Delta-CoMe: Training-Free Delta-Compression with Mixed-Precision for Large Language Models
Authors: Bowen Ping
Shuo Wang
Hanqing Wang
Xu Han
Yuzhuang Xu
Yukun Yan
Yun Chen
Baobao Chang
Zhiyuan Liu
Maosong Sun
Venue: NeurIPS 2024 poster (2024)
Keywords: Large Language Model
Delta Compression
Efficient Serving
Multi-Tenant Serving
Primary area: natural_language_processing

Abstract: Fine-tuning is a crucial process for adapting large language models (LLMs) to diverse applications. In certain scenarios, such as multi-tenant serving, deploying multiple LLMs becomes necessary to meet complex demands. Recent studies suggest decomposing a fine-tuned LLM into a base model and corresponding delta weights, which are then compressed using low-rank or low-bit approaches to reduce costs. In this work, we observe that existing low-rank and low-bit compression methods can significantly harm the model performance for task-specific fine-tuned LLMs (e.g., WizardMath for math problems). Motivated by the long-tail distribution of singular values in the delta weights, we propose a delta quantization approach using mixed-precision. This method employs higher-bit representation for singular vectors corresponding to larger singular values. We evaluate our approach on various fine-tuned LLMs, including math LLMs, code LLMs, chat LLMs, and even VLMs. Experimental results demonstrate that our approach performs comparably to full fine-tuned LLMs, surpassing both low-rank and low-bit baselines by a considerable margin. Additionally, we show that our method is compatible with various backbone LLMs, such as Llama-2, Llama-3, and Mistral, highlighting its generalizability.

Delta-CoMe: Training-Free Delta-Compression with Mixed-Precision for Large Language Models
Bowen Ping
Shuo Wang Hanqing Wang Xu H Yuzhuang Xu² Yukun Yan?
Yun Chen
Baobao Chang' Zhiyuan 2455. Maosong Sun 2.45.4,5f
Peking University Deptt. of Comp. Sci. & Tech., Tsinghua University, Beijing, China
Shanghai University of Finance and Economics
Innttittte for AI, Tsinghua University, Beijing, China
Beijing National Research Center for Information Science and Technology
Abstract
Fine-tuning is a crucial process for adapting large language models (LLMs) to diverse applications. In certain scenarios, such multi-tenant serving, deploying multiple LLMs becomes necessary to meet complex demands. Recent studies suggest decomposing fine-tuned LLM into a base model and corresponding delta weights, which are then compressed using low-rank or low-bit approaches reduce costs. In this work, we observe that existing low-rank and low-bit compression methods can significantly harm the model performance for task- specific fine-tuned LLMs (e.g., WizardMath for math problems). Motivated by the long-tail distribution of singular values in the delta weights, we propose delta quantization approach using mixeddpreeisiin representation for singular vectors corresponding to larger singular values. We evaluate our approach on various fine-tuned L.LMs, including math L.MMs, code LLMs, chat LLMs, and even VLMs. Experimental results demonstrate that our approach performs comparably to full fine-tuned LLMs, surpassing both low-rank and low-bit baselines by considerable margin. Additionally, we show that our method is compatible with various backbone LLMs, such as Llama-2, Llama-3, and Mistral, highlighting its generalizability.
This method employs higher-bit
Introduction
Large language models (LLMs) Touvroo et al. 2023 Jiang al. 2023 increasingly becoming the standard for a wide range of downstream tasks (Luo et al.) 2023a Yu et al.) 2023 Wei et al.
2023 Luo al. 2023b Liu et al. 2024a Wang al. 2023), significantly surpassing conventional small models. To meet the demands of various application domains and scenarios, many researchers
direct their attention to developing advanced alignment or adaptation algorithms together with diverse training data to learn aligned LLMs based on generally pre-trained models. For instance, Luo (2023a) propose reinforcement learning from evol-instruct feedback (RLEIF) method to construct LLMs with strong mathematical reasoning abilities. Similarly. Yu et al. (2023) employ bootstrapping method to diversify mathematical questions and then fine-tune open-source LLMMs to build mathematical models. For code generation, Luo et al. (2023b) adapt the evol-instruct method to the coding domain, resulting the WIZARDCODER model, which demonstrates superior coding abilities compared to generally trained LLMs. Additionally, Wei et al. (2023) enhance the capabilities of open-source code LLMs by using automatically generated high-quality instruction data based on
Equal contribution.
1 Corresponding authors.
Code will be publicly available at attppp:///gtabbbb.ca///b/lalllla
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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BiDeita
SVD low rank
Ours
1or
168t
00t
Sinuuar Wa
0.5
-... Singuls Value
Singuler Value
28t
301
E. 2
G
Sinpuar Value
1.2
10-
Singiar Vasee
Bits
0.3
Bits
12
Bits
0.1
2000
4000
1000
2020 3000
3000
4002
Singular Value Index
Singular
Singular Value
Figure/ 1: Left: illustration of BitDelta (Liu et al. 2024b), which employs 1-bit quantization for all the delta weights. Middle: illustration of low-rank compression (Ryu et al. 2023b), retaining the top-k singular values and the corresponding singular vectors. Right: illustration of the proposed Delta-CoMe method, which represents the singular vectors of larger singular values using high-bit vectors while compressing the singular vectors of smaller singular values into low-hit representations. This method is inspired by the long-tail distribution of singular values in delta weights.
existing code snippets. Wang et al. (2023) utilize various resources of mixed quality and design new conditioned reinforcement learning fine-tuning method to train the OPENCHAT model. Beyond the text modality, some studies propose fine-tuning pre-trained LLMs to understand other modalities. For instance, [iu et al. (2024a) construct a multi-modal instruction tuning dataset and develop the LLAVA model, which can understand both text and images.
Building on the aforementioned alignment approaches, LLMs are endowed with specialized capabili- ties that align with distinct user demands and application requirements (Liu et al. 2024b). In certain scenarios, deploying multiple L.MSs with different abilities is necessary. For example, in multi-tenant serving, different LLMs may be needed to satisfy various users. Additionally, some complex tasks consist of multiple sub-tasks, each requiring different model capabilities. To address these tasks, we should organize and deploy group of LLMs simultaneously. A straightforward question arises: why not use a single general LLM that encompasses all the necessary capabilities? For example, we could develop one model that can both understand images and generate code programs. To our knowledge, LLMs with various capabilities (e.g., GPT-4 typically have an enormous number of parameters, making them impractical for sssorrce-limite situations (e.g., edge-side scenarios).
In pursuit of this objective, field of research advocates for the minimization of expenses associated with multi-model serving. Delta-co emerges as a crucial and viable approach in this context, offering the potential to decrease both storage requirements and GPU memory utilization in scenarios involving multiple models. The primary objective of delta-compression is to minimize the size of the -AAAMM---CHAL and LLAMA-2), 2)) Ryu ett al. (2023b) identify the low-rank nature of delta weights and enhance storage efficiency through low-rank approximation. Alternatively, [iu et al. (2024b) propose a 1-bit quantization approach, termed BitDelta, to further reduce the size of delta weights. Thev validate the effectiveness of BitDelta across various chat models, including LLAMA- 2-CHAT (Touvron et al. 2023), VICUNA and WIZARDLM (Xu et al. (023). In this work, we reassess the performance of both low-rank and low-bit delta-comppession methods across a diverse range of aligned LLMs, encompassing mathematical, coding, chat, and multi-modal L.LMs. Our experimental results (e.g., Table 3 reveal that current low-rank and low-bit compression techniques may significantly degrade the performance of aligned LLMs. These results motivate us explore more advanced compressoonn methods capable of achieving performance nearly equivalent to the aligned LLMs before compression.
delta weights between aligned and pre-trained LLMs (e.g..
Inspired by the long-tail distribution of singular values, illustrated in Figure I we propose allocating bigher-bit representations for singular vectors associated with larger singular values, given their greater impact on the approximation of delta weights prior to compression. Conversely, for singular vectors associated with smaller singular values, we employ low-bit formats to reduce the delta size. For singular values that extremely small, omit the corresponding singular vectors altogether. The resulting method, which we term Delta-CoMe, can be viewed as a hybrid of low- rank and low-bit compression techniques. Delta-CoMe outperforms both the low-rank compression method and BitDelta. Moreover, our method achieves performance comparable to that of the full
httpe://chattptt.cmm
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aligned LLMs. For instance, Delta-CoMe attains an average score of 53.2 across eight representative tasks, closely matching the average score of 53.5 achieved by the aligned LLMs. In comparison, the scores of the low-rank and low-bit baselines are 47.8 and 49.3, respectively.
Further, we compare the performance of the involved delpa-comp methods LoRA (Hu et al. 2022). widely-used delta- delta-tuning approach (Wang et al. 2024). The primary distinction between delta-compression and delta-tuning that delta-compression first optimizes the full model and then converts the modified weights into a lightweight module, reducing inference costs in multi-model settings. In contrast, delta- titta-tuning primarily aims to lower training costs. Our experimental results demonstrate that the proposed Delta-CoMa method significantly outperforms LoRA, with scores of 41.9 versus 29.8. respectively. These results suggest that delta-compression can deliver superior performance in multi-model settings compared to delta-tuning.
Finally, Delta-CoMe achieve more than 10x GPU memory and disk storage savings, enabling the deployment of multiple models with limited resources. For practical application, we implement Triton (Tilect et al., 2019) kernel tailed for Delta-Come, achieving approximately a 3 speedup compared to the PyTorch implementation.
Our contribution can summarized as follows:
We propose a mixee-precision delta-compression method that employs varying bit-widths for different singular vectors based their singular values;
We validate the effectiveness of the proposed method across different types of aligned LLMs of varying sizes, including mathematical, coding, chat, and multi-modal LLMs;
We conduct in-depth analyses to understand the superior performance of our method over low-rank and low-bit baselines. Our method can also outperform selta-tuning approaches such as LoRA, demonstrating that the proposed cempr-comooosion method is more practical for multi-model serving scenarios.
We verify that the proposed method can achieve over 10x saving in GPU memory and disk storage. By constructing a Triton kernel, we can achieve approximately a 3x speedup. demonstrating the hardware compatibility of Delta-CoMe.
2 Related Work
2.1 Delta-Compression
Recently, celta-compression has garnered increasing interest in the LLM community due to its ability to substantially diminish the storage and inference expenses associated with serving multiple models. GPT-Zip extends the GPTQ approach (Frantar et al. 2023) to compress the delta weights between aligned models and the backbone model, successfully using 2-bit delta weights to approximate the model. Additionally, they sparsify the quantized delta weights to further reduce storage costs. However, the sparsification technique can hardly reduce GPU memory usage during inference. Similarly, Yu et al. (2024) find that dropping the majority of the delta weights has a limited effect on the performance of aligned LLMs. Ryu et al.] (2023a) identify the low-rank property of delta weights and propose reducing the storage requirements of aligned LLMs through low-rank approximation. Yao & Klimovic (003) adopt the concept of celta-compression to develop a multi-tenant serving system, DeltaZip. Most recently, [Lu et al. (2024b) introduced BitDelta, which successfully quantizes the delta weights into 1-bit. However, they only examined the performance of this compression method using chat LLMs, leaving wide range of other types of aligned LLMs unexplored. In this work, we propose leveraging the benefits of both low-rank and low-bit compression methods by using varying bit-wssths represent different components of the delta weights. We evaluate representative low-rank and low-bit bit delta-compressien methods across various types of aligned LLMs to provide a comprehensive comparison of these methods.
2.2 Model Compression with Mix-Precision
Using mixxedpprcisionn to compress the model weights is an effective technique that has been investigated many previous studies. SpQR (Dettmers et al. (023)) isolates small number of outlier weights and retains them with high-precision, while keeping the other weights at low-precision,
3

