Title: Spectral Adapter: Fine-Tuning in Spectral Space
Venue: NeurIPS 2024
Accepted: yes
Keywords: fine-tuning; peft; spectral decomposition; transfer learning in spectral space
Abstract: Recent developments in Parameter-Efficient Fine-Tuning (PEFT) methods for pretrained deep neural networks have captured widespread interest. In this work, we study the enhancement of current PEFT methods by incorporating the spectral information of pretrained weight matrices into the fine-tuning procedure. We investigate two spectral adaptation mechanisms, namely additive tuning and orthogonal rotation of the top singular vectors, both are done via first carrying out Singular Value Decomposition (SVD) of pretrained weights and then fine-tuning the top spectral space. We provide a theoretical analysis of spectral fine-tuning and show that our approach improves the rank capacity of low-rank adapters given a fixed trainable parameter budget. We show through extensive experiments that the proposed fine-tuning model enables better parameter efficiency and tuning performance as well as benefits multi-adapter fusion.  The source code will be open-sourced for reproducibility.

Spectral Adapter: Fine-Tuning in Spectral Space

Abstract

Recent developments in Parameter-Efficient Fine-Tuning (PEFT) methods for pretrained deep neural networks have captured widespread interest. In this work, we study the enhancement of current PEFT methods by incorporating the spectral information of pretrained weight matrices into the fine-tuning procedure. We investigate two spectral adaptation mechanisms, namely additive tuning and orthogonal rotation of the top singular vectors, both are done via first carrying out Singular Value Decomposition (SVD) of pretrained weights and then fine-tuning the top spectral space. We provide a theoretical analysis of spectral fine-tuning and show that our approach improves the rank capacity of low-rank adapters given a fixed trainable parameter budget. We show through extensive experiments that the proposed fine-tuning model enables better parameter efficiency and tuning performance as well as benefits multi-adapter fusion. Code is released at https://github.com/pilancilab/spectral_adapter .

1 Introduction

Size of language and vision model undergoes a drastic explosion in recent days and results in billions of parameters up to date. While fine-tuning has been used a lot for adapting pretrained large models to various downstream tasks, fine-tuning tasks become increasingly hard with current size of pretrained models due to the huge demand of computing resource. Meanwhile, exchange and storing of finetuned models are also expensive given their enormous size. To alleviate these rising problems for fine-tuning large pretrained models, a recent line of research has digged into the Parameter-Efficient Fine-Tuning (PEFT) model family and harnessed great attention. A high-level philosophy behind those PEFT methods is to train a reduced number of parameters compared to full fine-tuning, which instantly saves computing resource and enables light-weight fine-tuned model exchange. Among all PEFT methods, Low-Rank Adaptation (LoRA) [20] model is a huge success attributed to its simplicity and effectiveness. Specifically, LoRA proposes to tune an additive trainable low-rank matrix and brings zero inference latency after merging the adapter into pretrained model weights. Since its emergence, numerous variants of LoRA have been developed. For instance, AdaLoRA [65], IncreLoRA [62], and DyLoRA [54] propose to dynamically adjust LoRA rank distribution for improving tuning efficiency, QLoRA [10] combines LoRA with model quantization to further save computing resource, LoRA+ [16] and PrecLoRA [61] study the optimization landscape of LoRA training, and more recent variant DoRA [32] decomposes pretrained weights into magnitude and direction components and applies LoRA for direction tuning, see Apppendix A for a more comprehensive review of different LoRA variants. Other PEFT methods such as Orthogonal FineTuning (OFT) proposes to multiply pretrained weights by tunable orthogonal matrices for preservation of hypersphere energy between pretrained neurons. Though these different PEFT methods focus on improving fine-tuning efficiency with reduced parameters, rare attention has been paid to utilize pretrained model weights' information beyond its magnitude in the fine-tuning procedure.

Full

LoRA

DoRA

OFT

AdaLoRA

Training Loss (Orca Math)

Spectral Adapter(ours)

0.7

0.6

61%

0.5

59%

57%

0.4

55%

53%

0.3

Global Steps

51%

Global Steps

200

400

600

800

1000

1200

100

300

500

700

9001100

Prior research in statistical machine learning such as [36] has studied the Empirical Spectral Distribution (ESD) of deep models' weight matrices and found that the ESDs for larger model weights are usually more structured and contain indicative information to distinguish between different training stages. More recent work such as [4] investigates the "dark matter" effect of bottom spectral space of model weights and recognizes its critical role in attention sink phenomenon observed in [57]. Both work contributes to decrypting spectral information of model weights and sheds light on building insightful understanding of the connection between weight matrices' spectral information and model performance. In this work, we explore further the value of model weights' spectral pattern and unravel its effectiveness in enhancing fine-tuning tasks. We showcase via extensive empirical observation that integration of spectral information of pretrained model weights improves current PEFT methods' parameter efficiency, tuning effect, and arises as a natural solution to multi-adapter fusion problems. Moreover, the suggested fine-tuning model maintains better practicality compared to prior spectral tuning models, which will be investigated further below.

Though any technique for weight fine-tuning can be directly applied to fine-tune singular vector matrices of pretrained model weights, we investigate two specific forms of such extension, namely additive tuning and orthogonal rotating the top singular vector space, which we address as Spectral Adapter A and Spectral Adapter R respectively in later content. The spectral adaptation mechanisms being considered are formally depicted in Section 2. As a warmup, to show that incorporating spectral information is indeed helpful, Figure 1 displays the training loss of fine-tuning Llama3 8B model on HuggingFace Orca Math dataset and validation score on GSM8K benchmark, from which it can be clearly observed that Spectral Adapter A performs superior to recent variants of PEFT methods and behaves closest to full fine-tuning, here we follow experimental setup in [53], see Appendix F.1 for details and more investigation. In below, we first introduce the fine-tuning model being studied in Section 2 and we then provide some theoretic insights in Section 3. After that, we detail the advantage of our spectral adapter in enhancing fine-tuning result, improving model's parameter efficiency, and helping with multi-adapter fusion as well as address any concern with respect to practicality issues in Section 4. Conclusion and future work is discussed in Section 5. For sake of page limitation, literature review is deferred to Appendix A.

