Linear Combination of Saved Checkpoints Makes Consistency and Diffusion Models Better

## Abstract

Abstract Diffusion Models (DM) and Consistency Models (CM) are two types of popular generative models with good generation quality on various tasks. When training DM and CM, intermediate weight checkpoints are not fully utilized and only the last converged checkpoint is used. In this work, we find that high-quality model weights often lie in a basin which cannot be reached by SGD but can be obtained by proper checkpoint averaging. Based on these observations, we propose LCSC, a simple but effective and efficient method to enhance the performance of DM and CM, by combining checkpoints along the training trajectory with coefficients deduced from evolutionary search. We demonstrate the value of LCSC through two use cases: (a) Reducing training cost. With LCSC, we only need to train DM/CM with fewer number of iterations and/or lower batch sizes to obtain comparable sample quality with the fully trained model. For example, LCSC achieves considerable training speedups for CM (23 × \times × on CIFAR-10 and 15 × \times × on ImageNet-64). (b) Enhancing pre-trained models. Assuming full training is already done, LCSC can further improve the generation quality or speed of the final converged models. For example, LCSC achieves better performance using 1 number of function evaluation (NFE) than the base model with 2 NFE on consistency distillation, and decreases the NFE of DM from 15 to 9 while maintaining the generation quality on CIFAR-10. Our code is available at https://github.com/imagination-research/LCSC .

###### Keywords:

## 1 Introduction

Diffusion Models (DMs) [, , ] are a newly emerged paradigm of generative modeling and have gained widespread attention rapidly in the last several years. DM achieves surprising performance in various tasks like image generation [, , ], video generation [, ] and 3D generation [, ].
However, DM requires an iterative denoising process in the generation process, which could be slow.
Consistency Models (CMs) [] are proposed to handle this dilemma.
CM provides a better generation quality under one or few-step generation scenarios
and is applied in many practical cases  [, , ].

Figure: Figure 1: Comparison of LCSC and vanilla training. LCSC achieves more than 14$\times$ training speed up on CIFAR-10 with Consistency Distillation (CD) and more than 15$\times$ training speed up compared to official released model on ImageNet-64 with CD. LCSC can also enhance the final converged model significantly and achieves 1.7$\times$ inference speedup for DM.
Refer to caption: extracted/5522312/fig/speedup.png

In this paper, we investigate the training process of DM and CM. We find that checkpoints—the model weights periodically saved during the training process—have under-exploited potential in boosting the performance of DM and CM.
Specifically, there are numerous high-quality basins located near any given point along the optimization trajectory in the metric landscape. These high-quality basins *cannot* be reliably reached through Stochastic Gradient Descent (SGD), including its advanced variants like Adam []. However, we find that an appropriate linear combination of different checkpoints can locate these basins.

Inspired by the observation, we propose to find a Linear Combination of Saved Checkpoints (LCSC) that optimizes the model performance.
Specifically, for a set of checkpoints saved untill any point of the training process, LCSC searches for the optimal linear combination weights that optimize certain metrics (e.g., FID []) using an evolutionary algorithm.
Note that the widely used Exponential Moving Average (EMA) [] in DM and CM can be viewed as a special case of LCSC whose combination coefficients are determined heuristically and we will show in our experiments that EMA coefficients are sub-optimal. LCSC optimizes only in a low-dimension space and needs no back-propagation, thus could be faster than training. LCSC can optimize objectives or metrics whose gradients are hard to compute (e.g., FID), under which gradient-based optimization is not applicable. From a broader perspective, LCSC is a general method and may be used in other tasks and other neural networks.

We argue that LCSC is beneficial in all steps in the DM/CM production stage. For example, LCSC can be used to:
Reduce training cost. The training process of DM and CM is very costly. For example, training an EDM model [] on CIFAR-10 costs about 2 days with 8 V100 GPUs, and training a CM model [] costs about one week with 8 A100 GPUs. For datasets with higher resolution, the training cost is even larger. On ImageNet-64, SOTA DMs and CMs [, , ] are trained with a batch size of several thousand
on 32 or 64 A100 GPUs for about one week, consuming very huge computational resources.
*With LCSC applied at the end, we can train CM/DM with many fewer iterations or smaller batch sizes and reach similar generation quality with the fully trained model,
thereby reducing the computational cost of training.*
Enhance pre-trained models.
Assuming the full training is already done, LCSC can still be applied to get a model that is *better than any model in the training process*.
For model developers with saved checkpoints, LCSC can be directly applied.
For users who can only access the final released checkpoint, they can fine-tune the model for a few more iterations and apply LCSC on these checkpoints.
As DM/CM provides a flexible trade-off between generation quality and the number of generation steps, *the enhanced model from LCSC could lead to either better generation quality or faster generation.*

In conclusion, the contributions of this paper are as follows:

In Sec. 3, we analyze the metric landscape of DM and CM. We demonstrate that there are basins with better generation quality
that cannot be reached through traditional SGD but can be achieved by linear combinations of checkpoints. We further demonstrate the sub-optimality of the commonly used EMA, illustrating the importance of exploring better ways of weight combination.
In Sec. 4, we propose our simple but effective method LCSC to address the problem. LCSC employs evolutionary search to optimize linear combination coefficients of previous checkpoints. LCSC has two use cases: (1) reducing training cost and (2) enhancing pre-trained models. LCSC needs no back-propagation and has the potential to be applied to other tasks and models.
In Sec. 5, we validate the effectiveness of LCSC on DM and CM for the two use cases. LCSC accelerates the training process of Consistency Distillation (CD) and Consistency Training (CT) by 23$\times$ and 7$\times$ on CIFAR-10, and 15$\times$ and 10.4$\times$ on ImageNet-64. Additionally, we apply LCSC to enhance the pre-trained models. For DM, LCSC by combining pre-trained checkpoints decreases the NFE from 15 to 9 while keeping the sample quality. For CM, LCSC achieves better performance with 1 NFE compared to the vanilla final model with 2 NFE. With the same NFE, models from LCSC have much better sample quality.
In Sec. 6, we present and discuss the searched patterns of combination coefficients to inspire further research on weights combination. Then we discuss the reasons why LCSC works due to the unique properties of DM and CM.

Figure: Figure 2: A schematic diagram of LCSC. Given a set of checkpoints from training, LCSC use evolutionary search to find the optimal linear combination weights. LCSC can be applied on checkpoints from a training process with fewer training iterations or batch sizes and still gets similar performance (abbreviated as “Perf.”), thus reducing training cost and enabling faster training. LCSC can also enhance the final model in terms of generation quality or speed.
Refer to caption: extracted/5522312/fig/illustration.png

## 2 Background and Related Work

### 2.1 Diffusion Probabilistic Model

Let us denote the data distribution by $p_{data}$ and consider a diffusion process that perturbs $p_{data}$ with a stochastic differential equation (SDE) []:

$$ $d\mathbf{x}_{t}=\bm{\mu}(\mathbf{x}_{t},t)dt+\sigma(t)d\bm{w}_{t},$ (1) $$

where $\bm{\mu}(\cdot,\cdot)$ and $\sigma(\cdot)$ represent the drift and diffusion coefficients, respectively, $\bm{w}_{t}$ denotes the standard Brownian motion, and $t\in[0,T]$ indicates the time step. $t=0$ stands for the real data distribution. $\bm{\mu}(\cdot,\cdot)$ and $\sigma(\cdot)$ are designed to make sure $p_{T}(\bm{x})$ becomes pure Gaussian noise.

Diffusion models (DM) [, , , , , , ] undertake the reverse operation by initiating with $\bm{x}_{T}$ sampled from pure Gaussian noise and progressively denoising it to reconstruct the image $\bm{x}_{0}$. Importantly, SDE has its corresponding “probability flow” Ordinary Differential Equation (PF ODE) [, ], which delineates a deterministic pathway that yields the same distribution $p_{t}(\bm{x})$ for $\forall t$, thereby offering a more efficient sampling mechanism:

$$ $d\mathbf{x}_{t}=\left[\bm{\mu}(\mathbf{x}_{t},t)-\frac{1}{2}\sigma(t)^{2} \nabla\log p_{t}(\mathbf{x})\right]dt,$ (2) $$

where $\nabla\log p_{t}(\bm{x})$ is referred to as the score function of $p_{t}(\bm{x})$ []. Various ODE solvers have been introduced to further expedite the sampling process utilizing Eq. 2 or minimizing the truncation error [, , , ].

A pivotal insight within diffusion models is the realization that $\nabla\log p_{t}(\bm{x})$ can be approximated by a neural network $\bm{s}_{\bm{\theta}}(\bm{x}_{t},t)$, which can be trained using the following objective:

$$ $\mathbb{E}_{t\sim\mathcal{U}(0,T]}\mathbb{E}_{\mathbf{y}\sim p_{\text{data}}} \mathbb{E}_{\mathbf{x}_{t}\sim\mathcal{N}(\mathbf{y},\sigma(t)^{2}\mathbf{I})} \lambda(t)\|\mathbf{s}_{\bm{\theta}}(\mathbf{x}_{t},t)-\nabla_{\mathbf{x}_{t}} \log p(\mathbf{x}_{t}|\mathbf{y})\|,$ (3) $$

where $\lambda(t)$ represents the loss weighting and $\bm{y}$ denotes a training image.

### 2.2 Consistency Models

Drawing inspiration from DM theory, consistency models (CM) have been proposed to enable single-step generation [, ]. Whereas DM incrementally denoises an image, e.g., via the PF ODE, CM denoted by $f_{\bm{\theta}}$ is designed to map any point $\bm{x}_{t}$ at any given time $t$ along a PF ODE trajectory directly to the trajectory’s initial point $\bm{x}_{t}$ in a single step.