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resulting in significant performance improvement. Based on activations, Agile-Quant (Shen et al. 2024) utilizes token pruning achieve mixed-preeee sexeedprecision quantization of both weights and activations. Bablani et al. 2023) propose employing varying bit-widths for different layers of the model, while Yao al. )2021) propose quantizing activations and model weights with different precisions. In this work, we propose using mixed-precistie compression for different singular vectors of the delta model, marking the first method to introduce sixed-precision compression for delta weights.
Approach
Code
Math
Multi-Modal Modal
Aligned Models
Compressed Deltas
Backbone Model
Figure 2: Illustration of Delta-CoMe, where utilize varying bit-widths for singular vectors with different singular values. Singular vectors corresponding to larger singular values are assigned higher bit-widths. For extremely small singular values, we omit the singular vectors (i.e., 0-bit).
3.1 Preliminaries
For backbone LLM 6g, we can customize into an aligned model 0, for specific purpose using advanced alignment algorithms (Xu et al. 2023 Luo et al. [2023a] Yu al. 2023 Luo et 2023b Wei al.) 2023 Liu et al. (0224)). In some practical scenarios, mentioned in Section | may need to deploy multiple LLMs the same time. Formally, we should store and deploy a series of {o()) .... NoN) where N is the number of aligned models. The total size of the group of aligned models is N x M, where M is the size of one model. We use A to represent the delta weights between the aligned model and the backbone model, which given by
aligned LLMs
(n)
Ob.
(I)
where ((n) is the n-th aligned LLM. Note that the sizes of A (n) @(a) and O, are the same.
Delta-compressiee aims to compress the delta weights A (n) into Å (n), where the latter has signifi- cantly fewer parameters. After delta-compression, we can only maintain one backbone model and
N compressed delta models: 0,, (1), Á A()). The total size is decreased from N M to
(1 aN) x M, where is the compression ratio. During inference, can restore each aligned
LLM in the following way:
@(n) = 0, + À (s)
For good delta-compressson method, expect it can achieve smaller a, while making @(n) attain comparable performance with o(m). BitDelta (Liu al. 2024b), to our knowledge, is the most recent study that successfully quantizes delta weights into b-btt. which means that 1/16 when the

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original aligned model is represented by FPI6 or BFI6. In this work, we propose to improve the performance of compresomon methods by inducing mixed-precision quantization, which will be detailed in the following sub-sections.
3.2 Delta Decomposition
Previous works have investigated mixed-precision model compression methods at the token (Shen et al., 2024) layer level (Bablani al. (202).. For delta-con colta-compression, propose employing mixed precision for different singular vectors. We first use the SVD algorithm to decompose each delta matrix:
AW = UEVT,
(3)
where AW& E Rhout Xhin U E R Aout X hous, E E R hout x hin V E R Rhixkhn Intuitively, the singular vectors associated with larger singular values have greater impact on the approximation the delta matrix A AW, we thus spend more bits for these vectors to reduce the quantization error.
3.3
Mixed-Precision Quantization
Some representative quantization methods, such as GPTQ (Frantar et al. 2023), aims to minimize the following objective:
W = Quant (W, (Wan X) argmin WX Wx
W
where X E Righin is the input to the parameter W and W is the corresponding quantized parameter. We use Quant, to denote the k-bit quantization algorithm. In this work, we employ the widely-used GPTQ method with group_size = 128 for cases where k >> 1, and BitDelta for 1-bit quantization. For a certain group of singular vectors, let Tbegin and Tend represent the start and end indices, respectively. The quantization of the singular vectors be given by
Vi Tbegin Tena]' T = Quanty Tbegin Tena] X),
Üß: "begin /end]]
(5)
Quant (U. Tbegin Tendl I(rbegiu Tend: Pbegin read V[t, Pbegin Fend] X).
As illustrated in Figure 2 varying quantization bits for different groups of singular vectors. By employing different mixed-precision strategies, we can control the trade-off between achieving a small delta size and maintaining high performance. We will provide more details about the exploration of the mixing strategy in Section
Experimental Setup
To thoroughly investigate the proposed delta-comppessson method Delta-CoMe and the involved baselines, examine the performance of different methods across several tasks, which are typical applications of recent aligned LLMs.
4.1 Tasks
Mathematical Problem Solving Solving mathematical problems is a challenging task for modern LLMs. For this task, we employ GSM8K (Cobbe et al. 2021) and MATH (Hendrycks et al. (021) as the evaluation datasets, which are among the most popular mathematical benchmarks for L.MMs. The reported score accuracy, which estimated by comparing the ground-truth number with the result calculated by the model.
Code Generation The ability to process code is crucial for numerous practical applications, including data analysis and LLM--ased agents. For this task, we use HumanEval (Chen al.) 2021) and MBPP Austin et al. (021)) as the evaluation datasets, which are widely used in recent studies. The reported score is the pass rate, indicating that the model-generrted code can successfully run the test cases in one pass (i.c., pass @ 1).
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Chat The chat ability enables LLMs to intercct with users, providing helpful and safe suggestions or responses based on the user's requests. A good chat model expected to be well aligned with human preferences. For evaluating chat L.MMs, we select TruthfulQA (Lin et al., 2022) and SafehyBench (Zhang et al. 2023) as the evaluation datasets, which measure helpfulness and safety, respectively. The reported score is the accuracy, indicating that the choice of the model is correct.
Multi-Modal Chat Vision-language models (VLMs) are attracting increasing attention due to their ability to process both text and images. Most recent VLMs based pre-trained visual encoders and language models, with the language models fine-tuned to understand the visual signal. For this task, we GQA (Hudson & Manning (019) and TextVQA (Singh et al. 2019). The reported score the accuracy, indicating that the choice the model correct.
4.2 Models
Table Selected backbone and aligned models for the examined four tasks.
7B Models
13B Models
Task
Backbone
Aligned
Backbone
Aligned
Math
LLAMA-2
WIZARDMATH-V1.0
LLAMA-2
WIZARDMATH-VL.O
Code
CODELLAMA-PY
MAGICODERS-CL.
CODELLAMA-PY
SAAARDDCDER-PP-VIII
Chat
LLAMA-2
LLAMA-2-CHAL
LLAMA-2
LLAMA-2-CHA
Multi-Modal
VICUNA-V1.5
LLAVA-V1.5
VICUNA-VI.5
LLAVA-V1.5
For the four tasks, we provide the backbone and aligned models in Table All the model weights are open-soureed by the authors. We use both and 13B models make a thorough comparison between different delta-compression models. During inference, we use greedy search.
4.3 Baselines
We employ two representative baselines, including SVD-based low-rank compression and Bit- Delta (Liu al.) 2024b). For the low-rank baseline, we re-implement the method, while for BitDelta, we use the code open-sourced by the authors All methods are evaluated on NVIDIA A100 GPUs.
Experimental Results
5.1 Exploration of Mixed-Precision Strategies
Table 2: Comparison of different mixed- precision strategies.
To determine which bit-width to use and how many singular vectors quantize, conduct preliminary experiment using different mixed-precision strategies. We examine three types of strategies: single-precision, double-precision, and triple-precision settings. The size of the compressed delta remains consistent across settings. For single-precision compression, to O, and Tond is set to guarantee that the delta size is the same as BitDelta (Liu al.] (024b). In other words, the compression ratio a for all settings is 1/16. Formally, for delta matrix AW E Rhoa xhin, Tbegin and Tead set to satisfy the following equation:
# Precision
Setting
GSMSK
45.6
2
50.6
3
$1.8
Single
4
51.6
set Tbegin
47.8
16 3
52.5
++3
Double
4 +
52.2
3 2
52.3
& x (Tend - (hogin) hin) = 16 ahout/in (6)
16 + +
53.2
Triple
4 +
52.2
where is set to 1/16 in our experiments, which is the same BitDelta. In double-precisioe settings, Tbegin and Fond are set to 0 and 2, respectively, for the first precision. For the second precision, Tbegin
8+3+2 53.6

