image during inference, thereby damaging the generalization of the generated images. However, the paper lacks experiments to evaluate the generalization of generated images.\n\n2. Considering that all the unconditional generation results in the paper are trained on classification datasets, is the distribution of noise assigned by the proposed method strongly correlated with the category? Intuitively, after the proposed noise assignment, images of the same category should have closer noise in current batch.\n\nQUESTIONS:\nPlease refer to the Weaknesses\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}