CMs are usually trained through discretized time steps, so we consider segmenting the time span from $[\epsilon,T]$ into $K-1$ sub-intervals, with $\epsilon$ being a small value approximating zero.
Training of CM can follow one of two primary methodologies: consistency distillation (CD) or direct consistency training (CT). In the case of CD, the model $f_{\bm{\theta}}$ leverages knowledge distilled from a pre-trained DM $\phi$.
The distillation loss can be formulated as follows:

$$ $\mathbb{E}_{k\sim\mathcal{U}[1,K-1]}\mathbb{E}_{\bm{y}\sim p_{data}}\mathbb{E} _{\bm{x}_{t_{k+1}}\sim\mathcal{N}(\bm{y},t_{k+1}^{2}\bm{I})}\lambda(t_{k})d(f_ {\bm{\theta}}(\bm{x}_{t_{k+1}},t_{k+1}),f_{\bm{\theta}^{-}}(\hat{\bm{x}}_{t_{k }}^{\phi},t_{k})),$ (4) $$

where $f_{\bm{\theta}^{-}}$ refers to the target model and $\bm{\theta}^{-}$ is computed through EMA (see Eq. 6) of the historical weights of $\bm{\theta}$, $\hat{\bm{x}}_{t_{k}}^{\phi}$ is estimated by the pre-trained diffusion model $\phi$ through one-step denoising based on $\bm{x}_{t_{k+1}}$, and $d$ is a metric implemented by either the $\ell_{2}$ distance or LPIPS [].

Alternatively, in the CT case, the models $f_{\bm{\theta}}$ are developed independently, without relying on any pre-trained DM:

$$ $\mathbb{E}_{k\sim\mathcal{U}[1,K-1]}\mathbb{E}_{\bm{y}\sim p_{data}}\mathbb{E} _{\bm{z}\sim\mathcal{N}(\bm{0},\bm{I})}\lambda(t_{k})d(f_{\bm{\theta}}(\bm{y}+ t_{k+1}\bm{z},t_{k+1}),f_{\bm{\theta}^{-}}(\bm{y}+t_{k}\bm{z},t_{k})),$ (5) $$

where the target model $f_{\bm{\theta}^{-}}$ is set to be the same as the model $f_{\bm{\theta}}$ in the latest improved version of the consistency training [], i.e. $\bm{\theta}^{-}=\bm{\theta}$.

### 2.3 Weight Averaging

The integration of a running average of the weights by stochastic gradient descent (SGD) was initially explored within the realm of convex optimization [, ]. This concept was later applied to the training of neural networks []. Izmailov et al.[] suggest that averaging multiple weights over the course of training can yield better generalization than SGD.
Wortsman et al.[] demonstrate the potential of averaging weights that have been fine-tuned with various hyperparameter configurations. The Exponential Moving Average (EMA) employs a specific form that uses the exponential rate $\gamma$ as a smoothing factor:

$$ $\tilde{\bm{\theta}}_{n}=\gamma\tilde{\bm{\theta}}_{n-1}+(1-\gamma)\bm{\theta}_ {n},$ (6) $$

where $n=1,\dots,N$ denotes the number of training iterations, $\tilde{\bm{\theta}}$ represents the EMA model, and is initialized with $\tilde{\bm{\theta}}_{0}=\bm{\theta}_{0}$.

Practitioners often opt for advanced optimizers such as Adam [] for different tasks and network architectures, which might reduce the need for employing a running average. However, the use of EMA has been noted to significantly enhance the quality of generation in early DM studies [, , , , ]. This empirical strategy has been adopted in most, if not all, subsequent research endeavors. Consequently, CM have also incorporated this technique, discovering EMA models that perform substantially better.

Recently, several works of more advanced weight averaging strategies have been proposed for Large Language Models (LLMs). Most of them focus on merging models fine-tuned for different downstream tasks to create a new model with multiple capabilities [, , , ]. Different from these approaches, our work aims to accelerate the model convergence and achieve better performance in a standalone training process. Moreover, the methodology we employ for determining averaging coefficients is novel, thereby distinguishing our method from these related works. Further details can be found in Sec. 3 and Sec. 4.

### 2.4 Search-based Methods for Diffusion Models

Search algorithms are widely used across various domains like Neural Architecture Search (NAS) [, ], where they are employed to identify specific targets. Recently, many works studies have set discrete optimization dimensions for DMs and utilized search methods to unearth optimal solutions. For example, Liu et al. [], Li et al. [] and Yang et al. [] use search methods to find the best model schedule for DMs. Liu et al. [] apply search methods to find appropriate strategies for diffusion solvers. However, these works do not involve modification to model weights during search and are only applicable to DMs.

## 3 Motivation

Figure: (a) Metric Landscape of DM.
Refer to caption: x1.png

One notable challenge associated with DMs and CMs is their difficulty in training. We observe that the objectives for DMs and CMs tend to produce substantial variance in gradient estimations, leading to fluctuations in the quality of generation by the resulting models. For DMs (refer to Eq. 3), even if the networks $\bm{s}_{\bm{\theta}}$ are finely tuned to perfectly mirror the score function $\nabla\log p_{t}(\bm{x})$ for any given $\bm{x}_{t}$, the loss as defined does not reduce to zero. As a result, the batched gradient estimation for this objective exhibits high variance. For CMs (refer to Eqs. 4 and 5), at any given time $t$, the network $f_{\bm{\theta}}$ is solely optimized to align with the target model $f_{\bm{\theta}^{-}}$ from the previous time step, $t-1$. This implies that any loss of image information in the early time steps could result in inaccurate gradient estimation in subsequent time steps.

To explore the training challenges of DMs and CMs, we visualize the landscape of generation quality in a 2-dimension (2D) plane. Specifically, we select 3 checkpoints of the weights $\theta_{n_{0}},\theta_{n_{1}},\theta_{n_{2}}$ from the same training trajectory. We then sweep across the 2D space spanned by these three points to assess the FID scores.
The results are shown in the first row of Fig. 3. As we can see, there exist model weights with much higher generation quality than any of $\{\theta_{n_{0}},\theta_{n_{1}},\theta_{n_{2}}\}$ in the sub-space spanned by those three checkpoints for both DM and CM. Additionally, results at different training iterations indicate that this pattern of landscape remains stable in different training stages. These insights imply that there exists a substantial opportunity to enhance the model’s performance at any training phase through the linear combination of existing weight checkpoints.

EMA [] is simple yet widely used technique in the training of DMs and CMs. It operates by exponentially averaging the weights of all previous checkpoints (see Eq. 6), which stabilizes the training process of DMs and CMs to a certain extent. Nonetheless, EMA represents an empirical method of averaging, suggesting potential areas for enhancement. To investigate this hypothesis, we conducted an experiment where the three checkpoints from previous tests were substituted with three EMA weights, followed by a comprehensive re-sweep of all coordinates. The results are demonstrated in the second row of Fig. 3.
Notably, it is observed that a linear combination of the three EMA weights could achieve superior performance compared to each individual weight. It is important to highlight that the linear combination of three EMA weights at various training iterations cannot be replicated by simply adjusting EMA rates. Consequently, this suggests that EMA may indeed be a sub-optimal solution, indicating possibilities for further improvement in this area.

Based on the above insights and the observation, we propose LCSC, standing for Linear Combination of Saved Checkpoints, to enhance the training process. Since the basin in Fig. 3 extends beyond the convex hull of the model weights, we further expand the scope of the combination coefficients to encompass any real values and introduce an optimization method to search for the optimal linear combination of the weights.

## 4 Method

We discuss LCSC in this section. The search problem is defined in Sec. 4.1. In Sec. 4.2, we elaborate on our algorithm. Finally, we list several typical use cases of our method in Sec. 4.3.

### 4.1 Problem Definition

Let $\{1,2,\ldots,N\}$ denote the set of training iterations. Suppose $\{n_{1},n_{2},\ldots,n_{K}\}$ is a subsequence selected from $\{1,2,\ldots,N\}$. Define $\Theta$ as the set of checkpoints saved at these specific training iterations: $\Theta=\{\bm{\theta}_{n_{1}},\bm{\theta}_{n_{2}},\bm{\theta}_{n_{3}},\cdots,
\bm{\theta}_{n_{K}}\}$. Given $\Theta$, we aim to find a group of coefficients $\bm{\alpha}=\{a_{1},a_{2},a_{3},\cdots,a_{K}\}$. These coefficients are intended to linearly combine all the checkpoints in such a way that the resulting model achieves the optimal performance:

$$ $\displaystyle\mathop{\arg\min}\limits_{\begin{subarray}{c}a_{1},a_{2},a_{3}, \cdots,a_{K}\in\mathbb{R}\end{subarray}}\mathcal{F}(a_{1}\bm{\theta}_{n_{1}}+a _{2}\bm{\theta}_{n_{2}}+a_{3}\bm{\theta}_{n_{3}},\cdots,a_{K}\bm{\theta}_{n_{K }});\quad\text{s.t.}\quad\sum_{i=1}^{K}a_{i}=1,$ (8) $$

where $\mathcal{F}(\bm{\theta})$ denotes an evaluation function that measures the generation quality of $\bm{\theta}$ regarding a specific metric, indicating higher quality with a smaller value. The constraint $\sum_{i=1}^{K}a_{i}=1$ reduces the dimension of our search space by 1. However, we find it enables a more effective exploration for the search algorithm.