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is set to 2, and rend is adjusted so that the total delta size is 1/16 of the uncompressed delta. In triple-precision settings, Phegin and Lend are set and respectively, for the first precision. Thegin and r fend are set to 2 and 34, respectively, for the second precision. For the third precision, Pbegin is set to 34, and rend is adjusted so that the total delta size is 1/16 of the uncompressed delta. Since the diagonal matrix E occupies little storage, the averaged bit-width for triple-precision compression is approximately
hout hin
3 Σ
begia
(7)
hout hin
We conduct experiments on the math task, and the results are shown in Table 2 3-bit setting performs best among the Langle-precsstee settings. Therefore, keep the 3-bit setting and add other bit-widths to form double-precision settings. Among the cuable-precisoon settings, "8+3" achieves the highest score, which is then combined with additional bit-width form triple-preeee peele-preciioon settings. We find that the best double- peeeieiion setting can outperform the best single-precision setting, and the best triple-precision setting achieves the highest across all the examined settings. We use "8+342" as the default setting in the following experiments.
find that the
5.2 Main Results
Tables 3 and show the performance of different cella--ompresioonn methods on 7B and 13B models, respectively. Across all tasks, Delta-CoMe outperforms both baselines. While BitDelta (Liu al., 2024b) can achieve near lossless performance on chat models, it significantly degrades the performance of math and code LLMs, phenomenon not investigated by Liu et al. (2024b). Surprisingly, our method achieves good performance in the delta-compression of VLMs. To our knowledge, we are the first to investigate delta-compression for VLMs.
Table 3: The performance of different delpa-soopoessionn methods 7B aligned models.
WIZARDMATH
MAGICODERS-OO
-LAMA-2-CHAL
LLAVA-VI.5
Method
Ave.
GSMSK
MATH
HumanEval
MBPP
SafetyBunch
TrathfuIQ
GQA
TextVQ
Backbone
11.0
2.9
38.4
47.6
41.7
38.9
n/a
n/a
n/a
Aligned
55.2
10.9
70.7
69.2
59.5
44.6
62.0
58.2
53.5
Low-Rank
1/16
43.2
8.0
56.7
65.7
55.4
42.5
$7.7
53.3
47.8
BitDelta
1/16
45.6
8.6
57.3
65.9
59.3
41.1
59.7
56.9
49.3
Delta-CoMe
1/16
53.6
10.3
67.1
67.9
59.8
47.0
61.7
58.5
53.2
Table 4: The performance of different delta-compression methods 13B aligned models.
WIZARDMATH
WIZARDCODER
LLAMA-2-CHAT
LLAVA-V1.5
Method
Ave.
GSMSK
MATH
HumanEval
MBPP
SafehyBench
TruthfaQQ
GQA
TeaVQA
Backbone
17.8
3.9
43.3
49.0
55.0
37.3
n/a 61.3
Aligned
63.9
14.0
60.4
66.9
62.7
43.9
63.2
54.5
Low-Rank
1/16
54.2
9.4
53.0
62.3
60.2
58.3
$1.0
BitDelta 1/16
54.8
10.6
51.8
64.2
62.6
41.6
60.9
60.3
50.9
Delta-CoMe
1/16
58.9
12.8
57.9
67.2
62.9
44.1
63.1
61.2
53.5
5.3 Results on More Backbone Models
To investigate the generalization abilities of the delta-compression methods, we conduct experiments on aligned models based on other representative backbone LLMs. For additional backbones, utilize MISTRAL-7B-VO.. Jiang et al. 2023) and LLAMA-3-SE The corresponding aligned
http::///uugghagaaeccco/otttaal
Llama-2 3-

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Table 5: Results other representative backbones. The backbone of ODPENCHAT-3.5-000 WWagg et al. 2023) MISTRAL-7B--O,, (Jiang et al. 2023). Both MISTRAL-7B-VO.1 and LLAMA-3-8B
uidely-used open-source LLMs.
OPENCHAT SERHHATT-3.5-010 3.5-0106
LLAMA --AMA-3--85 CE--B-INSTRUC
Method
Ave.
GSMBK
HumanEval
TruthfelQA
SafetyBench
GSMSK
Humankval
TruthfulQA
SafetyBench
Backbone
52.2
28.7
61.0
42.1
44.8
33.5
43.6
43.9
43.7
Aligned
77.1
73.2
78.4
61.0
78.5
61.6
68.2
51.6
68.7
Low-Rank
1/16
50.5
52.4
76.9
68.3
67.5
51.3
57.8
BitDelta
1/16
70.3
54.9
78.4
50.0
67.6
56.1
68.6
50.2
62.0
Delta-Cokee
1/16
74.8
59.8
78.9
62.6
77.1
60.4
69.1
51.8
66.8
models are OPENCHAT-3.5 --500 (Wang et al., 2023 and -LLAM----BB-NNSTRUCTT respectively. As shown Table our proposed Delta-CoMe method maintains superior performance over the two baselines, demonstrating its generalization ability.
5.4 Delta-Compression vs. Delta-Tuning
closely related area to delta-compression delta-tuning. While delta-tuning primarily aims to reduce the training cost of LLMs, delta-compression focuses reducing the storage and inference cost for multi-model serving. remains unclear whether delta-compression outperforms delta-tuning when using the same delta size. To investigate this, trained LoRA (Hu et al. 2022) modules for all model parameters to compare delta- celta-compression with delta-tuning, We set the LoRA rank to 128 and the scale factor to 16, using cosine warmup schedule with warmup ratio of 0.04 and peak learning rate of le-4. For each task, we trained the LoRA for 3 epochs. For mathematical LoRA, the training dataset is from Yu al. (2023), which consists of 395K training examples. For code LoRA, the training set is from Wei et al. (2023), which contains 186K training examples. For fair comparison, fine-tune all model parameters using the same dataset used for LoRA training. We then apply different delta-comppessiin methods to both the fine-tuned mathematical and code L.MSs.
Table 6 shows the results of both delta-tunme tenitttningg and delta-compression methods. The results reveal that LoRA achieves superior performance compared to the low-rank com- pression approach and BitDelta in the mathematical task. However, when it comes to the coding task, LoRA exhibits lower performance than both low-rank compression and BitDelta. By contrast, proposed delta-compression method (i.e., Delta-CoMe) consistently out- performs LoRA across all four bench- marks. Specifically, the performance of our method is close to that of the uncompressed aligned models (41.9 vs. 43.0), while the average score of LoRA is only 29.8. These results imply that learning an aligned model and then compressing can achieve better results than delta-tuning.
Table 6: Comparison between LoRA and delta-compression methods.
Math
Code
Method
Ave.
GSMSK
MATH
HumunBval
MBPP
Backhone Aligned
11.0
2.9
10.5
17.7
10.5
65.4
18.6
43.2
44.9
43.0
LoRA
58.3
11.4
17.6
31.8
29.8
Low-Rank
54.8
5.5
26.2
42.6
32.3
BitDelta
47.8
10.7
26.2
41.9
31.7
Delta-CoMe
65.1
18.0
39.6
44.9
41.9
5.5 Inference Speed and Memory Cost
For practical applications, we also examine the inference speed and memory cost of Delta-CoMe. In terms of inference speed, we implement Triton kernel. 3 shows the inference time of the Py PyTorch and Triton implementation of Delta-CoMe. Overall, we can achieve approximately speedup across different settings. As Figure 3a shows, we first conduct an ablation experiment on varying batch sizes. Our implemented Triton kernel is consistently faster than the PyTorch implementation with different batch size settings. As Figure 33b depicts, we conduct an ablation experiment on hidden size to verify the adaptability of the Triton kernel to models of different sizes.

