A Unified View of Delta Parameter Editing in Post-Trained Large-Scale Models

## Abstract

Abstract Post-training has emerged as a crucial paradigm for adapting large-scale pre-trained models to various tasks, whose effects are fully reflected by delta parameters (i.e., the disparity between post-trained and pre-trained parameters). While numerous studies have explored delta parameter properties via operations like pruning, quantization, low-rank approximation, and extrapolation, a unified framework for systematically examining these characteristics has been lacking. In this paper, we propose a novel perspective based on Riemann sum approximation of the loss function to elucidate delta parameter editing operations. Our analysis categorizes existing methods into three classes based on their post-editing performance: competitive, decreased, and improved, explaining how they are expressed by the Riemann sum approximation term and how they alter the model performance. Extensive experiments on both visual and language models, including ViT, LLaMA 3, Qwen 2, and Mistral, corroborate our theoretical findings. Furthermore, we introduce extensions to existing techniques like DARE and BitDelta, highlighting their limitations in leveraging the properties of delta parameters and reorganizing them into general expressions to enhance the applicability and effectiveness of delta parameter editing in post-trained models.

## 1 Introduction

With the remarkable success of large-scale pre-trained models, post-training has emerged as the de facto standard paradigm for effective adaptations to various tasks (Han et al., ; Xin et al., ; Dodge et al., ; Zhao et al., ). Conceptually, post-training optimizes the parameters of pre-trained backbone on task-specific data, endowing models with diverse abilities like visual recognition (Chen et al., ; Sandler et al., ), instruction following (Rafailov et al., ; Ethayarajh et al., ), and mathematical reasoning (Luo et al., ; Tong et al., ). It has been noted that the impact of post-training is fully manifested in the delta parameters, which are defined as the difference between parameters of pre-trained and post-trained models (Ilharco et al., ; Yu et al., ).

Due to the inherent correlations between delta parameters and post-training, significant efforts have been made to investigate the properties of delta parameters through various editing operations in recent years. For instance, studies like DARE (Yu et al., ) and DELLA-Merging (Deep et al., ) showed that models can achieve comparable performance with only a small fraction of delta parameters, highlighting their extreme redundancy. BitDelta (Liu et al., ) demonstrated that delta parameters could be quantized to 1 bit with modest performance compromise. Twin-Merging (Lu et al., ) and TIES-Merging (Yadav et al., ) discovered that most of the benefits of post-training can be retained after executing singular value decomposition and magnitude-based pruning on delta parameters. EXPO (Zheng et al., ) observed that cheaply extrapolating delta parameters with a suitable scaling factor can even enhance the performance. However, a comprehensive framework for systematically discussing delta parameter characteristics and theoretically explaining how different operations impact model performance remains lacking.

In this work, we make a pioneering effort to provide a unified view of delta parameter editing in post-trained large-scale models. We formulate the editing operations of delta parameters based on Riemann sum approximation of the loss difference. By mathematically analyzing existing editing operations’ loss change, we elucidate why certain operations result in competitive, decreased, or improved performance. Specifically, we verify that: 1) methods such as DARE and DELLA-Merging can well keep the approximation term to zero through the random drop and rescale processes, ensuring equal loss between the edited and post-trained models and achieving competitive performance; 2) techniques including BitDelta, Twin-Merging, and TIES-Merging often result in decreased performance, with a positive approximation term introduced by quantization, low-rank approximation, and magnitude-based pruning;
3) EXPO-like methods, by extrapolating delta parameters, produce negative loss changes on alignment data, resulting in better-aligned models.
To validate our theoretical analysis, extensive experiments are conducted on large-scale visual models (ViT (Radford et al., )) and language models (LLaMA 3 (Dubey et al., ), and Mistral (Jiang et al., )), and the results strongly support our analysis.

Besides understanding existing delta parameter editing techniques in the proposed view, we further present several extensions to provide more general formats.
Firstly, we introduce a factor to handle the dropped parameters in DARE, effectively expanding methods like DARE.
Secondly, we extend the scope of quantification-based methods like BitDelta, identifying a broader area for effective quantification beyond reducing magnitude diversity to a single value.
Finally, we identify that extrapolation is not the key to the success of EXPO-like methods. Instead, we should determine whether to use extrapolation or interpolation based on the direction of the approximation term.
Experimental results also demonstrate the effectiveness of the proposed extensions.

## 2 Related Work

### 2.1 Post-training of Large-Scale Models

In recent years, with the rapid development of large-scale models, post-training has become an essential process for adapting the pre-trained backbone to a variety of tasks (Xin et al., ; Dodge et al., ; Zhao et al., ). Post-training realizes the adaptation via adjusting the pre-trained backbone’s parameters through full fine-tuning (Dosovitskiy et al., ; Liu et al., ; Devlin et al., ; Radford et al., ) or parameter-efficient fine-tuning (He et al., ; Houlsby et al., ; Li & Liang, ; Hu et al., ; Han et al., ) algorithms. It is straightforward to conclude that the effectiveness of post-training can be perfectly denoted by the delta parameters, which represent the difference between post-trained and pre-trained parameters (Ilharco et al., ; Yu et al., ). Given the close correlations between delta parameters and the post-training process, investigating the properties of delta parameters becomes particularly important. In this paper, we present a novel perspective to illustrate delta parameter characteristics of post-trained models.

### 2.2 Delta Parameter Editing for Post-Trained Models

Existing delta parameter editing techniques can be generally categorized as three aspects according to their post-editing performance, including competitive, decreased, and improved performance.

Delta Parameter Editing with Competitive Performance.
DARE (Yu et al., ) is a widely used approach to edit delta parameters without compromising the model performance. Technically, DARE can eliminate most (90% or even 99%) of the delta parameters with the random drop and rescale operations. Inspired by DARE, DELLA-Merging (Deep et al., ) presented a magnitude-aware drop to replace the random drop for achieving better performance, which ranks delta parameters by their magnitude and assigns higher dropout probabilities to those with lower ranks (i.e., corresponding to lower magnitudes). Yu et al. () and Deep et al. () explained that DARE and DELLA-Merging can work because they are able to approximate the original embeddings based on only a small fraction of delta parameters, thus maintaining the model performance.