### 4.2 Evolutionary Search

When the number of combined checkpoints $K$ is large (e.g., ranges approximately from 200 to 500 in our experiments), the grid search method used in Sec. 3 becomes inefficient. Consequently, we propose to use evolutionary search  [, , , , ] to solve the problem.

Our detailed algorithm can be found in Alg. 1. Given a set of checkpoints $\Theta$, we can apply this algorithm at any training iteration $n$ along the training trajectory. We initialize the population using EMA coefficients corresponding to several different rates. Then we conduct an evolutionary search which views a group of coefficients as an individual. In each search iteration, only the top-performing individuals are selected as parents to reproduce the next generation through crossover and mutation. Specifically, during crossover, we randomly mix the coefficients of the two parents or select one of them. For mutation, we introduce random Gaussian noise to each coefficient. This stochastic element ensures that positive adaptations in the coefficients are conserved and advanced to the subsequent generation, whereas detrimental modifications are discarded.
Finally, we normalize the new coefficients to adhere to the regularization condition in Eq. 8. We repeat this reproduction process for a predetermined number of times within each search iteration and update the whole population with all newly generated individuals. Upon the completion of the search process, we choose the best individual as the output of our search algorithm.

Figure: Algorithm 1 Evolutionary Search for Combination Coefficients Optimization

Given the absence of neural back-propagation in the search process, LCSC can be viewed as a gradient-free optimization method, akin to methods discussed in [, ].
This characteristic renders LCSC highly efficient in terms of computation, offering significant savings in GPU memory usage (applicable to both DM and CM) and computational time (specifically for CM, given that the inference process for DM is notably resource-intensive). Furthermore, LCSC possesses the unique advantage of optimizing models directly towards non-derivable metrics where gradient-based methods falter. It is also important to underscore that the applicability of LCSC extends beyond CM and DMs to various other tasks and different models.

### 4.3 Use Cases

In this section, we delve into the use cases of our method.
(1) Our method can be employed during the initial phases of the training process or in scenarios where the training is conducted with a smaller batch size. This application aims to expedite the training process; (2) At the final stage of training, when the model has essentially converged, our method can be utilized to further enhance the model’s performance. This involves refining the output quality of the already converged model and potentially reducing NFE.

Figure: Figure 4: Training curves on CIFAR-10 with CM and DM. P1, P2 represent the first phase and the second phase.
Refer to caption: x3.png

Decrease Training Cost.
The consumption of training resources is directly influenced by both the number of iterations and the batch size. Through our investigation, we have discerned that the training process for Diffusion Models (DM) and Consistency Models (CM) can be categorized into two distinct phases, as illustrated in Fig. 4. The initial phase is relatively brief, during which DM and CM rapidly acquire the capability to generate visually satisfactory images. In contrast, the second phase is characterized by a slower optimization of models, focusing on the enhancement of sample quality. Implementing LCSC at the onset of this second phase can facilitate an efficient identification of high-quality basins, potentially yielding performance that is on par with, or even surpasses, that of the final converged model. This strategy effectively reduces the total number of training iterations required. Regarding *batch size*, SOTA DMs and CMs necessitate training with large batch sizes [, , ], which places a substantial demand on hardware and computational resources. Decreasing the batch size often leads to worse convergence performance. LCSC can be used for low batch size training and achieve equivalent performance to models trained with full batch size.

Enhance Converged Models.
LCSC can also be applied to the checkpoints saved during the final stage of training to enhance the performance of converged models.
Furthermore, for users who have access to released pre-trained models, we suggest fine-tuning the model for a few iterations and saving the intermediate checkpoints. Then LCSC can be used to enhance the released model.

## 5 Experiments

In this section we present our experiments to validate the effectiveness of LCSC. In Sec. 5.1, we introduce the experimental settings. Then we demonstrate our results on CM and DM for the two use cases in Sec. 5.2.1. We finally ablate several important hyper-parameters in our workflow in Sec. 5.3.

### 5.1 Experimental Setup

For DM, we follow DDIM [] for the evaluation on CIFAR10 [] and iDDPM [] for the evaluation on ImageNet-64 []. To improve the inference efficiency, we adopt DPM-Solver []. For CM [], we evaluate LCSC with both CD and CT on CIFAR-10 and ImageNet-64. We train models with our own implementation. We follow all the settings reported by the official paper except ImageNet-64, on which we decrease the batch size to 256 on CM and 512 on DM due to the limited resources. We applied LCSC at different training stages.
Specifically, for any selected iteration, we utilize checkpoints from every predetermined interval of iterations within a defined window size.
Then we run a search process to find the optimal combination coefficients.

Evaluation. We choose the most commonly used metric, FID [], as the search objective. To evaluate the combination coefficients during the evolutionary search, we sample 5K images on ImageNet-64 with DM and 10K in other settings, using a fixed group of initial noise. For the final model evaluation, we generate 50K samples using another group of initial noises. To demonstrate that LCSC is not overfitted to FID but achieves overall improvement, we report on FID, IS [], precision [], and recall []. In Sec. 0.B.5, we further report FID@Clip to show that LCSC also brings improvements in other feature spaces.

Baselines. Since EMA is the default setting in almost all works of DM and CM, we report the performance of the EMA weights using the rate reported by official papers [, , ] as our main baselines. For CM models on ImageNet-64, we additionally download the official models trained with full batch size and test their performance for a complete comparison. We further conduct a grid search of EMA rate for the final model as a stronger baseline, denoted as EMA*, to prove the sub-optimality of EMA and that LCSC is a more effective method. More experimental details are provided in App. 0.B.

### 5.2 Results

Our results for CM are shown in Tab. 1 and Tab. 2, while for DM, the results are presented in Tab. 3. By conducting LCSC, we consistently find models that outperform the current EMA model across all settings. Although LCSC searches according to FID, consistent improvements are observed across different metrics. Notably, improvement is evident even at the early stages of training, e.g., 50K iterations, when the model is not yet fully trained. Below, we discuss our results under two use cases: decreasing training costs and enhancing the final converged model. Generated images are visualized in App. 0.E.

#### 5.2.1 Results on Decreasing the Training Cost

We conduct a series of experiments using CD and CT to validate that LCSC can significantly reduce training costs. Our reported speedup results have been adjusted to include the search cost on both CPU and GPU, with detailed information available in Sec. 0.B.2.

LCSC can be applied at an early training stage to decrease the number of training iterations. As illustrated in Tab. 1, for CIFAR-10, applying LCSC to the model with only 50K training iterations achieves a better FID than the final model trained with 800K iterations (3.18 vs. 3.65) while accelerating the training by approximately 14× for CD. For CT, we only achieve a converged FID of 9.87 with vanilla training and fail to fully reproduce the result of FID=8.70 reported by Song et al. []. However, by applying our method at 400K training iterations, we can still surpass the reported FID by Song et al. [] with 800K training iterations, achieving around a 1.9× speedup. On ImageNet-64, we decrease the batch size from 2048 to 256 due to limited resources. For CD, LCSC achieves better performance (6.56, 5.71 vs. 7.30 in FID) than the final converged model with only a quarter or half of the training iterations (150K, 300K vs. 600K). For CT, LCSC also significantly reduces the final converged FID (12.05 vs. 15.56) with fewer training iterations (600K vs. 800K).

LCSC can also contribute to reducing the training batch size. On CIFAR-10, we train another model with a lower batch size to validate LCSC’s performance under these conditions. As illustrated in Tab. 1, LCSC with a batch size of 128 outperforms the final converged model with a batch size of 512 for both CD and CT (3.34 vs. 3.65 with CD and 8.33 vs. 8.70, 9.87 with CT), achieving overall speedups of 23× and 7×, respectively. On ImageNet-64, we test the official models of Song et al. [], which were trained with a 2048 batch size for both CD and CT. With the assistance of LCSC, models trained with a 256 batch size consistently outperform the official models (5.71, 5.54 vs. 6.31 with CD and 12.05 vs. 13.07 with CT), achieving overall acceleration ratios of 15× and 10.4×.