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The Triton kernel can maintain substantial speedup across different hidden sizes, demonstrating its ability adapt to various models.
1.0
1.0
Pyterch
Pytorch
0.8
0.8
Interence Time (ms)
0.6
Interence
0.4
6
10
1000
2000
3000
4000
5000
Batch Size
Hidden Size
(a) Effect of batch size.
(b) Effect of hidden size.
Figure 3: Inference time of the PyTorch and Triton implementation of Delta-CoMe.
In Table we show the GPU memory cost of deploying multiple aligned models that fine-tuned from LLAMA- 2-7B. The model parameters are represented in BF16 single 80G GPU. Without delta compression, single GPU can not support models, let alone more models. Using our proposed celta-compression method, we can load up to 50 models into one GPU, significantly reducing the deployment cost.
Table 7: GPU memory cost (GB).
Num. of Models
w/o DC
DC
2
26.67
15.54
4
52.24
18.17
8
23.44
16
00X
33.95
32
00M
55.06
50
00M
78.70
6
Analysis
6.1
Analysis of Quantization Error
To better understand the performance of various delta-compression methods, we estimate the quanti- zation error defined Eq. (4). It is important note that the error we calculate differs from that of GPTQ. Specifically, we use the mean square error between the activations of the uncompressed aligned model and those of the combination of the backbone model and the compressed delta model. The error is estimated on the GSM8K test set using WIZARDMATH. 7B B-v1. 1.0 as the aligned model and LLAMA 2-BB as the backbone model. Since different layers have varying impacts on the final output (Wu et al. 2023), we distinguish low-, medium-, and high-layers when estimating the average quantization error. Specifically, the first 11 layers are designated as low-layers, the 12th to 22nd layers medium-layers. and the last 10 layers high-layers. Moreover, outliers play critical role in model compression (Dettmers et al. [2023 Lin et al., (023). we also calculate the average eπror outlier parameters. For each delta matrix AW. select the top 1% of columns with the largest absolute values as outliers. Table [ presents the results. We find that the average error of methods (i.e., "Single" and "Triple") is substantially lower than both the low-rank baseline and BitDelta. Furthermore, the error of "Triple" is consistently less than that of "Single," reaffirming the necessity of mixed- mixed-precision compression for delta weights.
6.2
Case Study
We also present detailed case study in Figure 4 Three delta- celta-compression methods are examined: BitDelta, single-precision compression, and triple-precision compression. The reference answer "104 hours". We observe that BitDelta makes mistakes initially, while single-precision compression generates an incorrect intermediate result at the second reasoning step. In contrast, our mixed- precision telta-compression method calculates the correct final answer.

---

Table 8: Approximation errors (x10-2) at the activation level for different model parameters. "Low", "Medium", "High" represent low-, medium-, and high-layers, respectively. "All" means the error averaged across all the parameters, while "Out." denotes the average error estimated only on outliers.
Param
Attn.Q_Proj
Attn.K_ P_oj
Layer
Low
Medium
High
Low
Medium
High
Type
Out.
Out.
Out.
All
Out.
All
Out.
All
Out.
Low-Rank
0.75
2.24
4.24
14.31
4.47
10.28
0.87
9.90
4.79
34.04
4.82
31,41
BitDelta
0.97
2.48
4.66
14.48
4.84
10.01
1.09
10.34
5.16
33.03
5.14
28.06
Single
0.20
0.74
1.37
5.11
1.24
3.36
0.23
3.19
1.52
11.30
1.36
8.48
Triple
0.13
0.28
0.54
1.07
0.71
0.88
0.15
0.56
0.58
1.99
0.73
2.10
Param
Attn.V Proj
Attn.O Proj
Layer
Low
Medium
High
Low
Medium
High
Type
Out.
All
Out.
Out.
All
Out.
All
Out.
All
Out.
Low-Rank
0.41
3.61
1.84
8.27
2.93
4.64
0.01
0.13
0.10
0.39
0.38
5.94
BitDelta
0.45
3.60
1.95
8.02
3.18
4.85
0.01
0.13
0.11
0.44
0.37
5.45
Single
0.14
1.42
0.65
3.58
0.79
1,45
0.00
0.04
0.03
0.10
0.10
1.60
Triple
0.04
0.12
0.21
0.35
0.52
0.61
0.00
0.01
0.02
0.05
0.06
0.92
Param
FFN.Up Proj
GFN.Gate Groe_rrojjjj
Layer
Low
Medium
High
Low
Medium
High
Type
All
Out.
All
Out.
All
Out.
All
Out.
All
Out.
All
Out.
Low-Rank
0.13
0.86
0.93
3.43
2.20
11.45
0.10
0.26
0.79
1.12
1.87
9.74
BitDelta
0.18
0.97
1.06
3.84
2.38
12,22
0.13
0.31
0,90
1.26
2.02
11.74
Single
0.03
0.17
0.27
1.08
0.56
3.10
0.02
0.06
0.23
0.35
0.47
2.14
Triple
0.03
0.11
0.15
0.49
0.39
2.01
0.02
0.03
0.14
0.15
0.35
1.64
Quession:: article average of hours, she wrote articles Monday and 2/5 times more articles uescay than Monday. On Wednesday, she wroce twice the number articles wrote on Tuesday. Calculate che total number of hours.
calculate number
FFirst, We know she wrots times more articles on Tuesday chan on Manday. Therefore, the number she articles more than on Monday. Thus, the total number articles, We know that she wrote twice the number articles she wrote Tuesday. Therefore, the number on Wadnesday is 2 = articles. The total number of articles is + 7 the total number of hours she spent writing articles hours.
Since she wrote 5 articles on Mondby, calculate the number
and Tumsday. she wrote articlis
Monday and 2/5
uuedday: 215 5 We can calculate the number of articles she wrote Wednesday:
arcicles on Tuesday than Monday. Therefore, the number on
2/3 5
Tuesday
Tuesday 215 = ardilees
the number on Monday and
Tuesday
Monday: articees Tuesday: 2
Calcubeee the total number.
Tuesday S+2 articless Next. calculate number
'Answert on Tuesday she wrote 2/5* 5 more articles. The tocal number Tuesday 5+2 =
articles Wednisday: articles
Wednesday, twice the number articles she wrote on Tuesday. which 292 articles, The total number of articles 7+4 articles. Therefore, the total number
Tocal: 4 articles. Since
takos avarage of hours blog article, the total number hours as follows: articles 4 hourslarticle = 52 hours, Tharefora, Meredith spent a total of 52 hours the chres days.
Wednesthy, the number was twice on Tuesday. so was 297 = 14 articles.
articles.
5+7 +14 26 articles. each article takes her 4 hours, she spent #*26 = 101 Hanl 104
three
spent writing articles the three days is 11 64 hours.
days
Therefore, the final answer is 104 hours.
Question and
BitDelta
Single-Precision
Triple-Precision
Ground- Truth Answer
Figure 4: Case study for different delta-compression methods, where only the triple-precision compression method proposed in this work can give the correct answer.
7 Conclusion
In this paper, we propose Delta-CoMe, a delta-compression method with mixeed-precision inspired by the long-tail distribution of singular values in the delta weights. Delta-Cole achieves near-lossless performance compared to uncompressed aligned models across various typical tasks, including math, code, chat, and multi-modal tasks. We validate the effectiveness of Delta-CoMe on several widely- used aligned LLMs, whose backbone pre-trained models include Llama-2, Llama-3, and Mistral. Experimental results demonstrate that Delta-CoMe outperforms several representative baselines by considerable margin. We believe the newly introduced Delta-CoMe method has significant value for many practical applications, such multi-tenant serving.
10