Delta Parameter Editing with Decreased Performance.
BitDelta (Liu et al., ) quantized delta parameters to only 1 bit according to the average magnitude scalar and sign bits. Twin-Merging (Lu et al., ) applied singular value decomposition (Klema & Laub, ) on delta parameters to extract exclusive knowledge for each specific task. TIES-Merging (Yadav et al., ) retained delta parameters with the largest magnitudes for reducing redundancy. All the above methods yield slightly worse results after executing the corresponding quantization, low-rank approximation, or pruning operations.

Delta Parameter Editing with Improved Performance.
EXPO (Zheng et al., ) extrapolated delta parameters calculated by two relatively weaker models with an appropriate scaling factor to construct a stronger model, which can enhance the model performance.

It can be concluded that current approaches utilizes distinct operations for editing delta parameter, lacking a comprehensive analysis of whether these editing operations are suitable and why different operations cause various influence on the model performance. In this work, we make the first attempt to introduce a unified view of delta parameter editing in post-training, which is supported both theoretically and empirically.

## 3 Preliminaries

### 3.1 Notations

Delta Parameters During Post-Training. Let $\bm{W}_{\text{PRE}}\in\mathbb{R}^{d\times k}$ denote the parameters of a pre-trained model, where $d$ and $k$ represent the output and input dimensions. A post-trained model with parameters $\bm{W}_{\text{POST}}\in\mathbb{R}^{d\times k}$ can be derived from the pre-trained backbone, yielding delta parameters $\Delta\bm{W}=\bm{W}_{\text{POST}}-\bm{W}_{\text{PRE}}\in\mathbb{R}^{d\times k}$. As delta parameters denote the alterations of parameters during the post-training process, investigating the characteristics of delta parameters can provide a deeper understanding of post-training.

Delta Parameter Editing. Let $\mathcal{F}$ represent the delta parameter editing function. The edited parameters $\Delta\widetilde{\bm{W}}_{\text{Edit}}=\mathcal{F}(\Delta\mathbf{W})$ is then combined with $\mathbf{W}_{\text{PRE}}$ to obtain the final edited parameter $\mathbf{W}_{\text{Edit}}=\mathbf{W}_{\text{PRE}}+\Delta\widetilde{\bm{W}}_{
\text{Edit}}$. Existing delta parameter editing methods can be categorized into three types based on their effects on model performance, i.e., competitive, decreased, and improved performance. These methods employ various techniques including pruning, quantization, low-rank approximation, and extrapolation. Notable works in this field include DARE, BitDelta, Twin-Merging, TIES-Merging, and EXPO, which are investigated in this paper.

### 3.2 A Unified View of Delta Parameter Editing

In this work, we introduce a unified view of delta parameter editing during the post-training process based on Riemann sum approximation.
Specifically, we represent the changes caused by existing editing methods by $\Delta\widetilde{\bm{W}}$ and aim to investigate their effects on performance via analyzing the loss difference. To better analyze the changes in loss, we introduce the Riemann sum approximation, which corresponds to the difference in loss made by the editing operation as follows,

$$ $\displaystyle\Delta\mathcal{L}$ $\displaystyle=\mathcal{L}(\bm{W}_{\text{POST}}+\Delta\widetilde{\bm{W}})- \mathcal{L}(\bm{W}_{\text{POST}})=\int_{0}^{1}\nabla\mathcal{L}(\bm{W}_{\text{ POST}}+t\Delta\widetilde{\bm{W}})\cdot\Delta\widetilde{\bm{W}}\,dt$ (1) $\displaystyle\approx\frac{1}{C}\sum\limits_{c=0}^{C-1}\langle\nabla\mathcal{L} (\bm{W}_{\text{POST}}+\frac{c}{C}\Delta\widetilde{\bm{W}}),\Delta\widetilde{ \bm{W}}\rangle=\frac{1}{C}\sum\limits_{c=0}^{C-1}\langle\nabla\mathcal{L}^{c}, \Delta\widetilde{\bm{W}}\rangle,$ $$

where $\mathcal{L}(\bm{W}):\mathbb{R}^{d\times k}\rightarrow\mathbb{R}$ denotes the loss function of a model with parameters $\bm{W}\in\mathbb{R}^{d\times k}$,
$\nabla\mathcal{L}(\bm{W})$ is the gradient of the loss function at $\bm{W}$, and $\langle\cdot,\cdot\rangle$ denotes the Frobenius inner product. $C$ denotes the number of subdivisions of the interval $[0,1]$. This expansion provides a linear approximation of the loss function in the neighborhood of $\bm{W}_{\text{POST}}$, allowing the analysis of the impact of parameter changes on the model performance.
In most cases, the loss difference can reflect the influence on performance, with a positive value indicating deterioration, zero indicating stability, and a negative value indicating improvement.
In section 4, section 5, and section 5, we respectively discuss editing operations that cause competitive, decreased, and improved performance, and derive the format of these operations when organizing them into the proposed unified paradigm.

To validate our theoretical analysis and the proposed extensions, we conducted experiments on LLaMA-3-8B-Instruct (Dubey et al., ), Mistral-7B-Instruct-v0.3 (Jiang et al., ), and ViT-B-32 (Radford et al., ). We evaluate text models on 8 tasks: 25-shot ARC Challenge (Clark et al., ), 5-shot GSM8K (Cobbe et al., ), 10-shot HellaSwag (Zellers et al., ), zero-shot HumanEval (Chen et al., ), zero-shot IFEval (Zhou et al., ), 5-shot MMLU (Hendrycks et al., ), zero-shot TruthfulQA (Lin et al., ), and 5-shot Winogrande (Sakaguchi et al., ), and evaluate vision models on 8 tasks: Cars (Krause et al., ), DTD (Cimpoi et al., ), EuroSAT (Helber et al., ), GTSRB (Stallkamp et al., ), MNIST (LeCun et al., ), RESISC45 (Cheng et al., ), SUN397 (Xiao et al., ), and SVHN (Netzer et al., ).