**Table 1: Generation quality of consistency models on CIFAR-10. The training speedup is compared against the standard training with 800K iterations and 512 batch size. Our results that beat or match the standard training (the “released” model for FID and IS, and our reproduced results for Prec. and Rec.) are underlined.**
| Model | Method | Training Iter | Batch Size | NFE | FID($\downarrow$) | IS($\uparrow$) | Prec.($\uparrow$) | Rec.($\uparrow$) | Speed($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| CD | EMA | 50K | 512 | 1 | 4.84 | 9.13 | 0.67 | 0.55 |  |
| 200K | 512 | 1 | 4.08 | 9.18 | 0.68 | 0.56 |  |  |  |
| 400K | 512 | 1 | 3.84 | 9.26 | 0.68 | 0.57 |  |  |  |
| 800K | 512 | 1 | 3.66 | 9.35 | 0.68 | 0.57 |  |  |  |
| 800K | 512 | 2 | 2.90 | 9.56 | 0.69 | 0.59 |  |  |  |
| 850K | 512 | 1 | 3.65 | 9.32 | 0.68 | 0.57 |  |  |  |
| 850K | 512 | 2 | 2.89 | 9.55 | 0.69 | 0.58 |  |  |  |
| (released) | 800K | 512 | 1 | 3.55 | 9.48 | - | - |  |  |
| (released) | 800K | 512 | 2 | 2.93 | 9.75 | - | - |  |  |
| EMA* | 800K | 512 | 1 | 3.51 | 9.37 | 0.68 | 0.57 |  |  |
| LCSC | 50K | 512 | 1 | 3.18 | 9.60 | 0.67 | 0.58 | $\sim$14$\times$ |  |
| 250K | 512 | 1 | 2.76 | 9.71 | 0.67 | 0.59 | $\sim$3.1$\times$ |  |  |
| 800K | 512 | 1 | 2.42 | 9.76 | 0.67 | 0.60 | - |  |  |
| 800+40K | 512 | 1 | 2.38 | 9.70 | 0.67 | 0.60 | - |  |  |
| 100K | 128 | 1 | 3.34 | 9.51 | 0.67 | 0.57 | $\sim$23$\times$ |  |  |
| CT | EMA | 200K | 512 | 1 | 16.4 | 8.22 | 0.64 | 0.42 |  |
| 400K | 512 | 1 | 12.1 | 8.52 | 0.67 | 0.43 |  |  |  |
| 600K | 512 | 1 | 10.8 | 8.67 | 0.68 | 0.42 |  |  |  |
| 800K | 512 | 1 | 9.87 | 8.81 | 0.69 | 0.42 |  |  |  |
| (released) | 800K | 512 | 1 | 8.70 | 8.49 | - | - |  |  |
| EMA* | 800K | 512 | 1 | 9.70 | 8.81 | 0.69 | 0.42 |  |  |
| LCSC | 400K | 512 | 1 | 8.60 | 8.89 | 0.67 | 0.47 | $\sim$1.9$\times$ |  |
| 800+40K | 128 | 1 | 8.05 | 8.98 | 0.70 | 0.45 | - |  |  |
|  | 450K | 128 | 1 | 8.33 | 8.67 | 0.69 | 0.44 | $\sim$7$\times$ |  |

**Table 2: Generation quality of consistency models on ImageNet-64. For CD, the speedup is compared against the standard training with 600K iterations and 2048 batch size. For CT, the speedup is compared against the standard training with 800K iterations and 2048 batch size. Our results that beat the standard training (“released”) are underlined.**
| Model | Method | Training Iter | Batch Size | NFE | FID($\downarrow$) | IS($\uparrow$) | Prec.($\uparrow$) | Rec.($\uparrow$) | Speed($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| CD | EMA | 150K | 256 | 1 | 8.24 | 36.2 | 0.66 | 0.61 |  |
| 300K | 256 | 1 | 7.70 | 37.0 | 0.67 | 0.62 |  |  |  |
| 600K | 256 | 1 | 7.30 | 37.2 | 0.67 | 0.62 |  |  |  |
| 650K | 256 | 1 | 7.17 | 37.7 | 0.67 | 0.62 |  |  |  |
| (released) | 600K | 2048 | 1 | 6.31 | 39.5 | 0.68 | 0.63 |  |  |
| EMA* | 600K | 256 | 1 | 7.17 | 37.7 | 0.67 | 0.62 |  |  |
| LCSC | 150K | 256 | 1 | 6.56 | 38.9 | 0.67 | 0.62 | $\sim$29$\times$ |  |
| 300K | 256 | 1 | 5.71 | 41.8 | 0.68 | 0.62 | $\sim$15$\times$ |  |  |
| 600+20K | 256 | 1 | 5.54 | 40.9 | 0.68 | 0.62 | $\sim$7.6$\times$ |  |  |
| CT | EMA | 200K | 256 | 1 | 20.3 | 28.8 | 0.61 | 0.53 |  |
| 400K | 256 | 1 | 17.9 | 29.8 | 0.62 | 0.54 |  |  |  |
| 600K | 256 | 1 | 16.6 | 30.6 | 0.62 | 0.54 |  |  |  |
| 800K | 256 | 1 | 15.8 | 31.1 | 0.64 | 0.55 |  |  |  |
| 1000K | 256 | 1 | 15.6 | 31.2 | 0.64 | 0.55 |  |  |  |
| (released) | 800K | 2048 | 1 | 13.1 | 29.2 | 0.70 | 0.47 |  |  |
| EMA* | 1000K | 256 | 1 | 15.6 | 31.2 | 0.64 | 0.55 |  |  |
| LCSC | 600K | 256 | 1 | 12.1 | 35.1 | 0.67 | 0.54 | $\sim$10.4$\times$ |  |
|  | 1000K | 256 | 1 | 11.1 | 35.7 | 0.65 | 0.57 | $\sim$6.3$\times$ |  |

#### 5.2.2 Results on Enhancing Pre-trained Models

Our experiments on both DM and CM show that LCSC can significantly boost the performance of final converged models and reduce the required sampling steps.

For CM, on CIFAR-10 with CD, CT and ImageNet-64 with CD, to simulate scenarios where users have access only to pre-trained models, we fine-tune the final converged model for a few iterations and then apply LCSC. As illustrated in the 800+40K LCSC row with CD and CT in Tab. 1 and the 600+20K LCSC row with CD in Tab. 2, this approach significantly enhances the performance of the already converged model. For model developers, we directly utilize the checkpoints close to the final iteration, as demonstrated in the 800K LCSC row with CD in Tab. 1 and 1000K LCSC row with CT in Tab. 2. LCSC leads to notable improvements in all these scenarios. Notably, on CIFAR-10, LCSC even outperforms the 800K and 850K models that use 2-step sampling, achieving better performance with 1-step sampling (2.42, 2.38 FID vs. 2.89, 2.90 FID), thereby doubling the inference speed.

For DM, when applying LCSC at the final stage of training, we can identify a model that performs significantly better than the final model obtained through EMA, using the EMA rate selected by the baseline. While a better EMA model may be discovered through an extensive grid search, the outcomes achieved by LCSC consistently surpass those obtained via the EMA strategy, thereby underscoring the superiority of our method. Moreover, we observe that the model identified through our approach can match the performance of the best EMA model with fewer NFE during inference. Specifically, on CIFAR-10, our method requiring NFE=9 is on par with EMA* using NFE=15. Similarly, on ImageNet-64, our method with NFE=12 is competitive with EMA* utilizing NFE=15. This highlights the potential for reducing inference costs while still achieving the baseline’s performance level.

**Table 3: Generation quality of diffusion models. Our results that beat the standard training are underlined.**
| Dataset | Method | Training Iter. | NFE | FID($\downarrow$) | IS($\uparrow$) | Prec.($\uparrow$) | Rec.($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| CIFAR10 | EMA | 50K | 15 | 80.9 | 5.51 | 0.52 | 0.30 |
| 150K | 15 | 6.28 | 8.74 | 0.61 | 0.59 |  |  |
| 350K | 15 | 4.45 | 9.23 | 0.63 | 0.59 |  |  |
| 800K | 15 | 4.16 | 9.37 | 0.64 | 0.60 |  |  |
| EMA* | 800K | 15 | 3.96 | 9.50 | 0.64 | 0.60 |  |
| LCSC | 50K | 15 | 10.8 | 8.35 | 0.55 | 0.58 |  |
| 150K | 15 | 4.76 | 8.97 | 0.61 | 0.59 |  |  |
| 350K | 15 | 3.56 | 9.35 | 0.64 | 0.60 |  |  |
| 800K | 15 | 3.18 | 9.59 | 0.64 | 0.61 |  |  |
| 800K | 12 | 3.36 | 9.56 | 0.64 | 0.60 |  |  |
| 800K | 9 | 3.97 | 9.50 | 0.63 | 0.60 |  |  |
| ImageNet | EMA | 50K | 15 | 59.8 | 11.0 | 0.40 | 0.38 |
| 150K | 15 | 22.3 | 15.0 | 0.55 | 0.45 |  |  |
| 350K | 15 | 20.0 | 16.4 | 0.58 | 0.59 |  |  |
| 500K | 15 | 19.8 | 16.9 | 0.58 | 0.59 |  |  |
| EMA* | 500K | 15 | 18.1 | 17.3 | 0.59 | 0.59 |  |
| LCSC | 50K | 15 | 27.2 | 12.9 | 0.47 | 0.56 |  |
| 150K | 15 | 19.1 | 15.3 | 0.56 | 0.56 |  |  |
| 350K | 15 | 16.8 | 16.9 | 0.59 | 0.59 |  |  |
| 500K | 15 | 15.3 | 17.6 | 0.59 | 0.59 |  |  |
| 500K | 12 | 17.2 | 17.2 | 0.57 | 0.59 |  |  |
| 500K | 10 | 20.5 | 16.5 | 0.55 | 0.58 |  |  |

### 5.3 Hyperparameter Study

The hyperparameters of LCSC includes the window size ($n_{K}-n_{1}$) for retrieving the historical weights, the interval ($n_{k}-n_{k-1}$) for two adjacent weight checkpoints, the number of samples for computing FID and the number of search iterations.
Table 4 llustrates the impact of varying the number of samples, search iterations, and the size of the interval. We observe that increasing the number of samples and search iterations leads to higher performance, though this comes at the expense of increased search cost. In practice, we find that 10K samples and 2K search iterations can efficiently identify strong models. With a fixed window size, the interval between two checkpoints determines the dimension of the search space. Our findings suggest that search performance generally improves as the search space expands, but limiting the search dimension to fewer than 200 can detrimentally affect search performance. A more comprehensive study of the hyperparameters is provided in Sec. 0.B.3.