---

Acknowledgments and Disclosure of Funding
This work is supported by the National Key R&D Program of China NN.220222D0111311)) National Natural Science Foundation of China (No. 62236004, No. 62236011) I), and Institute Guo Qiang at Tsinghua University.
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A Limitation and Broader Impact
For limitations, on the one hand, we carried out extensive experiments to verity Delta-CoMe is near lossless in delta compression. However, we haven't explored mixed-precision in model compression. Recently, mixed-precision applied widely in model compression and Delta-CoMe can provide new perspective for model compression. On the other hand, our kernel is trivial, Wei et al. (2024) and
and Guo et al. (2024) have implemented more advanced kernels. We can draw on their methods to achieve higher acceleration ratios.
For broader impacts, this paper presents Delta-CoMe that mainly focuses on compression which can not only boost efficiency but also save GPU memory, may bring benefits to society.
B Genetic Search for Bits Settings
In Section 5.2, we have elaborated the setting of different bits. All models employ the same configuration and have demonstrated near loss-less performance, which illustrates robustness.
Allocating different numbers to different bits (e.g. 16-bit, 8-bit) is multi-objective optimization problem. We implemented genetic algorithm to achieve more fine-grained search. We use the following objective function,
min (PL Z2, I4. 275)
where 21, 22, F4, indicating the number of 16-bit, 8-bit, 4-bit, 3-bit, 2-bit and PPI.(.) means we calculate perplexity using samples randomly chosen form C4 dataset. Table illustrates the results, particularly in code tasks, where genetic search shows significant improvement compared to greedy search. The average performance of genetic search across all tasks even surpasses that of the original salf-precision models. However, the time and storage overhead of genetic search is much greater than that greedy search.
Table 9: The performance of different bits allocate methods on 7B aligned models. "Greedy search" represents the method in Section 5.1.
WIZARDMATH
MAGICODERS CL
LLAMA-2-CHAL
LLAVA-VI.5
Method
Ave.
GSMSK
MATH
HamanEval
MBPP
SafetyBench
TrothfalQA
GQA
TeaVQA
Backhone Aligned
11.0
2.9
47.6
41.7
38.9
n/a
n/a
n/a
55.2
10.9
70.7
69.2
59.5
44.6
62.0
58.2
53.5
Greedy S.
1/16
53.6
10.3
67.1
67.9
59.8
46.9
61.7
58.5
53.2
Genetic S.
1/16
53.6
10.3
69.5
68.9
59.9
47.3
61.7
58.5
53.7
C Delta-CoMe Combine with Low-bit Backbone
Quantization methods (e.g., GPTQ, AWQ) have been widely used for quantizing backbones. It is of great significance for us to verify whether Delta-CoMe can still maintain good performance in low-bit backbone scenarios.
We evaluated the performance of Delta-CoMe using various backbones across multiple tasks in Table I0] We utilized GSM8K for math tasks, MBPP for code, TruthfulQA for chat, and TextVQA for multi-modal tasks. Table TO has demonstrated that even when backbones are in low precision, Delta-CoMe can achieve performance similar to the original, indicating that Delta-CoMe can be further applied backbones of various precision levels.
D Exploring the boundary of Delta-CoMe
We have shown that Delta-CoMe can maintain near lossless performance under 16x compression ratio. In the following, we attempt to explore the compression limits of Delta-CoMe. We employ
14

---

Table 10: Performance drop 4-bit and 16-bit backbone across different tasks.
Precision
Backbone
Tasks
Delta
WazardMath 4-bit
49.36
n/a
4-BIT BACKBONE
Llama2 4-bit + 1bit delta
47.01
-2.3
WizardMath 16-bit
55.2
16-BIT BACKRONE
Llama2 16-bit 1bit delta
53.6
-1.6
Magicoder 4-bit
66.2
n/a
4-BTT BACKBONE
Codellama-pythhon 4-bit + 1bit delta
65.4
-0.8
Magicoder 16-bit Coytllama-pythhn 16-bit 1bit delta
66.7
n/a
16-BIT BACKBONE
67.2
+0.3
WizardMath 4-bit Llama2 4-bit + 1bit delta
49.36
n/a
4-BIT BACKBONE
47.01
-2.3
WizardMath 16-bit Llama2 16-bit + 1bit delta
55.2
n/a
16-BIT BACKBONE
53.6
-1.6
Llava-v1. 4-bit Vicuna 4-bit 1bit delta
57.68
n/a
4-BIT BACKBONE
57.58
-0.1
Llava-v1.5 16-bit Vicuna 16-bit + 1bit delta
58.2
n/a
16-BIT BACKBONE
58.5
+0.3
WizardMath-7B in GSM8K task to carry out experiments which is shown in 11 experiments, the rank share the same setting.
For all the
When the compression ratio is within 20x, Delta-CoMe still performs well. However, at a compression ratio of 32x, there is noticeable decline in performance, but still outperforms low-rank and low-bit methods, which only achieve 16x compression ratio.
Table 11: Performance under different compression ratios for WizardMath-7B
Model
w/o Comp.
1/16
1/18
1/20
1/22
1/26
1/32
WithrdMath-7B
55.2
53.6
52.2
51.9
51.2
50.1
48.8
15