## 4 Unifying Editing Operations with Competitive Performance

As a widely-used approach for delta parameter editing, DARE (Yu et al., ) presents the random drop and rescale process to remove 90% or even 99% delta parameters without compromising the model performance. Following this line, many follow-up works have been proposed. For example, DELLA-Merging (Deep et al., ) modifies the drop operation in DARE from random to magnitude-aware. In this section, we select DARE for analysis because it is the most representative method among those that can retain the original model performance after editing delta parameters.

### 4.1 Express DARE with Approximation Term

Mathematically, the editing process of delta parameters in DARE is denoted by

$$ $\begin{split}&\bm{W}_{\text{DARE}}=\bm{W}_{\text{POST}}+\Delta\widetilde{\bm{W }}_{\text{DARE}}=\bm{W}_{\text{PRE}}+\Delta\bm{W}+\Delta\widetilde{\bm{W}}_{ \text{DARE}}\\ =\bm{W}_{\text{PRE}}+0\cdot\bm{M}&\odot\Delta\bm{W}+\frac{1}{1-p}\cdot(1-\bm{M })\odot\Delta\bm{W}=\bm{W}_{\text{PRE}}+\frac{1}{1-p}\cdot(1-\bm{M})\odot \Delta\bm{W},\end{split}$ (2) $$

where $p\in\mathbb{R}$ represents the drop rate and $\odot$ denotes the element-wise Hadamard product. $\bm{M}\sim\text{Bernoulli}(p,\Delta\bm{W})\in\mathbb{R}^{d\times k}$ is a mask matrix sampled from Bernoulli distribution according to $p$, whose shape is identical to that of $\Delta\bm{W}$. From Equation (LABEL:equ:dare_computation), we can derive that

$$ $\Delta\widetilde{\bm{W}}_{\text{DARE}}=\frac{p-\bm{M}}{1-p}\odot\Delta\bm{W}.$ (3) $$

Referring to Equation (1), we obtain

$$ $\begin{split}\Delta\mathcal{L}_{\text{DARE}}&\approx\frac{1}{C}\sum\limits_{c= 0}^{C-1}\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}\frac{p-M_{ij}}{1-p}\cdot \Delta W_{ij}\cdot\nabla\mathcal{L}^{c}_{ij}\\ &=\frac{1}{C}\sum\limits_{c=0}^{C-1}\left(\frac{p}{1-p}\cdot\sum\limits_{M_{ij }=0}\Delta W_{ij}\cdot\nabla\mathcal{L}^{c}_{ij}-\sum\limits_{M_{ij}=1}\Delta W _{ij}\cdot\nabla\mathcal{L}^{c}_{ij}\right).\end{split}$ (4) $$

Due to the vast number of parameters in large-scale models, we can use the Law of Large Numbers to approximate the summations by their expected values. Additionally, because of the randomness of the drop operation in DARE, $M_{ij}$ and $\Delta W_{ij}\cdot\nabla\mathcal{L}^{c}_{ij}$ can be considered approximately independent random variables. It is straightforward to deduce that

$$ $\begin{split}\sum\limits_{M_{ij}=0}\Delta W_{ij}\cdot\nabla\mathcal{L}^{c}_{ij }&\approx(1-p)\cdot\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}\Delta W_{ij} \cdot\nabla\mathcal{L}^{c}_{ij},\\ \sum\limits_{M_{ij}=1}\Delta W_{ij}\cdot\nabla\mathcal{L}^{c}_{ij}&\approx p \cdot\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}\Delta W_{ij}\cdot\nabla \mathcal{L}^{c}_{ij}.\end{split}$ (5) $$

Substituting Equation (5) into Equation (4), we derive

$$ $\begin{split}\Delta\mathcal{L}_{\text{DARE}}\approx(\frac{p}{1-p}\cdot(1-p)-p) \cdot\frac{1}{C}\sum\limits_{c=0}^{C-1}\sum\limits_{i=1}^{d}\sum\limits_{j=1}^ {k}\Delta W_{ij}\cdot\nabla\mathcal{L}^{c}_{ij}=0.\end{split}$ (6) $$

To this end, we can conclude that after editing delta parameters with DARE, the loss $\mathcal{L}(\bm{W}_{\text{DARE}})$ is approximately equal to $\mathcal{L}(\bm{W}_{\text{POST}})$, independent of the specific dataset, explaining why DARE can achieve competitive performance even when most delta parameters are eliminated.

To verify the above analysis, we used the DARE method to construct models on LLaMA3-8B-Instruct and computed the approximation term on the GSM8K dataset. The results are shown in Figure 3. We use the drop-only (w/o rescale) as the reference. As can be seen, models with DARE constructed consistently achieved lower average loss, and with a smaller drop rate, the approximation term calculated across different parts of the model remained relatively small. This validates our theoretical derivation above.

### 4.2 Extension of DARE

We further present a more general format of delta parameter editing operations that can achieve competitive performance. In particular, instead of dropping delta parameters, we introduce a term $k$ to adjust them and rescale the remaining ones with $(1-k\cdot p)/(1-p)$. Similar to the deduction in Equation (LABEL:equ:dare_computation) to Equation (6), we obtain

$$ $\begin{split}\bm{W}_{\text{COMP}}=&\bm{W}_{\text{PRE}}+\Delta\bm{W}+\Delta \widetilde{\bm{W}}_{\text{COMP}}=\bm{W}_{\text{PRE}}+k\cdot\bm{M}\odot\Delta \bm{W}+\frac{1-k\cdot p}{1-p}\cdot(1-\bm{M})\odot\Delta\bm{W},\\ \Delta\widetilde{\bm{W}}_{\text{COMP}}\,&=\,\frac{(k-1)(\bm{M}-p)}{1-p}\odot \Delta\bm{W},\\ \Delta\mathcal{L}\,&\approx\,\frac{1}{C}\sum\limits_{c=0}^{C-1}\left(\frac{p \cdot(1-k)}{1-p}\cdot\sum\limits_{M_{ij}=0}\Delta W_{ij}\cdot\nabla\mathcal{L} ^{c}_{ij}+(k-1)\cdot\sum\limits_{M_{ij}=1}\Delta W_{ij}\cdot\nabla\mathcal{L}^ {c}_{ij}\right)\\ &\approx\left(\frac{p\cdot(1-k)}{1-p}\cdot(1-p)+(k-1)\cdot p\right)\cdot\frac{ 1}{C}\sum\limits_{c=0}^{C-1}\sum_{i=1}^{d}\sum_{j=1}^{k}\Delta W_{ij}\cdot \nabla\mathcal{L}^{c}_{ij}=0.\end{split}$ $$