**Table 4: Hyperparameter study of LCSC with (a) DM and (b) CM. The models are trained on CIFAR10 with 250K iterations. We evaluate three values of each hyperparameter and compare them with our adopted setting highlighted in gray. The varied hyperparameter is in bold. For DM the window size is 50K, for CM it is 40K.**
| Samples | Search Iters. | Interval | FID($\downarrow$) | IS($\uparrow$) |
| --- | --- | --- | --- | --- |
| 10K | 2K | 200 | 3.87 | 9.27 |
| 2K | 2K | 200 | 9.23 | 9.11 |
| 5K | 2K | 200 | 4.04 | 9.09 |
| 10K | 1K | 200 | 4.01 | 9.17 |
| 10K | 4K | 200 | 3.79 | 9.28 |
| 10K | 2K | 100 | 3.82 | 9.15 |
| 10K | 2K | 500 | 4.41 | 9.11 |

## 6 Discussion

Figure: Figure 5: Visualization of weight combination coefficients obtained using LCSC compared to those from the default EMA on CIFAR10.
Refer to caption: x4.png

In this section, we briefly discuss (1) Search results obtained by LCSC and (2) The reasons why LCSC performs better than traditional SGD.

Analysis of Search Patterns. We visualize several searched combination coefficients in Fig. 5. First, we observe a small subset of weights characterized by coefficients of large magnitude, whereas the coefficients for the majority of weights are nearly zero. This pattern suggests that the basin of the metric landscape likely occupies a subspace of lower rank compared to our entire search space. Additionally, the presence of multiple significant negative coefficients highlights that certain weights act as critical negative examples. This finding further implies that traditional weight averaging methods, such as EMA, are suboptimal, given that their results are confined to the convex hull of the weights, which excludes the discovered basin. More examples illustrating different scenarios are available in Sec. 0.C.1.

LCSC vs.  Graient-based Methods. Our experiments in Sec. 5 have shown the promising benefits of LCSC. Specifically, it can find a basin in the metric landscape based on the weights obtained in early or middle training stages. An intriguing question arises regarding why gradient-based methods do not converge to the basin found by LCSC. In this work, we offer some insights from our research and defer a more comprehensive theoretical investigation to future studies. First, the objective of LCSC is different from gradient-based method. In this work, we employ FID as the search objective as it turns out to be efficient. In principle LCSC can search any objective while gradient-based methods require the objective to be differentiable. It is worth noting that minimizing $\ell_{2}$ loss, which is commonly used in gradient-based optimization, does not necessarily lead to higher generation quality [, ]. Second, DM and CM have a large variance in their gradient estimation as discussed in Sec. 3. During training, they may not be able to stay stationary within the basin we have found if the basin is small. A more detailed discussion is provided in Sec. 0.C.1.3.

## 7 Conclusion

This work proposes a simple but effective method, LCSC, to enhance the training process of DMs and CMs. Based on observations of the metric landscape, we propose to linearly combine saved checkpoints during training to achieve better performance. We further use evolutionary search to find the optimal group of combination coefficients. Our method can be used to decrease the training iteration and batch size, or enhance the final converged models.

## Acknowledgement

This work was supported by National Natural Science Foundation of China (No. 62325405, 62104128, U19B2019, U21B2031, 61832007, 62204164), Flemish Government (AI Research Program) and the Research Foundation - Flanders (FWO) through project number G0G2921N, Tsinghua EE Xilinx AI Research Fund, and Beijing National Research Center for Information Science and Technology (BNRist). We thank for all the support from Infinigence-AI.

## APPENDIX L inear C ombination of S aved C heckpoints Makes Consistency and Diffusion Models Better

In this appendix, we provide additional examples of the metric landscape in App. 0.A. We further elaborate on our experimental setup and present additional results in App. 0.B. A more in-depth discussion of our work is provided in App. 0.C. Potential directions for future work are explored in App. 0.D. Finally, we offer visualizations of the generated images in App. 0.E.
We will open-source our code to promote the application of LCSC.

## Appendix 0.A Additional Examples of the Metric Landscape

We previously introduced the metric landscape of the DM and CD models in Sec. 3. In this section, we extend our analysis by evaluating additional metric landscapes using the FID. Specifically, we incorporate an additional intermediate training iteration and explore the metric landscape of the CT model. The comprehensive landscapes of the DM, CD, and CT models are depicted in Fig. 6.

Figure: (a) Metric landscapes for DM.
Refer to caption: x5.png

## Appendix 0.B Experimental Details and Additional Results

### 0.B.1 Experimental Details

#### 0.B.1.1 Training Setting

Since LCSC employs weight checkpoints, we endeavored to replicate the training processes of the baseline models. For DMs, we were able to reproduce the results reported in the original papers successfully. However, for CMs, our training outcomes on CIFAR10 were slightly inferior to those documented in the original papers. Additionally, due to resource constraints, we opted for a smaller batch size than the original configuration when training on ImageNet. In the results tables, we present both our training outcomes and the results reported in the original papers, with the latter indicated as *released*. The specifics of the experimental setups are detailed below.

For the experiments with DM, we utilize the DDIM [] codebase ([https://github.com/ermongroup/ddim](https://github.com/ermongroup/ddim)) on CIFAR-10, adhering to the default configuration settings. For ImageNet-64, we employ the iDDPM [] codebase ([https://github.com/openai/improved-diffusion](https://github.com/openai/improved-diffusion)), setting the batch size to 512, the noise schedule to cosine, and maintaining other hyperparameters at their default values. During sampling, the DPM-Solver [] ([https://github.com/LuChengTHU/dpm-solver](https://github.com/LuChengTHU/dpm-solver)) is applied, conducting 15 timesteps of denoising using the default configuration for the respective setting. The reproduced results are on par with or slightly surpass those achieved using Euler integration, as reported in the original papers.

For the experiments involving CM, we adhere closely to the configurations detailed in Table 3 of the original CM paper []. On CIFAR-10, instead of using the NCSNPP model [] from CM’s official implementation, we opt for the EDM [] architecture. This decision stems from the official CM code for CIFAR-10 being implemented in JAX while our experiments are conducted using PyTorch. For further details, refer to [https://github.com/openai/consistency_models_cifar10](https://github.com/openai/consistency_models_cifar10). This difference in implementation may partly explain the discrepancies between our replication results and those reported in Table 1 of CM’s original paper []. On ImageNet-64, we follow the official implementation, employing the ADM [] architecture. Due to resource constraints, we train the model with a smaller batch size of 256 instead of 2048 as used in the original study, resulting in worse outcomes. However, as indicated in Tab. 2, applying LCSC to models trained with a reduced batch size can still achieve performance comparable to models trained with a larger batch size.

#### 0.B.1.2 Search Setting

At each selected training iteration, historical weights are leveraged within designated window sizes and intervals. For CM, checkpoints have a window size of 40K with an interval of 100 on CIFAR-10, and a window size of 20K with an interval of 100 on ImageNet-64. DM is assigned a window size of 50K and an interval of 200 for both datasets. FID calculation for DM on ImageNet-64 utilizes a sample of 5K images, while 10K images are used for all other configurations. An evolutionary search spanning 2K iterations is applied consistently across all experimental setups.

### 0.B.2 Search Cost Estimation

We detail the estimation of search costs, covering both CPU and GPU time consumption. Saved checkpoints are loaded into CPU memory and then transferred to the GPU after averaging. Consequently, CPU time comprises the duration for merging weight checkpoints and calculating the FID statistic. Meanwhile, GPU time is dedicated to the sampling and evaluation processes involving the diffusion U-Net and the Inception network. All experiments are performed on a single NVIDIA A100 GPU, paired with an Intel Xeon Platinum 8385P CPU.

**Table 5: Search time consumption per search iteration (in seconds). Results marked with * denote the CPU time cost, which is excludable from the overall time cost through parallel processing. “U-Net” refers to the denoising sampling process, “Inception” refers to the computation of inception features, “Merging” refers to the averaging of weights, and “FID” refers to the computation of FID statistics.**
| Model | Dataset | U-Net | Inception | Merging* | FID* |
| --- | --- | --- | --- | --- | --- |
| CM | CIFAR-10 | 7.34 | 4.28 | 5.57 | 2.49 |
| ImageNet-64 | 34.7 | 4.32 | 12.0 | 2.37 |  |
| DM | CIFAR-10 | 40.9 | 4.39 | 4.61 | 2.49 |
| ImageNet-64 | 114.5 | 2.24 | 13.8 | 2.53 |  |

First, we profile the durations of each function across all experimental settings and document the time consumed in a single search iteration in Tab. 5. We note that the CPU time cost is significantly lower than that on the GPU. Therefore, while conducting sampling with the current merged weights on the GPU, we can simultaneously perform the FID statistic computation from the previous iteration and merge the checkpoints for the next iteration on the CPU. As a result, the CPU time cost can be effectively excluded from the overall time cost.

Next, we examine the overall training and search time costs as outlined in Tab. 6, and we delve into the speedup ratio detailed in Tabs. 7 and 8. Given that DMs necessitate multiple timesteps for sampling, the overall search cost for DM proves to be non-negligible when compared to the training cost. This results in no observable speedup in convergence upon applying LCSC. However, this search cost is considered manageable and warranted for the anticipated improvements to the final model convergence.

In contrast, for CMs, the search cost is substantially lower than the training cost, which often results in a markedly faster convergence when combining training with LCSC than with training alone. The precise speedup ratios for CM training are detailed in Tabs. 7 and 8. It should be noted that the figures in these tables may exhibit slight discrepancies from those reported in Sec. 5 due to the latter being profiled in an environment with more variables and potential fluctuations.