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
the abstract and introduction, we mention that singular values follow long- long-tailed distribution. Based this observation, we propose mixed-pre mixed-precision quantization, assigning more bits to singular vectors corresponding to larger singular values, and provide experimental results.
Justification:
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is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
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5. Open access to data and code
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Reviews:
[{"review_id": "340EBFXFms", "reviewer": "Reviewer_vBME", "summary": "This paper introduces Delta-CoMe, a delta compression method for LLM. It proposes to combine low-rank compression and low-bit compression together to achieve better performance. Specifically, it applies mixed-precision quantization for different singular vectors based on their singular values. The experimental results demonstrate good performance of the proposed method.", "questions": "1. In equation 5, why the quantization of $V$ is $Quant_k(V, X)$ without $U$ and $\\Sigma$ as in the quantization of $U$?\n2. In Section 5.1, how are $r_{begin}$ and $r_{end}$ set to those values?", "limitations": "N/A", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The proposed method is easy and straight to implement for other LLMs, based on existing open-source tools.\n2. The motivation and the proposed idea is clear and supported by the experiment results.", "weaknesses": "1. The bit-width in mixed-precision seems to be a multi-objective optimization problem, and the authors adopt a greedy search in this paper. In Table 2, different settings vary a lot in performance. This suggests that it requires to decide the bit-width carefully, making the method not easy to generalize well to other tasks/models. Performance may not be guaranteed before doing a search.\n2. The value of $r_{begin}$ and $r_{end}$ seem to be set intentionally and empirically without explanation or study. This would also impact the generalization of the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your questions\n# For question 1\nIn delta-compression, we consider compressing $U$ and $V^{T}$. For $U$ and $V^{T}$, their inputs are crucial for adjusting weights during the quantization process, as illustrated in GPTQ.  In a forward pass $Y = W X + (UΣV^{T}) X$, the input of $V^{T}$ is $X$, while the input of $U$ is $ΣV^{T}X$. Therefore, the quantization of $V$ is $Quant(V, X)$, while the quantization of $U$ is $Quant(U, ΣV^{T}X)$.\n# For question 2\nDue to limited pages, we did not include the detailed process of the decision of the number of different bit-widths in the current version. When setting $𝑟_{𝑏𝑒𝑔𝑖𝑛}$ and $𝑟_{𝑒𝑛𝑑}$, we decided based on minimizing the error between the activations of the compressed model and the original model. When searching for \"Double Precision\", using two 8-bit singular values results in the smallest error, while for \"Triple Precision\", 32 3-bit singular values results in the smallest error.\n\n#### Ablation on 8-bit in \"Double Precision\", changing the number of 8-bit from 1-8\n\n| Num. of 8-bit | 1   | 2    | 3    | 4    | 5    | 6    | 7    | 8    |\n|---------------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.85 | **0.81** | 0.84 | 0.86 | 0.88 | 0.85 | 0.95 | 0.94 |\n#### Ablation on 3-bit in \"Triple Precision\", changing the number of 3-bit from 8-64\n\n| Num. of 3-bit | 8   | 16   | 24   | 32   | 40   | 48   | 56   | 64   |\n|---------------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.77 | 0.77 | 0.76 | **0.74** | 0.75 | 0.76 | 0.78 | 0.77 |\n\nBased on the reviewers' insights, we regard the mixed precision issue as a multi-objective optimization problem, considering single precision to be a special case of mixed precision. We developed a genetic algorithm to address this problem, using the bit count of single precision as the initial solution. We use the following objective function, $f = min PPL(x1, x2, x3, x4, x5) $ where x1, x2, x3, x4, x5 indicating the number of 16-bit, 8-bit, 4-bit, 3-bit, 2-bit and $PPL(.)$ means we calculate perplexity in 128 samples randomly chosen form C4 dataset.  For each aligned model, we can automatically determine the mixing strategy through the genetic algorithm. The results demonstrate that the genetic algorithm can yield better results than greedy search, making our method easy to be applied to many different aligned models.\n| Models                | WizardMath      |                     | magicoder-S-CL   |             | Llama-2-7b-chat  |             | Llava-v1.5   |         |   Ave.   \n|---------------|-----------------|---------------------|------------------|-------------|------------------|-------------|----------|--------|-----|\n| Tasks                 | GSM8K           | Math    | HumanEval        | Mbpp   | SafetyBench   | TruthfulQA    | GQA        | TextVQA   |     \n| loss-based greedy search | 53.6        | 10.24    | 67.1            | 67.9       | 59.8     | 46.9       | 61.7            | 58.5       |    53.2\n| genetic search        | 53.6           | 10.24       | **69.5**        | **68.9**   | **59.9**   | **47.3**       | 61.7            | 58.5  |   **53.7**\n\nWe also conducted experiments on the 13B model, where the genetic algorithm yielded better performance.\n| Models    | WizardMath      |         |\n|-------------|---------|-------------|\n| Tasks        | GSM8K   | Math    |\n| loss-based greedy search | 58.8   | 12.8    |\n| genetic search        | **59.4**       | **12.9**  |\n The genetic algorithm demonstrated better performance, with the average performance on the 7B model even slightly surpassing \"Aligned models\". However, even with the same settings without genetic algorithm, our method's performance is close to that, indicating its generalization ability.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ryanI2SBRx", "reviewer": "Reviewer_Wicf", "summary": "This paper proposes an improved way to apply delta-compression for aligned language models (compact representations of the difference in weights between the pretrained and finetuned language models), which is just as effective w.r.t compression as the most extreme existing binary quanitzation strategy (BitDelta), but resolving many existing quality issues w.r.t. math and code tasks over previous methods. The key insight is that it is possible to combine the best of both worlds between low-rank methods (selecting the top singular values) and quantization methods (e.g. BitDelta), and represent more singular values by using decreasing precision for decreasing singular values: a mixed precision technique overall. The authors call this technique Delta-CoMe. When comparing low-rank, BitDelta, and Delta-CoMe at a fixed compression rate (equal to the binarization / BitDelta rate, which is quite aggressive), Delta-CoMe consistently outperforms all baselines and is close to matching performance with the full precision aligned model. Experiments take place over a diverse set of tasks (math, code, QA, safety, multimodal) as well as a diverse set of models at 7B and 13B scales and aligned versions (Llama 2, WizardMath, MagicCoders, Llama-2-Chat, LLAVA v1.5, OpenChat, Llama-3-8b-Instruct). The authors also provide quality comparisons between delta-compression methods and delta-tuning (i.e. LoRA), demonstrating that Delta-CoMe outperforms LoRA w.r.t quality.", "questions": "1. What is the intuition behind choosing the singular vector bands for the different precisions the way that you do in Section 5.1?\n\n1. Do you think the ideas here could apply to mixed-precision approaches more generally?\n\n1. Why is the approach named Delta-CoMe? It appears to stand for delta compression method which seems very generic? And has nothing to do with the details of the approach. \n\n1. Are there any practical limitations e.g. hardware with mixed-precision delta-compression not covered in this paper?", "limitations": "The authors do clearly point out that a major limitation of the work is the lack of optimization for how the mixed precision is assigned. This makes the approach more of a proof of concept that even a naive solution works, rather than an optimized method. This is perfectly fine.\n\nIt would be valuable to understand though whether theres actually limitations in deploying this type of approach on actual hardware with multi-tenant hosting. \n\nIt would also be interesting to understand the boundaries to when this approach works or does not work, e.g. at what compression ratio would it break, whether theres a model scale too small or too large, etc.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. While neither low-rank or mixed precision quantization techniques are particularly novel, the combination of the two for delta-compression is novel and clever in the way that the framework smoothly models the trade-off between representing everything with low (binary) precision, and representing the most important features with full precision. Although this is not explored deeply in this paper (the authors recognize that this is a proof of concept of the effectiveness rather than an optimized solution), this tradeoff could be tuned to suit a variety of cases. This idea could also potentially be applied in other areas, such as PEFT or general mixed-precision work, as previous works used quite different heuristics to select which weights to have mixed precision on.\n\n2. The results presented are significant. The approach significantly outperforms the baselines in all aspects (math, code, chat, multimodal), with the biggest improvements in math and code. The approach is the only one to come close to matching the non-compressed aligned model overall over all capabilites. This will be of interest to anyone interested in delta-compression literature or even PEFT techniques. It is the first work to apply mixed-precision to delta weights.\n\n3. The experiments was done on a wide variety of models (math specialized tunings, code specialized tunings, chat tunings, multimodal tunings) and settings (7/8B, 13B, over base models like Llama2, Llama3 and Mistral) enough to demonstrate confidence in the ability of the technique. The authors also provide nice additional analysis of quantization error, and comparison in quality against vanilla LoRA as additional evidence for their approach.