Figure: Figure 1: The performance of LLaMA3-8B-Instruct on the GSM8K, TruthfulQA, and HumanEval datasets under varying $p$ and $k$.
Refer to caption: x1.png

It has been verified that $\Delta\mathcal{L}$ is approximately 0, which indicates the validity of the proposed format. Note that in DARE, the drop operation can be realized by setting $k$ to 0. Thus, our format is an extension of DARE with broader settings of $k$.

Figure: Figure 2: The performance of ViT-B-32 on the DTD, EuroSAT, and GTSRB datasets under varying $p$ and $k$.
Refer to caption: x2.png

We conducted validation experiments for the extension of DARE across a wide range of models and tasks. The representative results for LLaMA3-8B-Instruct and ViT-B-32 are shown in 1 and Figure 2, while the complete results for all models and tasks are presented in the Appendix B.1.
Specifically, on threee representative text datasets—GSM8K, TruthfulQA, and HumanEval, when both the rescale rate $k$ and sign change rate $kp$ are small (e.g., less than 0.5), the performance of our adjusted model is very close to that of the original post-trained model and significantly outperforms the pre-trained model.
Regarding the weight scalar $k$ introduced in our extension, we observed that, compared to the setting where $k=0$ (which reverts to the original DARE configuration), using $k\neq 0$ generally yields competitive performance across different datasets. This demonstrates the effectiveness of our extension.
For the ViT model, the results on the DTD, EuroSAT, and GTSRB datasets are more consistent with our expectations. Regardless of the rescale and sign change rates, the performance of the adjusted model is almost identical to that of the original post-trained model.

Interestingly, when $k<0$, indicating that the delta parameters are flipped in sign, the constructed model still achieves competitive performance to the post-trained model. This challenges the prior assumption that the sign of delta parameters is critical for performance (Yadav et al., ; Liu et al., ), suggesting that what truly matters during post-training is not the specific directional adjustments of individual parameters, but rather a more collective behavior of the entire delta parameters.

### 4.3 Further Discussions on DARE

Yu et al. () and Deep et al. () claim that DARE and DELLA-Merging are effective because the random drop of delta parameters ensures an approximation of the original embeddings, thereby preserving model performance. However, according to equation 5, we argue that random drop of delta parameters is a sufficient but not necessary condition for maintaining model performance. Furthermore, we contend that ensuring randomness in the element-wise product of delta parameters and approximation term is the necessary and sufficient condition.

**Table 1: Validation of the discussion on DARE. The leftmost column shows the random drop in DARE. The middle column illustrates the approach of multiplying all negative delta parameters by $k$ and all positive delta parameters by $\frac{1-k\cdot p}{1-p}$. The rightmost column demonstrates the method of first calculating the product of delta parameters and gradients, and then multiplying all negative products by $k$ and all positive products by $\frac{1-k\cdot p}{1-p}$.**
| k | Random | Biased $\Delta W$ | Biased $\Delta W\cdot\nabla L$ |
| --- | --- | --- | --- |
| 0.5 | 76.35 | 74.15 | 0.0 |
| 0.7 | 75.89 | 75.36 | 0.0 |
| 0.9 | 76.19 | 76.04 | 26.76 |
| 1.1 | 75.89 | 75.59 | 0.15 |
| 1.3 | 75.36 | 74.91 | 0.0 |
| 1.5 | 75.59 | 74.83 | 0.0 |

To verify the above analysis, we conduct two experiments on GSM8K dataset.
First, we disrupt the randomness of the delta parameter drop operation by multiplying all negative delta parameters by $k$ and all positive delta parameters by $(1-k\cdot p)/(1-p)$. The results are shown in the middle column of Table 1, illustrating that the model performance remains intact. This validates that the randomness of the delta parameter dropout operation is a sufficient but not necessary condition for maintaining model performance.
Furthermore, we disrupt the randomness of the dropout operation on the approximation term by multiplying all negative products by $k$ and all positive products by $(1-k\cdot p)/(1-p)$. The results, as depicted in the rightmost of Table 1, show a significant decline in model performance. This validates that the randomness of the dropout operation on the product of delta parameters and approximation term is a necessary and sufficient condition for maintaining model performance.

Figure: Figure 3: Validation of our theoretical derivation of DARE, BitDelta, Twin-Merge(sparsity rate=0.9), and Ties-Merge.
Refer to caption: x3.png

## 5 Unifying Editing Operations with Decreased Performance

This section discusses three delta parameter editing operations that incur reduced results, including quantization, low-rank approximation, and pruning. We respectively choose BitDelta (Liu et al., ), Twin-Merging (Lu et al., ), and TIES-Merging (Yadav et al., ) as typical works.