**Table 6: Overall search time and training time consumption (in hours).**
| Model | Dataset | Search | Training |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| Iteration | Time | Iteration | Batch Size | Time |  |  |
| CD | CIFAR-10 | 2K | 6.45 | 800K | 512 | 818 |
| ImageNet-64 | 2K | 21.7 | 600K | 2048 | 7253 |  |
| CT | CIFAR-10 | 2K | 6.45 | 800K | 512 | 640 |
| ImageNet-64 | 2K | 21.7 | 800K | 2048 | 7040 |  |
| DM | CIFAR-10 | 2K | 25.2 | 800K | 128 | 29.3 |
| ImageNet-64 | 2K | 64.8 | 500K | 512 | 372 |  |

**Table 7: Accurate training speedup of consistency models on CIFAR-10. The speedup is compared against the standard training with 800K iterations and 512 batch size.**
| Model | Method | Training Iter | Batch Size | Speedup($\uparrow$) |
| --- | --- | --- | --- | --- |
| CD | EMA | 800K | 512 | - |
| LCSC | 50K | 512 | 14.27$\times$ |  |
| 250K | 512 | 3.12$\times$ |  |  |
| 100K | 128 | 25.55$\times$ |  |  |
| CT | EMA (released) | 800K | 512 | - |
| LCSC | 400K | 512 | 1.96$\times$ |  |
| 450K | 128 | 6.64$\times$ |  |  |

**Table 8: Accurate training speedup of consistency models on ImageNet-64. For CD, the speedup is compared against standard training with 600K iterations and 2048 batch size. For CT, the speedup is compared against standard training with 800K iterations and 2048 batch size.**
| Model | Method | Training Iter | Batch Size | Speedup($\uparrow$) |
| --- | --- | --- | --- | --- |
| CD | EMA (released) | 600K | 2048 | - |
| LCSC | 150K | 256 | 29.20$\times$ |  |
| 300K | 256 | 15.27$\times$ |  |  |
|  | 620K | 256 | 7.55$\times$ |  |
| CT | EMA (released) | 800K | 2048 | - |
| LCSC | 600K | 256 | 10.33$\times$ |  |
|  | 1000K | 256 | 6.27$\times$ |  |

### 0.B.3 Hyperparameter Study

In Sec. 5.3, we investigate the sensitivity of several hyperparameters, including the number of search iterations, the number of samples, and the interval between two weight checkpoints $(n_{k}-n_{k-1})$. This section focuses on the impact of window size $(n_{K}-n_{1})$ on the performance of our search algorithm. Given that DM operates iteratively during sampling and a larger NFE (i.e. the number of timesteps for the ODE solver) typically leads to higher generation quality, we also examine the effects of varying NFE during the search within the DM context. The findings are detailed in Table 9.

The results reveal that, despite earlier models being further from convergence, a sufficiently large window for accessing historical weights proves advantageous. Furthermore, conducting a lower NFE during the search in the DM context results in a similar reduction in FID but a smaller improvement in IS. This suggests that the search output’s generality across different metrics diminishes when the generated samples during the search are less accurate, i.e., exhibit significant truncation error.

**Table 9: Hyperparameter analysis of LCSC, supplementing the study in Tab. 4. Models are trained on CIFAR10 for 250K iterations. We assess three different values for each hyperparameter, contrasting these with our chosen setting, which is highlighted in gray. The hyperparameter under variation is indicated in bold. For DM, the interval between checkpoints is set to 200, whereas for CM, it is 100. We perform 2K search iterations and sample 10K images to compute the FID score at each iteration.**
| Method | Window Size | NFE | FID($\downarrow$) | IS($\uparrow$) |
| --- | --- | --- | --- | --- |
| CM | 40K | 1 | 2.76 | 9.71 |
| 10K | 1 | 2.89 | 9.60 |  |
| 50K | 1 | 2.73 | 9.66 |  |
| DM | 50K | 15 | 3.87 | 9.27 |
| 10K | 15 | 4.41 | 9.11 |  |
| 30K | 15 | 3.82 | 9.15 |  |
| 50K | 7 | 3.91 | 9.10 |  |
| 50K | 10 | 3.84 | 9.17 |  |

### 0.B.4 Detailed Results of EMA Rate Grid Search

For the final training model of each experimental configuration, we conduct a comprehensive sweep across a broad range of EMA rates, presenting the optimal outcomes in Sec. 5. The exhaustive results are detailed in Tab. 10. In the majority of scenarios, the default EMA rate employed in the official implementations of our baseline models [, , ] yields slightly inferior performance compared to the best EMA rate identified. Nevertheless, the findings demonstrate that LCSC consistently surpasses all explored EMA rates, underscoring the limitations of EMA as a strategy for weight averaging.

**Table 10: EMA rate grid search outcomes. Asterisks (*) indicate the results using the default rates from the official paper. Listed FID scores for each EMA rate correspond to fully trained models: CIFAR-10 models at 800K iterations; ImageNet-64 CD/CT/DM models at 600K/1000K/500K iterations, respectively. Iteration counts for models employed by LCSC are shown in parentheses.**
| Model | Dataset | EMA rate | LCSC |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 0.999 | 0.9995 | 0.9999 | 0.999943 | 0.99995 | 0.99997 | 0.99999 |  |  |  |
| CD | CIFAR-10 | 4.35 | 4.13 | 3.66* | 3.56 | 3.58 | 3.51 | 3.54 | 2.42 (800K) |
| ImageNet-64 | 7.45 | 7.38 | 7.19 | 7.17* | 7.17 | 7.22 | 7.41 | 5.54 (620K) |  |
| CT | CIFAR-10 | 9.80 | 9.78 | 9.70* | 9.70 | 9.69 | 9.71 | 9.77 | 8.60 (400K) |
| ImageNet-64 | 15.7 | 15.7 | 15.6 | 15.6* | 15.6 | 15.6 | 15.7 | 12.1 (600K) |  |
| DM | CIFAR-10 | 5.73 | 5.18 | 4.16* | 3.99 | 3.96 | 4.04 | 5.04 | 3.18 (800K) |
| ImageNet-64 | 23.1 | 20.3 | 19.8* | 19.0 | 18.1 | 18.1 | 18.5 | 15.3 (500K) |  |

### 0.B.5 Evaluation with FID@CLIP

In this section, we present the results of FID@CLIP (i.e., FID computed within the feature space of a CLIP model) to demonstrate that LCSC facilitates improvements beyond the inception feature space. We utilize the visual component of a CLIP model based on ViT-B/32. Detailed outcomes are documented in Tabs. 11 and 12. The findings clearly indicate that LCSC consistently surpasses the performance of the EMA, as measured by FID@CLIP. Notably, models trained for fewer iterations or with smaller batch sizes experience enhancements through LCSC, achieving better performance than the fully converged EMA model as gauged by FID@CLIP.

**Table 11: FID on CIFAR10 calculated within the visual feature space of a CLIP model, denoted as FID@CLIP.**
| Model | Method | Training Iter | Batch Size | NFE | FID($\downarrow$) | FID@CLIP($\downarrow$) |
| --- | --- | --- | --- | --- | --- | --- |
| CD | EMA | 50K | 512 | 1 | 4.84 | 28.1 |
| 250K | 512 | 1 | 4.08 | 20.3 |  |  |
| 800K | 512 | 1 | 3.66 | 17.7 |  |  |
| 800K | 512 | 2 | 2.90 | 13.3 |  |  |
| 850K | 512 | 1 | 3.65 | 17.3 |  |  |
| 850K | 512 | 2 | 2.89 | 13.1 |  |  |
| LCSC | 50K | 512 | 1 | 3.18 | 21.4 |  |
| 250K | 512 | 1 | 2.76 | 14.3 |  |  |
| 800K | 512 | 1 | 2.42 | 14.8 |  |  |
| 800+40K | 512 | 1 | 2.38 | 14.1 |  |  |
| CT | EMA | 400K | 512 | 1 | 12.08 | 43.0 |
| 800K | 512 | 1 | 9.87 | 35.8 |  |  |
| LCSC | 400K | 512 | 1 | 8.60 | 29.5 |  |
| 450K | 128 | 1 | 8.33 | 25.5 |  |  |
| DM | EMA | 150K | 128 | 15 | 6.28 | 46.4 |
| 350K | 128 | 15 | 4.45 | 41.9 |  |  |
| 800K | 128 | 15 | 4.16 | 42.7 |  |  |
| LCSC | 150K | 128 | 15 | 4.76 | 43.1 |  |
| 350K | 128 | 15 | 3.56 | 36.1 |  |  |
| 800K | 128 | 15 | 3.18 | 36.4 |  |  |

**Table 12: FID on ImageNet-64 calculated within the visual feature space of a CLIP model, denoted as FID@CLIP.**
| Model | Method | Training Iter | Batch Size | NFE | FID($\downarrow$) | FID@CLIP($\downarrow$) |
| --- | --- | --- | --- | --- | --- | --- |
| CD | EMA | 300K | 256 | 1 | 7.70 | 30.0 |
| 600K | 256 | 1 | 7.30 | 29.1 |  |  |
| 650K | 256 | 1 | 7.17 | 28.4 |  |  |
| (released) | 600K | 2048 | 1 | 6.31 | 25.5 |  |
| LCSC | 300K | 256 | 1 | 5.71 | 28.2 |  |
| 600+20K | 256 | 1 | 5.54 | 24.9 |  |  |
| CT | EMA | 600K | 256 | 1 | 16.6 | 49.8 |
| 1000K | 256 | 1 | 15.6 | 47.5 |  |  |
| (released) | 800K | 2048 | 1 | 13.1 | 47.5 |  |
| LCSC | 600K | 256 | 1 | 12.1 | 38.3 |  |
| DM | EMA | 250K | 512 | 15 | 20.8 | 46.6 |
| 500K | 512 | 15 | 19.8 | 44.2 |  |  |
| LCSC | 250K | 512 | 15 | 16.3 | 44.8 |  |
| 500K | 512 | 15 | 15.3 | 41.5 |  |  |

## Appendix 0.C More Insights and Analysis

### 0.C.1 Analysis of Search Patterns

#### 0.C.1.1 More Examples of Search Patterns

We demonstrate search results on ImageNet-64 in Fig. 7, which share similar patterns with results on CIFAR-10 (refer to Fig. 5). Our analysis in Sec. 6 also holds in these cases.