\n\n4. The paper is overall well structured and well written, which makes the problem easy to understand and the author's intuition and investigation relatively easy to follow. There are some improvements that could be made (below), but it is nice overall.", "weaknesses": "1. In Table 4, it does seem like on the larger 13B models WizardMath models, the performance recovery in GSM8k and MATH is not as strong as on 7 or 8B scale models. It does bring into question whether there are limitations to this approach not discussed in this work that may limit general adoptability. Whether that means not working as well on larger scales, or for the certain type of post-training happening in WizardMath. \n\n1. Although I understand the author's motivation of using greedy decoding to encourage reproducibility and decrease noise in the evaluations, many models especially on chat tasks use sampling based-decoding, and having results here would improve the overall robustness of the experiments to verify that the techniques still work well. \n\n1. Presentation a couple points in the paper could be improved :\n\n    a. While the authors do note that the greedy search strategy to allocate mixed-precision to the singular values presented in Section 5.1 represents a proof of concept to show that even such an unoptimized approach can achieve good results, it would be valuable to understand how some of the design choices here were informed. For example, in the 3-precision setting, the range of singular vectors for the 2nd precision is set to be between 2 and 34, which feels arbitrary. What is the intuition behind this or were there ablations ran here? Being that Section 5.1 is the core description of the mixed-precision strategy, more detail and clarity here would be appreciated.\n\n    b. (Minor) The introduction could be cleaned up a bit, with lines 21-36 being oddly detailed. The claim on line 44-46 to motivate the work is also not super well founded -- there are models on LMSys showing general ability with ~20B parameters, not too much bigger than those considered in this work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your questions, weaknesses, and limitations which can bring much improvement to our paper.\n\n# For question 1.\nDue to limited pages, we did not include the detailed process of the decision of the number of different bit-widths in the current version. When setting $𝑟_{𝑏𝑒𝑔𝑖𝑛}$ and $𝑟_{𝑒𝑛𝑑}$, we decided based on minimizing the error between the activations of the compressed model and the original model.\n#### Ablation on 8-bit in \"Double Precision\", changing the number of 8-bit from 1-8\n\n| Num. of 8-bit | 1 | 2 | 3  | 4  | 5  | 6  | 7  | 8  |\n|------|-----|----|----|----|---|----|----|----|\n| Error (× 10⁻²) | 0.85 | **0.81** | 0.84 | 0.86 | 0.88 | 0.85 | 0.95 | 0.94 |\n#### Ablation on 3-bit in \"Triple Precision\", changing the number of 3-bit from 8-64\n\n| Num. of 3-bit | 8  | 16 | 24 | 32  | 40  | 48 | 56  | 64  |\n|--------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.77 | 0.77 | 0.76 | **0.74** | 0.75 | 0.76 | 0.78 | 0.77 |\n# For question 2.\nBased on the reviewers' insights, we regard the mixed precision issue as a multi-objective optimization problem, considering single precision to be a special case of mixed precision. We developed a genetic algorithm to address this problem, using the bit count of single precision as the initial solution. We use the following objective function, $f = min PPL(x1, x2, x3, x4, x5) $ where x1, x2, x3, x4, x5 indicating the number of 16-bit, 8-bit, 4-bit, 3-bit, 2-bit and $PPL(.)$ means we calculate perplexity in 128 samples randomly chosen form C4 dataset.  For each aligned model, we can automatically determine the mixing strategy through the genetic algorithm. The results demonstrate that the genetic algorithm can yield better results than greedy search, making our method easy to be applied to many different aligned models.\n| Models     | WizardMath      |           | magicoder-S-CL   |      | Llama-2-7b-chat  |   | Llava-v1.5   |         |   Ave.   \n|----------|---------|------------|-------|--------|-------------|--------|-----|--------|-----|\n| Tasks       | GSM8K    | Math    | HumanEval        | Mbpp   | SafetyBench   | TruthfulQA    | GQA        | TextVQA   |     \n| loss-based greedy search | 53.6   | 10.24    | 67.1     | 67.9  | 59.8     | 46.9       | 61.7    | 58.5       |    53.2\n| genetic search        | 53.6      | 10.24   | **69.5**   | **68.9**   | **59.9**   | **47.3**   | 61.7    | 58.5  |   **53.7**\n\nWe also conducted experiments on the 13B model, where the genetic algorithm yielded better performance.\n| Models    | WizardMath   |    |\n|-------------|---------|-------------|\n| Tasks    | GSM8K   | Math    |\n| loss-based greedy search | 58.8   | 12.8    |\n| genetic search        | **59.4**   | **12.9**  |\n\n The genetic algorithm demonstrated better performance, with the average performance on the 7B model even slightly surpassing \"Aligned models\". However, even with the same settings without genetic algorithm, our method's performance is close to that, indicating its generalization ability.\n\nRecently, mixed precision is being more widely used such as Apple intelligence (Gunter, Tom, et al. ) allocate different bits to different layers which achieves 3.5-bit on average and Jinhao Li et al. employ mixing 2-bit, 4-bit and 16-bit that ultimately attains an average of 2-bit. Recently, mixed precision has received more attention than before, and the use of mixed precision methods has also been widely applied to model compression. We believe that our method could also be further applied to model compression, which is the direction of our next exploration.\n# For question 3.\n Thanks for your great question. We will give a more proper title.\n# For question 4.\n  Ablation on batch, we set sequence length to 128, our method achieves 3x speedup than Pytorch. Further, Han Guo et al. and Jianyu et al. have implemented more advanced kernels. We can draw on their methods to achieve higher acceleration ratios.\n #### Ablation on batch, we set sequence length to 128, our method achieves 3x speedup than Pytorch.\n| bsz | Linear (ms) | Our (ms) |\n|-----|-------------|----------|\n| 2   | 0.5995      | 0.1488   |\n| 4   | 0.6024      | 0.1995   |\n| 6   | 0.6136      | 0.2134   |\n| 8   | 0.6656      | 0.2561   |\n #### Ablation on hidden_size, we set batch 8.\n| hidden_size | Linear (ms) | Our (ms) |\n|-------------|--------------|----------|\n| 1024        | 0.5369       | 0.1206   |\n| 2048        | 0.5586       | 0.1654   |\n| 3072        | 0.6011       | 0.2046   |\n| 4096        | 0.6656       | 0.2561   |\n| 5120        | 0.6897       | 0.2788   |\n\n# For limitation\n  Thanks for your great insight, we conducted a more granular exploration, compressing the model up to 32×. We use WizardMath-7B in GSM8K task and results are shown in the following table. When the compression ratio is within 20×, our method still performs well. However, at a compression ratio of 32×, there is a noticeable decline in performance, but it still outperforms low-rank and low-bit methods, which only achieve a 16× compression ratio.\n|                | w/o Comp.    | 1/16    | 1/18    | 1/20    | 1/22    | 1/26    | 1/32 |\n|----------------|---------|---------|---------|---------|---------|---------|-----------|\n| WizardMath-7B  | 55.2  | 53.6  | 52.2  | 51.9  | 51.2  | 50.1   | 48.8     |\n\n# For weakness\nWe have used sample decoding algorithm and set decoding temperature = 0.2.  We run 5 times and calculate the mean value and confidence interval shown in the table.\n|          | GSM8K           | truthfulQA      | MBPP            | TextVQA         |\n|----------|------------------|-----------------|-----------------|-----------------|\n| Low-rank | 42.6 (41.9, 43.3)| 42.2 (42.0, 42.4)| 65.4 (65.1, 65.9)| 52.9 (52.5, 53.6)|\n| Bitdelta | 45.2 (44.0, 46.4)| 40.8 (40.4, 41.3)| 65.7 (65.4, 65.9)| 56.5 (56.2, 57.1)|\n| Ours     | 53.8 (53.0, 54.6)| 46.9 (46.3, 47.6)| 68.0 (67.6, 68.5)| 58.3 (57.9, 58.9)|", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XlMIOgTWf8", "reviewer": "Reviewer_PxGh", "summary": "In the context of SVD compression of delta weight, the paper employs higher bitwidth for singular vectors corresponding to larger singular values. The available bitwidth 8, 3, 2 are empirically chosen. Once the bitwidths are assigned to the singular vectors, the vectors are group-wise quantized by GPTQ.", "questions": "Why is there an advantage to SVD the delta weights as opposed to SVD the aligned model weights directly?", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is generally well-written and the experiments are very thorough. The combination of delta-compression and mixed-precision quantization appears to be novel. The method is straightforward and well-motivated.", "weaknesses": "Something is wrong with the latex encoding of this pdf. Sections are not detected properly by pdf readers, and there are odd reference errors such as the ones on line 26 and 30. \n\nThere are three steps of optimization being done: the number of largest singular vectors chosen, the bitwidth assigned to the singular vectors, and the quantization algoritm performed on the singular vectors. In the paper, the three steps are being done sequentially, leading to arbitrary, empirically-driven and possibly suboptimal decisions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your comment. \n# For weakness\nDue to limited pages, we did not include the detailed process of the decision of the number of different bit-widths in the current version. When setting $𝑟_{𝑏𝑒𝑔𝑖𝑛}$ and $𝑟_{𝑒𝑛𝑑}$, we decided based on minimizing the error between the activations of the compressed model and the original model. When searching for \"Double Precision\", using two 8-bit singular values results in the smallest error, while for \"Triple Precision\", 32 3-bit singular values results in the smallest error.\n\n#### Ablation on 8-bit in \"Double Precision\", changing the number of 8-bit from 1-8\n\n| Num. of 8-bit | 1   | 2    | 3    | 4    | 5    | 6    | 7    | 8    |\n|---------------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.85 | **0.81** | 0.84 | 0.86 | 0.88 | 0.85 | 0.95 | 0.94 |\n#### Ablation on 3-bit in \"Triple Precision\", changing the number of 3-bit from 8-64\n\n| Num. of 3-bit | 8   | 16   | 24   | 32   | 40   | 48   | 56   | 64   |\n|---------------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.77 | 0.77 | 0.76 | **0.74** | 0.75 | 0.76 | 0.78 | 0.77 |\n\nBased on the reviewers' insights, we regard the mixed precision issue as a multi-objective optimization problem, considering single precision to be a special case of mixed precision. We developed a genetic algorithm to address this problem, using the bit count of single precision as the initial solution. We use the following objective function, $f = min PPL(x1, x2, x3, x4, x5) $ where x1, x2, x3, x4, x5 indicating the number of 16-bit, 8-bit, 4-bit, 3-bit, 2-bit and $PPL(.)$ means we calculate perplexity in 128 samples randomly chosen form C4 dataset.  For each aligned model, we can automatically determine the mixing strategy through the genetic algorithm. The results demonstrate that the genetic algorithm can yield better results than greedy search, making our method easy to be applied to many different aligned models.