### 5.1 Express BitDelta with Approximation Term

BitDelta quantizes delta parameters down to 1 bit, utilizing the sign bit matrix and a high-precision scalar, where the latter is initially computed by the average magnitude of delta parameters. Specifically, BitDelta can be represented by

$$ $\begin{split}&\bm{W}_{\text{BitDelta}}=\bm{W}_{\text{POST}}+\Delta\widetilde{ \bm{W}}_{\text{BitDelta}}=\bm{W}_{\text{PRE}}+\Delta\bm{W}+\Delta\widetilde{ \bm{W}}_{\text{BitDelta}}\\ =\bm{W}_{\text{PRE}}+&\frac{1}{d\cdot k}\sum\limits_{i=1}^{d}\sum\limits_{j=1} ^{k}|\Delta W_{ij}|\cdot\text{Sign}(\Delta\bm{W})=\bm{W}_{\text{PRE}}+\text{ AVG}(|\Delta\bm{W}|)\cdot\text{Sign}(\Delta\bm{W}),\end{split}$ (7) $$

where $|\cdot|$ denotes the operation of taking magnitudes. $\text{AVG}(|\Delta\bm{W}|)$ represents the average magnitude of $\Delta\bm{W}$. Since $\Delta\bm{W}=|\Delta\bm{W}|\odot\text{Sign}(\Delta\bm{W})$, based on Equation (LABEL:equ:bitdelta_computation), we can further obtain

$$ $\Delta\widetilde{\bm{W}}_{\text{BitDelta}}=(\text{AVG}(|\Delta\bm{W}|)-|\Delta \bm{W}|)\odot\text{Sign}(\Delta\bm{W}).$ (8) $$

Based on Equation (1), we get

$$ $\begin{split}\Delta\mathcal{L}_{\text{BitDelta}}\approx\frac{1}{C}\sum\limits_ {c=0}^{C-1}\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}(\text{AVG}(|\Delta\bm{W} |)-|\Delta W_{ij}|)\cdot\text{Sign}(\Delta W_{ij})\cdot\nabla\mathcal{L}^{c}_{ ij}.\end{split}$ (9) $$

Though $\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}((\text{AVG}(|\Delta\bm{W}|)-|\Delta
W
_{ij}|)=d\cdot k\cdot\text{AVG}(|\Delta\bm{W}|)-\sum\limits_{i=1}^{d}\sum
\limits_{j=1}^{k}|\Delta W_{ij}|=0$, it is hard to conclude that Equation (9) equals 0 due to the multiplication of $\text{Sign}(\Delta W_{ij})\cdot\nabla\mathcal{L}^{c}_{ij}$. Based on the approximation of the loss in Figure 3, it can be observed that the loss of BitDelta on GSM8K is greater than 0, which is consistent with its performance degradation on GSM8K compared to post-trained model.

### 5.2 Express Twin-Merging and TIES-Merging with Approximation Term

Twin-Merging employs singular value decomposition on delta parameters to derive task-specific knowledge. TIES-Merging preserves delta parameters with the highest magnitudes to minimize redundancy. Their computation processes are

$$ $\begin{split}\bm{W}_{\text{Twin}}&=\bm{W}_{\text{POST}}+\Delta\widetilde{\bm{W }}_{\text{Twin}}=\bm{W}_{\text{PRE}}+\Delta\bm{W}+\Delta\widetilde{\bm{W}}_{ \text{Twin}}=\bm{W}_{\text{PRE}}+\bm{U}_{r}\bm{\Sigma}_{r}\bm{V}_{r}^{T},\\ \bm{W}_{\text{TIES}}&=\bm{W}_{\text{POST}}+\Delta\widetilde{\bm{W}}_{\text{ TIES}}=\bm{W}_{\text{PRE}}+\Delta\bm{W}+\Delta\widetilde{\bm{W}}_{\text{TIES}} =\bm{W}_{\text{PRE}}+\bm{M}\odot\Delta\bm{W},\end{split}$ (10) $$

where rank $r\leq\min(d,k)$ denotes the number of linearly independent columns (or rows) in $\Delta\bm{W}=\bm{U}\bm{\Sigma}\bm{V}^{T}$. $\bm{U}_{r}\in\mathbb{R}^{d\times r}$ consists of the first $r$ columns of $\bm{U}$ (whose columns are the left singular vectors of $\Delta\bm{W}$). $\bm{\Sigma}_{r}$ is the $r\times r$ diagonal matrix containing the top $r$ singular values. $\bm{V}_{r}\in\mathbb{R}^{k\times r}$ includes the first $r$ columns of $\bm{V}$ (whose columns are the right singular vectors of $\Delta\bm{W}$). $\bm{M}\in\mathbb{R}^{d\times k}$ is a binary mask matrix where an entry of 1 indicates that the corresponding delta parameter is among the top-$n$ percent in magnitude. $n$ is the proportion of delta parameters to be retained.
According to Equation (10), we derive

$$ $\begin{split}\Delta\widetilde{\bm{W}}_{\text{Twin}}&=\bm{U}_{r}\bm{\Sigma}_{r} \bm{V}_{r}^{T}-\Delta\bm{W},\\ \Delta\widetilde{\bm{W}}_{\text{TIES}}&=\bm{M}\odot\Delta\bm{W}-\Delta\bm{W}=- \neg\bm{M}\odot\Delta\bm{W},\end{split}$ (11) $$

where $\neg\bm{M}$ is the element-wise NOT operation. Based on Equation (1), we get

$$ $\begin{split}\Delta\mathcal{L}_{\text{Twin}}&\approx\frac{1}{C}\sum\limits_{c= 0}^{C-1}\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}({\bm{U}_{r}\bm{\Sigma}_{r} \bm{V}_{r}^{T}}_{ij}-\Delta W_{ij})\cdot\nabla\mathcal{L}^{c}_{ij},\\ \Delta\mathcal{L}_{\text{TIES}}&\approx-\frac{1}{C}\sum\limits_{c=0}^{C-1}\sum \limits_{i=1}^{d}\sum\limits_{j=1}^{k}\neg M_{ij}\cdot\Delta W_{ij}\cdot\nabla \mathcal{L}^{c}_{ij}.\end{split}$ (12) $$

We exploit the value of the approximation term through experiments.
Models were constructed using LLaMA3-8B-Instruct, and the approximation term was calculated on the GSM8K dataset. As shown in Figure 3, for TIES-Merging, when the drop rate is relatively low, the approximation term is also lower. However, as the drop rate increases to a certain level (e.g., 0.9), the performance begins to degrade compared to DARE, which is consistent with the observations in DARE. For Twin-Merging, the approximation loss is greater than zero, which aligns with the observed performance degradation on the GSM8K dataset.