Figure: Figure 7: Visualization of weight combination coefficients obtained using LCSC compared to those from the default EMA on ImageNet-64.
Refer to caption: x8.png

#### 0.C.1.2 Consistency of Search Results

We evaluate the consistency of search patterns by comparing results across different random seeds. As indicated in Tab. 13, both seeds produce similar outcomes in terms of FID and IS metrics. Yet, Fig. 8 reveals that each seed’s dominant coefficients map to distinct subsets of weight checkpoints, suggesting the presence of multiple high-quality basins within the weight checkpoint subspace. This diversity underscores LCSC’s ability to identify and enhance model performance within one of these basins through a limited number of search iterations (e.g., 2K in our experiments), demonstrating its efficiency and effectiveness.

Figure: Figure 8: Visualization of linear combination coefficients obtained by LCSC with different random seeds.
Refer to caption: x9.png

**Table 13: Performance of LCSC with different random seeds.**
| Model | Method | Training Iter | Batch Size | NFE | Seed | FID($\downarrow$) | IS($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| CD | LCSC | 250K | 512 | 1 | 1 | 2.76 | 9.71 |
| 2 | 2.69 | 9.69 |  |  |  |  |  |

#### 0.C.1.3 The Critical Role of Search Space Configuration

As explored in Sec. 6, restricting averaged weights to the convex hull of candidate checkpoints, where all combination coefficients are non-negative and their sum equals 1, might limit search efficacy. To test this hypothesis, we perform searches under convex combination conditions by clipping coefficients to non-negative values and normalizing their sum to 1. Table 14 presents these findings, with “w/o” indicating no restriction on coefficients (allowing values below zero) and “w” representing the convex hull restriction. The results clearly show superior performance without the convex restriction, underscoring its limiting effect on search outcomes. Furthermore, we discover that normalizing the coefficient sum to 1 is vital for effective search exploration, a practice we continue even after lifting the convex condition, as detailed in Eq. 8.

**Table 14: Performance of LCSC with or without restriction to the convex hull of all saved checkpoints. “w/o” in the column “Restriction” is our default setting in main experiments, while “w” means the restriction holds.**
| Model | Method | Training Iter | Batch Size | NFE | Restriction | FID($\downarrow$) | IS($\uparrow$) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| CD | LCSC | 250K | 512 | 1 | w/o | 2.76 | 9.71 |
| w | 3.38 | 9.36 |  |  |  |  |  |
| DM | LCSC | 350K | 128 | 15 | w/o | 3.56 | 9.35 |
| w | 3.59 | 9.29 |  |  |  |  |  |

### 0.C.2 Discussion on the Training Variance of DM and CM

As discussed in Sec. 3, we are motivated by the observation that the training objectives of DM and CM tend to introduce substantial variance in gradient estimations. We further discuss the potential reasons or hypothesis of such phenomenon as follows.

#### 0.C.2.1 Unbiased Estimation as Objective

The objectives used in DM and CT are not accurate for any single batch. Instead, they serve as an unbiased estimation. Specifically, for DM, the neural network learns the score function at any $(\bm{x}_{t},t)$, denoted as $\bm{s}_{\theta}(\bm{x}_{t},t)$. The ground truth score function $\nabla\log p_{t}(\bm{x}_{t})$ is given as []:

$$ $\nabla\log p_{t}(\bm{x}_{t})=\mathbb{E}_{\bm{y}\sim p_{\text{data}}, \varepsilon\sim\mathcal{N}(\mathbf{0},\sigma(t)^{2}\mathbf{I})}(-\frac{ \varepsilon}{\sigma_{t}}|\alpha_{t}\bm{y}+\sigma_{t}\varepsilon=\bm{x}_{t})$ (9) $$

This formula implies that every sample $\bm{y}$ in the dataset contributes to the ground truth score function at any $(x_{t},t)$. However, the training objective Eq. 3 provides only an unbiased estimation of the ground truth score Eq. 9. At every iteration, the current target for the network is not identical to the ground truth score function but is determined by the sample $\bm{y}$ randomly drawn from the dataset and the noise level $\sigma_{t}$. Thus, even if $\bm{s}_{\theta}(\bm{x}_{t},t)$ exactly matches $\nabla p_{t}(\bm{x}_{t})$ for $\forall(\bm{x}_{t},t)$, its gradient estimation using a mini-batch of data does not converge to zero. The approximation of ground truth score function can only be obtained through the expectation over many training iterations. For CT, the situation is very similar. To simulate the current ODE solution step without a teacher diffusion model, Song et al. [] utilize the Monte Carol estimation $-\frac{\bm{x}_{t}-\bm{y}}{t^{2}}$ to replace the ground truth score function $\nabla\log p_{t}(\bm{x}_{t})=\mathbb{E}_{\bm{y}\sim p_{\text{data}},\bm{x}_{t}
\sim\mathcal{N}(\bm{y},t^{2}\mathbf{I})}(-\frac{\bm{x}_{t}-\bm{y}}{t^{2}}|\bm{
x}_{t})$ under the noise schedule from EDM [] as denoted in Eq. 5. Therefore, the objective is also not accurate for any single batch of data and the model has to learn to fit the target model output after solving the current step with the expectation of score function among many training iterations. These types of objectives, which are not accurate for any single batch, introduce high variance to the gradient estimation.

#### 0.C.2.2 Model across Different Time steps

For both DM and CM, the neural network has to be trained at various time steps. Since the model input and output follow different distributions, the gradients at different time steps may conflict with each other. This phenomenon of negative transfer between timesteps has been studied in several previous work, where they train different diffusion neural networks at different timesteps and achieve better performance [, ]. The misalignment between the objectives of different time steps may also contribute to the high variance in the gradient estimation.

#### 0.C.2.3 Error Accumulation of CM

For CM, the model learns to approximate the output of the target model at the previous step. This could potentially introduce the problem of error accumulation []. Consequently, any noise introduced during training at early time steps is likely to lead to inaccuracies in the target model, which may be magnified in subsequent timesteps. This property of CM amplifies the high training variance, particularly for one-step sampling.

## Appendix 0.D Future Work

LCSC represents a novel optimization paradigm, indicating its potential for widespread application. We recommend future investigations focus on three key areas:

- $\bullet$
*Expanded Search Space:* Presently, LCSC applies a uniform coefficient across an entire model. Considering that different model layers might benefit from distinct combination coefficients, partitioning model weights into segments for unique coefficient assignments could enhance the search space. For DMs, adopting variable coefficients across different timesteps may also offer further improvements.
- $\bullet$
*Efficient Optimization Methods:* The current reliance on evolutionary methods, characterized by their dependency on randomness, limits efficiency and risks convergence to local optima. Investigating more effective optimization strategies presents a promising avenue for enhancing LCSC.
- $\bullet$
*Broader Application Scope:* While the initial motivation for LCSC stems from managing the high training variance observed in DMs and CMs, its utility is not confined to these models. Exploring LCSC’s applicability to other domains, such as fine-tuning language models or additional vision models, could unlock new performance gains.

## Appendix 0.E Visualization

In this section, visualizations of images generated by DM (Figs. 9 and 10), CD (Figs. 11 and 12), and CT (Figs. 13 and 14) are presented. Many images produced by the EMA model are observed to be similar to those produced by the model derived from our LCSC when using the same noise input. This is expected given that both models are based on weights from the same training cycle. To more clearly highlight the distinctions between EMA and LCSC,
we generate 50K images from EMA and LCSC using the same set of noise inputs, order the image pairs according to their Euclidean distances in the inception feature space, and randomly select the images with large distances.
In general, images generated by the LCSC model are found to display enhanced sharpness, diminished noise, and more distinct object representation.

Figure: (a) EMA model (FID: 4.16).
Refer to caption: extracted/5522312/fig/dm_cifar_ema.png

Figure: (a) EMA model (FID: 19.8).
Refer to caption: extracted/5522312/fig/dm_imagenet_ema.png

Figure: (a) EMA model (FID: 3.66).
Refer to caption: extracted/5522312/fig/cd_cifar_ema.png

Figure: (a) EMA model (FID: 7.30).
Refer to caption: extracted/5522312/fig/cd_imagenet_ema.png

Figure: (a) EMA model (FID: 12.08).
Refer to caption: extracted/5522312/fig/ct_cifar_ema.png

Figure: (a) EMA model (FID: 16.6).
Refer to caption: extracted/5522312/fig/ct_imagenet_ema.png

## Peer Reviews

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: - Currently, the experiments seems to be limited to image generation tasks. Would it be possible to demonstrate this to another diffusion task such as audio or video?
- Have you studied how these coefficients change over the course of training? There perhaps maybe an interpretability study to understand the importance of different stages of training on the best performance
- On a similar front, can it be extended to layers? That is, composition of weights on layers may differ from the composition of overall weights and perhaps lead to a better result?
soundness: 4
strengths: - The simple methodology presented seems to have a well-motivated theoretical basis, and is quite effective in improving the training of diffusion and consistency models. Further, this method does not require backprop and side steps the need for differentiable loss functions through the evolutionary search.
- The analysis presented on the method along with the visualizations of the landscape to demonstrate that optimal model weights lie in basins which are inaccessible by optimization but may be accessible through checkpoint averaging.
- It has clear value in reducing compute time and improving model performance as demonstrated through the experiments. Seems like there's speedups by an order of magnitude which is quite impressive.
summary: This paper proposes LCSC (Linear Combination of Saved Checkpoints), a method to improve the performance and efficiency of diffusion and consistency models. The underlying idea is to use an optimal linear combination of model checkpoints saved during training. These optimal coefficients are determined using an evolutionary search method. The authors demonstrate that this approach can not only decrease the training cost but also enhance the performance of pre-trained models. The experiments which were performed on CIFAR-10 and ImageNet-64 provably demonstrate that this approach results in an increase in training time and improvements in evaluation metrics.
weaknesses: See questions

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 2
questions: (1) Can LCSC reduce the training cost of DM (not CM)? I only find experiments about reducing training cost with consistency distillation and consistency training but not vanilla DDPM. 