\n| Models                | WizardMath      |                     | magicoder-S-CL   |             | Llama-2-7b-chat  |             | Llava-v1.5   |         |   Ave.   \n|---------------|-----------------|---------------------|------------------|-------------|------------------|-------------|----------|--------|-----|\n| Tasks                 | GSM8K           | Math    | HumanEval        | Mbpp   | SafetyBench   | TruthfulQA    | GQA        | TextVQA   |     \n| loss-based greedy search | 53.6        | 10.24    | 67.1            | 67.9       | 59.8     | 46.9       | 61.7            | 58.5       |    53.2\n| genetic search        | 53.6           | 10.24       | **69.5**        | **68.9**   | **59.9**   | **47.3**       | 61.7            | 58.5  |   **53.7**\n\nWe also conducted experiments on the 13B model, where the genetic algorithm yielded better performance.\n| Models    | WizardMath      |         |\n|-------------|---------|-------------|\n| Tasks        | GSM8K   | Math    |\n| loss-based greedy search | 58.8   | 12.8    |\n| genetic search        | **59.4**       | **12.9**  |\n The genetic algorithm demonstrated better performance, with the average performance on the 7B model even slightly surpassing \"Aligned models\". However, even with the same settings without genetic algorithm, our method's performance is close to that, indicating its generalization ability.\n\n# For question\n1. Compress aligned models directly using SVD can easily achieve bad performance in low-bit setting. As the following table using SVD compress the model into 1-bit or 2-bit settings resulting in a performance drop to 0 in math and code tasks.\n| Tasks         | GSM8K | Math | HumanEval | Mbpp |\n|---------------|-------|------|-----------|------|\n| Compress 1-bit| 0     | 0    | 0         | 0    |\n| Compress 2-bit| 0     | 0    | 0         | 0    |\n2. Recently Onebit (Xu et al.) find compress aligned model needs more sophisticated algorithms and  post-training is also needed.  Even in this carefully designed setting, there is still a significant performance drop. For example,  in Table 2 of OneBit, the average performance on 6 benchmarks decreased from 64.06 to 51.33 and from 66.39 to 55.17 on Llama-2-7B and Llama-2-13B, respectively.\n3. In our experiments, delta exhibits a long-tail distribution characteristic, and using SVD compression algorithms can retain the model in a near-lossless manner, compressing it to an equivalent of 1-bit. Therefore, compressing delta-weights can achieve better performance.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bBhJxothKM", "reviewer": "Reviewer_ZM7A", "summary": "This paper proposes an application of the mixed precision quantization technique to the singular vectors of the delta weights, which is encountered when serving multiple aligned LLMs.", "questions": "My review is as follows:\n\n1) As a strength, this paper is well-written and makes it easy for the reader to follow the ideas.\n2) The proposed method is a somewhat straightforward extension of mixed precision quantization to the singular vectors of the delta weights. The mixed precision quantization technique is well-known and commonly used by practitioners. I view the contribution of this paper as an extension of that. Because of that, I am not sure if the contributions of the paper could be considered novel. Having said that, it is good to see that the method performs well in multiple scenarios. \n3) I understand that studying the compression of delta weights is important. In practice, the base model’s weights need to be quantized/compressed too for real-world situations. I don’t think the compression of base model weights and delta weights are completely orthogonal. I wonder what the authors think on the interdependency of the compression of base model weights and delta weights. In my opinion, it would improve the paper to include some results/discussion on this.\n4) I’m wondering why the authors haven’t considered a more thorough search algorithm in Section 5.1 (or if they did, it would be great to include a discussion on it). \n\nMinor:\n\n5) Typo in Table 1: “Aligend”\n6) Typo in Table 6: “HuamnEval”", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Please see the “Questions” section.", "weaknesses": "Please see the “Questions” section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your questions.\n\n# For question 2.\nRepresentative works on mixed precision include Yao et al., Agile-Quant, and SpQR, which focus on applying mixed precision to model weights and activations, and Bablani et al. who apply mixed precision across different layers, while we adopt different precisions across different singular values in the feature space. Recently, Apple intelligence has also applied mixed precision across different layers. As the reviewer Wicf said, \"While neither low-rank nor mixed precision quantization techniques are particularly novel, the combination of the two for delta-compression is novel and clever.\" As far as we know, we are the first to propose mixed precision compression in the feature space and in multi-modal settings.    \n\n# For question 3\nWe compress our backbone into 4-bit, and test their performance shown in the following table.  Our method can still retain performance though the backbone is compressed in low-bit format. \n|                   |                   | GSM8K  | delta  |\n|-------------------|-------------------|--------|--------|\n| 4-bit backbone    | WizardMath 4-bit  | 49.36% |  |\n|                   | Llama2 4-bit + 1bit delta | 47.01% |  -2.3%       |\n| 16-bit backbone   | WizardMath 16-bit | 55.2%  |  |\n|                   | Llama2 16-bit + 1bit delta | 53.6%  |  -1.6%       |\n\n|                   |                   | MBPP  | delta  |\n|-------------------|-------------------|--------|--------|\n| 4-bit backbone    | Magicoder 4-bit  | 66.2% |   |\n|                   | Codellama-python 4-bit + 1bit delta | 65.4% |    -0.8%    |\n| 16-bit backbone   | Magicoder 16-bit | 66.7%  |   |\n|                   | Codellama-python 16-bit+ 1bit delta | 67.2%  |    +0.3%    |\n\n|                   |                   | TruthfulQA  | delta  |\n|-------------------|-------------------|--------|--------|\n| 4-bit backbone    | WizardMath 4-bit  | 49.36% |  |\n|                   | Llama2 4-bit + 1bit delta | 47.01% |   -2.3%      |\n| 16-bit backbone   | WizardMath 16-bit | 55.2%  |   |\n|                   | Llama2 16-bit + 1bit delta | 53.6%  |  -1.6%      |\n\n|                   |                   | TextVQA  | delta  |\n|-------------------|-------------------|--------|--------|\n| 4-bit backbone    | Llava-v1.5 4-bit  | 57.68% |  |\n|                   | Vicuna 4-bit + 1bit delta | 57.58% |    -0.1%     |\n| 16-bit backbone   | Llava-v1.5 16-bit | 58.2%  |   |\n|                   | Vicuna 16-bit + 1bit delta| 58.5%  |  +0.3%      |\n# For question 4 \nDue to limited pages, we did not include the detailed process of the decision of the number of different bit-widths in the current version. When setting $𝑟_{𝑏𝑒𝑔𝑖𝑛}$ and $𝑟_{𝑒𝑛𝑑}$, we decided based on minimizing the error between the activations of the compressed model and the original model. When searching for \"Double Precision\", using two 8-bit singular values results in the smallest error, while for \"Triple Precision\", 32 3-bit singular values results in the smallest error.\n\n#### Ablation on 8-bit in \"Double Precision\", changing the number of 8-bit from 1-8\n\n| Num. of 8-bit | 1   | 2    | 3    | 4    | 5    | 6    | 7    | 8    |\n|---------------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.85 | **0.81** | 0.84 | 0.86 | 0.88 | 0.85 | 0.95 | 0.94 |\n#### Ablation on 3-bit in \"Triple Precision\", changing the number of 3-bit from 8-64\n\n| Num. of 3-bit | 8   | 16   | 24   | 32   | 40   | 48   | 56   | 64   |\n|---------------|-----|------|------|------|------|------|------|------|\n| Error (× 10⁻²) | 0.77 | 0.77 | 0.76 | **0.74** | 0.75 | 0.76 | 0.78 | 0.77 |\n\nBased on the reviewers' insights, we regard the mixed precision issue as a multi-objective optimization problem, considering single precision to be a special case of mixed precision. We developed a genetic algorithm to address this problem, using the bit count of single precision as the initial solution. We use the following objective function, $f = min PPL(x1, x2, x3, x4, x5) $ where x1, x2, x3, x4, x5 indicating the number of 16-bit, 8-bit, 4-bit, 3-bit, 2-bit and $PPL(.)$ means we calculate perplexity in 128 samples randomly chosen form C4 dataset.  For each aligned model, we can automatically determine the mixing strategy through the genetic algorithm. The results demonstrate that the genetic algorithm can yield better results than greedy search, making our method easy to be applied to many different aligned models.\n| Models                | WizardMath      |                     | magicoder-S-CL   |             | Llama-2-7b-chat  |             | Llava-v1.5   |         |   Ave.   \n|---------------|-----------------|---------------------|------------------|-------------|------------------|-------------|----------|--------|-----|\n| Tasks                 | GSM8K           | Math    | HumanEval        | Mbpp   | SafetyBench   | TruthfulQA    | GQA        | TextVQA   |     \n| loss-based greedy search | 53.6        | 10.24    | 67.1            | 67.9       | 59.8     | 46.9       | 61.7            | 58.5       |    53.2\n| genetic search        | 53.6           | 10.24       | **69.5**        | **68.9**   | **59.9**   | **47.3**       | 61.7            | 58.5  |   **53.7**\n\nWe also conducted experiments on the 13B model, where the genetic algorithm yielded better performance.\n| Models    | WizardMath      |         |\n|-------------|---------|-------------|\n| Tasks        | GSM8K   | Math    |\n| loss-based greedy search | 58.8   | 12.8    |\n| genetic search        | **59.4**       | **12.9**  |\n The genetic algorithm demonstrated better performance, with the average performance on the 7B model even slightly surpassing \"Aligned models\". However, even with the same settings without genetic algorithm, our method's performance is close to that, indicating its generalization ability.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Thank all the reviewers for the constructive suggestions. We will take into account the advice to improve the manuscript!

Meta-review: This paper presets a novel approach to apply delta-compression for aligned language models (from pretrained and finetued language models). The proposed approach try to combine the existing low-rank and low-bit compression methods together to achieve better performance. More specially, the approach applies mixed precision quantization for different SV based on their values. All reviews agree the paper is novel and the experiments indeed show the merit of the proposed approach.
Decision: Accept (poster)