### 5.3 Extension of BitDelta

We also extend the applicability of BitDelta by offering a more general form. Firstly, in addition to selecting the signs of delta parameters, we hypothesize that the effectiveness of BitDelta may stem from its choice of a holistic statistic that reflects the properties of the delta parameters.
Specifically, BitDelta utilizes the average magnitude of delta parameters to achieve the best approximation error in the $L_{2}$ norm.
To validate this, we conduct an experiment where we alter the holistic statistic selected by BitDelta, introducing varying degrees of noise to the average value.
As illustrated in the ”Degenerate” line of Figure 5, using the true average magnitude of the delta parameters yields nearly optimal performance on GSM8K, TruthfulQA, and HumanEval.
The performance changes along the degenerate line are quite steep, and slight modifications to this average value may result in a degradation of model performance.

Figure: Figure 4: Effectiveness of increasing the number of bits in BitDelta. The left subplot shows the performance of LLaMA3-8B-Instruct and Mistral-7B-Instruct-v0.3 on the GSM8K dataset as the number of bits increases. The right subplot shows the performance on the TruthfulQA dataset. In each subplot, we use the dashed line to represent the performance of the original post-trained model.
Refer to caption: x4.png

Secondly, instead of using a single value, we sample delta parameter magnitude matrices from both standard normal and uniform distributions, with the average magnitude serving as the mean. The experimental results, as depicted in Figure 5, demonstrate that even when these parameters are randomly sampled from distributions, the model performance remains on par with a statistic value used in BitDelta. This further underscores the significance of selecting an appropriate holistic statistic for the delta parameters.

Finally, while preserving the relative magnitude relationships of delta parameters, we enhance the effectiveness of BitDelta by employing multiple bits.
Specifically, we divide the delta parameters into $M$ blocks based on their magnitude, from smallest to largest.
Each block is then represented by the average value of the delta parameters within that block. When $M=1$, this approach corresponds to BitDelta, and when $M$ equals the total number of parameters in the model, it degenerates to the original post-trained model. The number of bits used is given by $\log_{2}M$.
As shown in Figure 5, increasing the number of bits significantly improves the model performance. When the number of bits is 4, the performance already surpasses that of the original post-trained model. This again highlights the redundancy in the delta parameters and demonstrates the potential for further advancements by expanding the bit representation in BitDelta.

## 6 Unifying Editing Operations with Improved Performance

Figure: Figure 5: Validation of the extension of BitDelta. The degenerate curve at 1.0 represents the original BitDelta. The full results on 8 datasets are shown in Figure 12.
Refer to caption: x5.png

EXPO (Zheng et al., ) is a recent method to extrapolate delta parameters, which can boost LLMs’ alignment. This section chooses EXPO as the representative approach for illustration.

### 6.1 Express EXPO with Approximation Term

Technically, EXPO first computes delta parameters between an aligned model and its initial fine-tuning checkpoints, and then extrapolates delta parameters with a suitable scaling factor for obtaining a better-aligned model. The calculation procedure is

$$ $\bm{W}_{\text{EXPO}}=\bm{W}_{\text{POST}}+\Delta\widetilde{\bm{W}}_{\text{EXPO }}=\bm{W}_{\text{PRE}}+\Delta\bm{W}+\Delta\widetilde{\bm{W}}_{\text{EXPO}}=\bm {W}_{\text{PRE}}+\Delta\bm{W}+\alpha\Delta\bm{W},$ (13) $$

where $\alpha$ controls the extrapolation length. Based on Equation (13), we derive

$$ $\Delta\widetilde{\bm{W}}_{\text{EXPO}}=\alpha\Delta\bm{W}.$ (14) $$

Referring to Equation (1), we obtain

$$ $\Delta\mathcal{L}_{\text{EXPO}}\approx\frac{\alpha}{C}\cdot\sum\limits_{c=0}^{ C-1}\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{k}\Delta W_{ij}\cdot\nabla\mathcal {L}^{c}_{ij}.$ (15) $$

Figure: Figure 6: Validation of our theoretical analysis of EXPO. we can observe that the approximation term first decreases and then increases as alpha changes, indicating that optimal performance is achieved at the trough.
Refer to caption: x6.png

An intuitive explanation for the improvements that EXPO achieves is that the DPO/RLHF training process of these models is suboptimal, which leads to the direction of loss reduction (the negative gradient) still aligning with the direction of the delta parameters, causing Equation (15) to be negative. Consequently, the loss of the edited model on alignment dataset is lower than that of the original post-training model, resulting in enhanced performance on alignment benchmarks.

We validated the aforementioned hypothesis by conducting experiments on Zephyr-7B-DPO-Full (trained by EXPO). We calculated the gradient of the models using DPO loss on UltraFeedback (Cui et al., ). As shown in Figure 6, when $\alpha$ is relatively small, the value of the loss approximation term gradually decreases, reflecting that the model is indeed suboptimal. Moving further in this direction decreases the loss and improves performance accordingly. However, as $\alpha$ increases, the loss term gradually increases until it exceeds zero, which is consistent with the observation in EXPO that there is an optimal value for $\alpha$.

### 6.2 Futher Discussions on EXPO

Figure: Figure 7: Comparison of Extrapolation and Interpolation Performance on LLaMA3-8B-Instruct. The performance gap represents the difference between the model’s performance after extrapolation or interpolation and the original performance.
Refer to caption: x7.png

EXPO claims that extrapolating delta parameters leads to better models. However, based on the derivation in Equation (15), we believe that whether to use extrapolation or interpolation primarily depends on the direction of the gradient, which is influenced by the specific data. Specifically, for LLaMA3-8B-Instruct, we uniformly selected $\alpha$ in the range of -1.0 to 1.0 at intervals of 0.1, performing both interpolation and extrapolation of the model’s delta parameters. As show in Figure 7, on most datasets, interpolation outperformed extrapolation, except for the IFEval dataset, where extrapolation significantly improved performance. This confirms that whether to interpolate or extrapolate is not a fixed formula but depends on the specific data.We also conducted experiments on Qwen2-7B and Mistral-7B(shown in Section B.3), and the results indicate that even for the same task, whether extrapolation or interpolation is required can vary across different models.