(2) It seems that most experiments are based on CM. Do you think LCSC is more suitable for CM and CD compared with DM? And why?

(3) Could you provide the CLIP-score for text-to-image generation?
soundness: 3
strengths: (1) The motivation is clear and easy to understand. The authors also provide theoretical analysis about why LCSC is better than EMA.

(2) The method is simple and straightforward, which uses the evolutionary search to linearly combine checkpoint weights. It seems that LCSC can be easy to implement and applied with different diffusion models.

(3) As shown in the experiment results, LCSC can significantly reduce the training cost of a diffusion model and enhance the pre-trained diffusion models. 

(4) The authors provide a lot of experiment results to study LCSC from different aspects.
summary: The paper proposes a method named LCSC for training diffusion models by linearly combining the intermediate weight checkpoints using an evolutionary algorithm. LSCS can significantly reduce the training cost of a diffusion model without sacrificing the generation quality. In addition, a pre-trained diffusion model can achieve better performance by fine-tuning with LCSC with few training iterations. The numerous experiment results are provided to demonstrate the effectiveness of LCSC in different datasets.
weaknesses: (1) It seems that the experiments about reducing the training cost only focus on datasets with small resolution (CIFAR-10 and ImageNet-64). I understand that the cost for training a diffusion model on a dataset with large resolution (e.g, LSUN-bedroom, COCO) from scratch is more expensive. I believe that such a set of experiments can make the LCSC more attractive.

(2) In my opinion, the experiment results about text-to-image generation are not convincible enough to validate the effectiveness of LCSC. The number of training iterations for fine-tuning LCM with LoRA is very small, which is only 6K with the batch size of 12. The authors should use a larger number of training iterations, or show that the generation quality of LCM will not improve with further training. I'm wondering whether the experimental settings for evaluating LCM with LCSC is suitable, since the generated images (Figure 2 and Figure 10) are in anime style while CC12M and MS-COCO are not. Besides the LCM, the authors could also fine-tune the pre-trained Stable Diffusion v1.5 for a few iterations with LCSC to see if LCSC can further enhance it.

In conclusion, I think LCSC is a good work for its motivation and method. While the experiments could be further improved and organized.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Potential Overfitting of Weights (Fig. 8): The weights depicted in Fig. 8 exhibit large absolute values (greater than 6), which might indicate overfitting. Could the authors address the following points?
- Regularization: Were any regularization techniques employed to constrain the magnitude of the coefficients?
- Interpretability: What is the theoretical or intuitive justification for assigning such large weights to certain checkpoints?
2. Convergence Curves Post-LCSC Application: Visualizing the convergence behavior after applying LCSC can provide a clearer understanding of its impact on training dynamics. Including convergence curves would help illustrate how LCSC influences the training process compared to baseline methods like EMA.
3. Convergence Curve of LCSC itself. How many iteration of evolution does it undertake before achieving a satisfactory solution?
4. Add experiments for further assessing the robustness of LCSC-Derived Coefficients.
5. Enhancing Interpretability of Coefficients: Understanding the role and significance of each coefficient can aid in demystifying the model's behavior. Implementing constraints or regularization techniques to limit the magnitude of coefficients may improve their interpretability. Alternatively, providing a theoretical justification or empirical analysis explaining the distribution of coefficient values can enhance comprehension.
soundness: 3
strengths: 1. Impressive Experimental Results: The experiments, particularly those presented in Tables 1 and 2, convincingly demonstrate the efficacy of LCSC for both CD and CT, showcasing significant reduction in training costs.
2. Innovative Findings: Figures 4 and 6 highlight the potential for significantly better linear combinations of checkpoints than the naive EMA solution, indicating a promising direction for future research.
3. Theoretical Foundation: The theoretical analysis providing insights into the convergence behavior of EMA models adds depth and credibility to the proposed method.
4. Practical Implications: By achieving speedups of up to an order of magnitude or more, LCSC offers tangible benefits in reducing computational resources and training time, which is highly relevant in the context of large-scale model training.
summary: The authors introduce a novel approach to enhance the performance and reduce the training costs of Diffusion Models (DM) and Consistency Models (CM) by learning a linear combination of saved checkpoints using evolutionary algorithms. The paper builds upon existing theoretical frameworks, demonstrating that Exponential Moving Average (EMA) models converge faster than their last-tier counterparts under specific conditions. Through a motivational experiment involving a grid search over numerous linear combinations of three checkpoints (with and without EMA), the authors reveal the existence of superior solutions compared to the traditional EMA approach. The proposed method, termed Linear Combination of Saved Checkpoints (LCSC), is further validated experimentally, showing significant reductions in training costs via fewer iterations and smaller batch sizes, as well as improvements in generation quality when applied to fine-tuned checkpoints.
weaknesses: 1. Search Cost Disparity Between DM and CM: Tables 8 and 9 indicate that the search cost is more substantial for Diffusion Models (DM) compared to Consistency Models (CM), suggesting that LCSC is less effective in reducing training costs for DM.
2. The coefficients derived by the evolutionary algorithm (EA) lack interpretability, with some being significantly large (\>6) and others near zero (Fig. 8). Additionally, coefficients learned from different seeds show considerable disagreement, despite yielding similar performance improvements (as seen in Fig. 9). The lack of interpretability may hinder the understanding of the model's behavior and its generalizability. It also raises concerns about the possibility of overfitting.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 4
questions: 1. How to get such smooth metric landscapes?
2. How to prove "a small subset of weights is characterized by coefficients of large magnitude"?
3. Why do the models at some iterations have negative coefficients?
soundness: 3
strengths: 1. The finding that proper checkpoint merging can significantly improve the training convergence and final performance is novel and interesting to me.
2. The theoretical analyses for EMA are promising and solid.
3. The experiments are extensive and convincing.
summary: This paper targets the efficient training problem for diffusion models and consistency models. They find that proper checkpoint merging can significantly improve the training convergence and final performance. Therefore they develop an evolutionary search algorithm for linearly combining the checkpoints. The experiments showcase their LCSC can reduce the training cost and enhance the performance of pre-trained diffusion models.
weaknesses: 1. The details for how to get the metric landscapes are missing. Do you apply a grid search for all $x,y$ with the formula $\theta_{(x, y)}=\theta_{n_0}+x\left(\theta_{n_1}-\theta_{n_0}\right)+y\left(\theta_{n_2}-\theta_{n_0}\right)$. If so, what is the grid interval? And what is the interval between $n_0$, $n_1$ and $n_2$? It is quite strange that the optimal points are always located inside of the regions surrounded by $n_0$, $n_1$, and $n_2$. Does it mean the models are always oscillating? Please provide more explanation.
2. The claim in lines 528-529 that "a small subset of weights is characterized by coefficients of large magnitude" is interesting but needs more evidence. 
3. Please provide more insights about why the coefficients of some iterations in Figure 5 are negative. It is clear that the negative coefficients mean the models at these iterations have a negative effect on the final performance but why does this happen? Is it because the models at these iterations have a relatively lower FID?

Meta-review: This paper proposes to leverage an optimal linear combination of intermediate model checkpoints during training to improve the efficiency (training cost) of diffusion and consistency models while maintaining or improving their performance.

Four knowledgeable referees reviewed this submission and acknowledged that the approach was well motivated from a theoretical perspective (K43d, Enc2,AvHi), and presented in a clear and easy to follow way (odMx). The reviewers also found the method novel (AvHi), interesting (AvHi), and simple (K43d, odMx), and valuable to reduce the training costs while improving model performance (K43d, odMx)

The main concerns raised by the reviewers were:
1. Experiments were not found fully convincing: limited experiments to image generation tasks (K43d), low resolution datasets (odMx), unclear whether generation quality would improve with additional training (odMx), some claims requiring additional evidence (AvHi)
2. Missing analysis of how coefficients change over the course of training (K43d), and of their interpretability (Enc2)

The reviewers also asked clarifying questions related to the metrics landscape (AvHi), and the extension of the proposed approach to layers (K43d). During rebuttal and discussion, the authors discussed image generation as a proof of concept for their approach, described the challenges in analyzing the evolution of coefficients and gave some intuitions, showcased the potential of the method to be applied layer-wise through one experiments, presented results with increased numbers of iterations, added model finetuning experiments, added comparisons based on CLIPscore, and added experiments to address concerns w.r.t. the magnitude of the coefficients. The discussion period addressed most of the reviewers' concerns, who unanimously lean towards acceptance. The AC agrees with the reviewers' assessment and therefore recommends to accept.

Decision: Accept (Poster)