## 7 Conclusion and Discussions

Post-training is a core step in the training of large models. In recent years, significant efforts have been directed towards editing the delta parameters of post-training to achieve improvements in either performance or efficiency. However, while previous work has shown some effectiveness, the complexity of large model parameters has led to a fragmented understanding of delta parameter editing, with different studies focusing on different aspects of its effectiveness, lacking a unified perspective.

In this paper, we provide a unified perspective on the previous work related to post-training delta parameter editing using Riemann sum approximation.
We find that the changes in model capability after altering the delta parameters can be analyzed through the loss differences approximated using Riemann sums.
By analyzing this approximation term, we can infer the reasons why the existing delta parameter editing methods lead to maintained, improved, or reduced model performance.

Our work offers a concise, unified, and powerful explanation for many previous work in the field of post-training delta parameter editing. We validate our hypothesis through numerical experiments. From our conclusions, several potential applications emerge for future work in this direction:
(1) Model Quantization: By finding an edit that sets the approximation term to zero while using lower precision, we can achieve nearly lossless compression of the model.
(2) Model Enhancement: By analyzing the approximation term, we might be able to find ways to enhance the model’s capabilities without additional training data.
(3) Post-training Mechanism Analysis: Since the model’s capability remains almost unchanged when the approximation term is zero, we can construct more concise post-training delta parameters.
This simplifies the parameter changes during the post-training phase, enabling a more effective analysis of the parameter mechanisms in this stage.

Additionally, our work highlights a critical observation: the analysis of parameter changes during the post-training phase should not be limited to specific parameters, such as knowledge neurons, but should consider the overall distribution of parameters. This is because the key constraint of the approximation term being zero does not depend on the changes in a specific parameter during post-training but requires a comprehensive consideration of all parameter deltas. This suggests that trying to infer the impact on the global model parameters from changes in a single or a few local parameters is likely futile.

## Appendix A Experimental Details

For the loss estimation experiments, we set the constant $C=5$ to calculate the approximation term. We interpolate the $\Delta\widetilde{\bm{W}}$ parameter at values of 0.2, 0.4, 0.6, and 0.8 to generate different models. For all large language model evaluations in Chapters 4 and 5, we employ the lm-eval framework for assessment. Further details on the tested models and datasets can be found in the original paper.

## Appendix B Full Experimental Results

### B.1 Extension of DARE

We conduct a thorough experimental validation on the extension of DARE. The results of LLaMA3-8B-Instruct, Mistral-7B-Instruct-v0.3, and ViT-B-32 across eight benchmarks are presented in Figure 8, Figure 9, Figure 10, and Figure 11, respectively.

Figure: Figure 8: The performance of LLaMA3-8B-Instruct on the all benchmarks under varying $p$ and $k$.
Refer to caption: x8.png

Figure: Figure 9: The performance of Mistral-7B-Instruct-v0.3 on the all benchmarks under varying $p$ and $k$.
Refer to caption: x9.png

Figure: Figure 10: The performance of Qwen2-7B-Instruct on the all benchmarks under varying $p$ and $k$.
Refer to caption: x10.png

Figure: Figure 11: The performance of ViT-B-32 on the all benchmarks under varying $p$ and $k$.
Refer to caption: x11.png

### B.2 Extension of BitDelta

The results of LLaMA3-8B-Instruct across eight benchmarks are presented in Figure 12.

Figure: Figure 12: Validation of the extension of BitDelta on LLaMA3-8B-Instruct.
Refer to caption: x12.png

### B.3 Discussion on EXPO

In Figure 13, we present the comparision of interpolation and extrapolation.

Figure: Figure 13: Comparison of Extrapolation and Interpolation Performance.
Refer to caption: x13.png

## Peer Reviews

confidence: 4
contribution: 1
initial_rating: 1
presentation: 3
questions: see weaknesses
soundness: 1
strengths: 1. The paper is clearly written.
summary: Authors propose using an approximation term to evaluate various methods to compress the model. In particular, authors use Riemann sum to establish the connection between delta W and delta L. Authors discuss different cases where approximation term (delta L) is equal to, larger than, or smaller than 0.
weaknesses: 1. The derivation in 4.1 is mathematically trivial due to this Riemann sum assumption. Either the locally constant assumption of Riemann sum is too strong or the expectation term derived in (5) is too strong. The math shows delta L is 0 regardless of p. If p=0.999, should the loss still be zero? In addition, I cannot see a connection between 4.1’s experiment and theory. The theory shows L is zero.
2. For the same reason, the math derivation in 4.2 is trivial. Adding a k does not have any effect on the proof.
3. Authors derived delta L in section 5 (larger than 0) and section 6 (smaller than 0), but there is no theoretical implication about why delta L in section 5 is positive and why delta L in section 6 is negative. The experiment results are empirical and whether it’s positive or negative has already been studied in BitDelta and EXPO.
4. A clear contradiction is: in section 4 when delta L is derived as zero, the experiment’s delta is 1e-5. While in section 6 when delta L is derived as non-zero, the absolute value of delta L is 1e-6, which is an order of magnitude smaller than the loss that is derived as zero. The experiment results also imply that the derivation in section 4 is false.
5. Overall I don’t see any value in the math derivation of this paper. The experiment part is also mostly expected after reading the paper that the respective section is referring to.

---

confidence: 5
contribution: 2
initial_rating: 3
presentation: 2
questions: See the weaknesses section.
Please let the reviewer know if there is any misunderstanding about the paper.
soundness: 1
strengths: - Present a novel unified view of existing model merging methods that is lacking so far.
- The proposed Riemann sum approximation of loss difference based analysis is interesting and gives insights for future work on theoretical understanding of model merging.
summary: This paper provides a unified view of weight-space editing methods (a.k.a. model merging) through the lens of approximated loss difference. The authors categorize existing methods into three classes -- maintained performance, increased performance, and decreased performance, and generalize the existing methods by analyzing the crucial factors in the proposed loss difference approximation framework.
weaknesses: - **Eq. (1), the main proposed theoretical framework in this work, has a severe technical flaw**.
  - Specifically, the authors analyze $\Delta L = L(W_{POST} + \Delta \tilde{W})-L(W_{POST})$ to discuss the performance of existing model merging methods.
  - However, all the existing model merging methods mentioned in this work such as DARE [1], TIES-Merging [2], BitDelta [3], and so on, applied the edited delta parameter to the $W_{PRE}$ rather than $W_{POST}$. 
  - Therefore, the desired analysis should be conducted on the loss term such as $L(W_{PRE}+\Delta \tilde{W})-L(W_{POST})$ or $L(W_{PRE}+\Delta \tilde{W})-L(W_{PRE})$ rather than the current form ($L(W_{POST}+\Delta \tilde{W})-L(W_{POST})$) to make any claims on the final downstream performance of existing merging methods.
- **Limited contributions**
  - Although the authors provide some generalization of existing methods, e.g., multiplies a magnitude hyperparameter $k$ in DARE framework, the novelty and innovativeness of these generalizations are too limited and the implications are also not surprising and uninformative. It seems like just reporting a result from engineering. Presenting more rigorous generalizations and providing profound implications from the proposed unified framework will improve the quality of this work significantly.
- **Unreasonable experiment setup**
  - Regarding the EXPO [4] method, the authors claim that the relative effectiveness of extrapolation and interpolation depends on the dataset, which shows the performances of interpolation and extrapolation over some NLP downstream tasks.
  - However, the motivation of EXPO is focused on the alignment for enhancing the instruction-following capability of large language models, and the authors should conduct the experiment about EXPO on that kind of benchmark such as AlpacaEval 2.0 adopted in the EXPO paper.
- **Bad presentation and validity of claims**
  - The quality of some presentations is not good enough for the purpose of publication. For example, see Figure 3. It is much better to omit the pre-train models' performance here to highlight the more important parts -- a comparison between varying $k$ values.
  - Moreover, although the authors make an argument based on some bar plots (Figure 1, Figure 4, Figure 7) they state the differences have some trend, and the absolute scale is too small among the comparison participants, which raises concerns about the statistical significance. 



> Reference
1. Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch, Yu et al. 2024
2. TIES-Merging: Resolving Interference When Merging Models, Yadav et al. 2023
3. BitDelta: Your Fine-Tune May Only Be Worth One Bit, Liu et al. 2024
4. Weak-to-Strong Extrapolation Expedites Alignment, Zheng et al. 2024

---

confidence: 4
contribution: 1
initial_rating: 3
presentation: 1
questions: see weakness
soundness: 2
strengths: 1. Delta parameter editing is an important topic in LLM efficiency.
summary: The authors propose a framework using Riemann sum approximation to analyze delta editing (pruning, compression ..) methods based on their effects on model loss.
weaknesses: The authors attempt to cover a broad scope in their analysis, but the exploration is often shallow, with certain aspects appearing incremental or inaccurate. 
1. Assumptions weaken the generalizability and accuracy of their conclusions:

    a.   In Equation (4), the authors address the randomness in DARE [1] by asserting that “it is straightforward to deduce that,” leading to Equation (5). This is wrong as this need to rely on $\Delta W_{ij}$ and $\Delta \mathcal{L}_{ij}$ being uniformly distributed, which is generally not the case, making the equality invalid. 

    b. In BitDelta [2] , the authors state that it is difficult to conclude that Equation (9) equals zero due to the interaction between $\text{sign}(\Delta W_{ij}) \Delta \mathcal{L_{ij}}$. This approach is inconsistent, as they assume uniformity of $\Delta W_{ij}$ and $\Delta \mathcal{L}_{ij}$  when proving Equation (5) for DARE, but not for BitDelta in Equation (9). If the uniformity also assumed for BitDelta, the $\mathcal{L}=0$ should hold. Consequently, the conclusion that BitDelta performs worse than DARE is questionable.

2. Question on EXPO[3]:

    a. Incremental Analysis and Use of EXPO Framework:  The paper uses the similar framework as EXPO, particularly referencing Equation (2) in EXPO, where EXPO use first-order Taylor Expansion with an alignment objective (which functions similarly to the loss used in the current paper). This similarity can be seen as an incremental extension rather than a substantial innovation.

    b. Claim on Gradient Correlation with Delta Parameter: In section 2.2 of EXPO, EXPO already established that the success of their approach depends on a positive correlation between the gradient and the delta parameter, highlighting a direct relationship with the approximation term. Given this established finding, it appears that the new paper is building on known results rather than offering a novel insight in this area.
   
   c. Claims on EXPO Limitations: According to EXPO, extrapolation can improve performance when a positive correlation between the gradient and the delta parameter, a point that the new paper seems to question. However, if EXPO already addressed this with clear justification, the claim in the current paper may not hold strong novelty or accuracy. If they misunderstand EXPO’s stance on extrapolation, it could weaken their argument about EXPO’s limitations.

3.  I feel the logic in DARE section is unclear. First, the extension of DARE lacks a clear connection to the theorem in Section 4.1, making the motivation for introducing k unclear. I though authors will give motivation in Section 4.3, but cannot found it. Also, in section 4.3, the authors claim that DARE overlooks delta loss. However, the original DARE analysis of random pruning considers both delta parameters and the input. Specifically, the authors’ analysis in Equation (4) focuses on delta parameters and delta loss, and due to the linear approximation, delta loss can be proportional to the input x. This resemblance to DARE makes it inappropriate to claim that DARE disregards delta loss (represented by x in DARE’s case).

4. I also find the logic in analyzing BitDelta [2] unclear. Similar to DARE, the motivation for introducing noise to mean magnitude lacks a clear connection to the theorem in Section 5.1. Additionally, the original BitDelta [2] already demonstrates that calibrating the scaling factors can improve performance as their contribution. This resemblance to BitDelta makes it inappropriate to claim that BitDelta overlooks this issue, which limits the novelty of the current approach.



[1]  Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch

[2]  BitDelta: Your Fine-Tune May Only Be Worth One Bit

[3] Weak-to-strong extrapolation expedites alignment.