To summarize, the proposed spectral adaptation mechanism demonstrates the first attempt to fine-tune spectral space of pretrained model weights in a parameter-efficient and storage-economic way which improves current PEFT methods from aspects involving tuning results, parameter efficiency, and multi-adapter fusion. We hope this work serves as a building block and motivates further and deeper insightful investigation for exploring spectral structure of pretrained model weights, which becomes increasingly meaningful especially in current large model regime.

2 Spectral Adapter: Incorporating Spectral Information into Fine-Tuning

Motivated by the intrinsic low-rank of weight shifts in fine-tuning procedure studied in [2], LoRA [20] proposes to add a low-rank factorized trainable matrix to pretrained model weights and tune only these additive parameters for downstream task adaptation, which usually injects far fewer trainable

SVD

Add

Merge

n

U

h

pretrained

V

U

S

X

A

十

weights

S

Pretrained

Weights

B=0

S

Spectral Adapter

A

WERdxd

A = N(0,α²)

SVD

OrthogonallyRotate

Merge

U

U

V

x

pretrained

U

×

S

V

LoRA

weights

V

S

S

Spectral Adapter R

parameters compared to full fine-tuning and results in light-weight tuned adapters. LoRA serves as an outstanding representative of PEFT family and is now widely-used for different fine-tuning tasks. Inspired by the parameter efficiency of LoRA and the close connection between matrix rank and its spectral representation, here we study two spectral fine-tuning mechanisms, both are completed via first carrying out Singular Value Decomposition (SVD) of pretrained model weights and then finetuning the top columns of singular vector matrices obtained via the SVD. More precisely, consider a pretrained weight matrix with its spectral representation of form W = USV T , we define additive spectral adapter as

and correspondingly the rotational version

where U 1 , V 1 denote the topr columns of U and V and U 2 , V 2 denote the rest of the columns. A = ( A U , A V ) consists of trainable matrices of shape same as ( U 1 , V 1 ) and R = ( R U , R V ) consists of two trainable orthogonal matrices of shape r by r such that R T U R U = R T V R V = I . As we show in later sections, the orthogonality constraint is efficiently handled with the Cayley parameterization, see Section 4.3 for details. The proposed fine-tuning model architecture can be visualized from Figure 2. Here Spectral Adapter A more resembles LoRA as it is of additive form while Spectral Adapter R more resembles prior Orthogonal Fine-Tuning (OFT) method which we compare further in Section 4. To ensure zero initialization as often done for PEFT methods, we initialize A U and A V both at zero. For rotational spectral adapter, we initialize R U and R V as identity matrices.

A more thorough literature review suggests that prior work considering tuning model weights' spectral representation (FSGAN[47], SVDiff [15]) has been proposed for alleviating overfitting when fine-tuning different vision models. These methods only look at tuning the singular values of flattened CNN weights and thus have fixed amount of trainable parameters. Moreover, these methods require storing all U,S and V during training while only the diagonal vector of S is tuned, which nearly doubles the storage requirement compared to pretraining when fine-tuning on downstream tasks. Contrarily, we consider incorporating spectral information in generic fine-tuning procedure for different layers (flattened CNN weights, dense linear weights, etc.) and our method enables flexible parameter budget choices by varying values of r . Methodology-wise, we consider tuning the topr columns of U and V by additive and rotational tuning, both requiring only these top columns to be stored additionally and the left part can be merged into a single weight matrix. See Section 4.4 for more investigation on practicality of the proposed method.

3 Theoretical Insights

After introducing the model architecture of spectral adapter we consider, the main question now remains whether tuning the spectral representation of pretrained weights is indeed an improvement over existing PEFT methods. Before we step into our empirical observations, we first provide

some theoretical insights for the proposed spectral adaptation mechanism. In this section, we show advantage of our spectral adapter method compared to LoRA from two theoretic perspectives by analyzing both the rank capacity of the adapters (Section 3.1) and the subspace alignment of pretrained weight matrices (Section 3.2). Specifically, we will see that Spectral Adapter A has larger rank capacity than LoRA adapter, which indicates the tuned weight has more adaptation freedom and thus is more desirable. Moreover, the dominant spectral direction of pretrained weight matrix identifies more ideal neuron alignment under the setting we consider in Section 3.2, which justifies the robustness of tuning top singular vectors in our spectral adapter. In Appendix D, we show that Spectral Adapter A is approximately equivalent to DoRA [32] for vector-form weights.

3.1 Adapter Rank Capacity

For any pretrained weight matrix W , suppose that the adapter is given by the parameterization f θ ( W ) where θ represents trainable weights. For instance with LoRA adapter, f θ ( W ) = W + AB T , where θ = { A,B } is trainable. We define the rank capacity of an adapter f θ ( W ) as follows:

which describes the range of matrix ranks the tuned weight can achieve given a specific adapter form. Then, the following lemma shows that Spectral Adapter A has twice the rank capacity of LoRA adapter under an equal number of trainable parameters.

Lemma 3.1. Suppose that W ∈ R n × m is an arbitrary full row-rank matrix and n ≤ m without loss of generality. Consider rank-r LoRA and rank-r additive spectral adapter, which have an equal number of trainable parameters. We have

See Appendix B for proof. Therefore when pretrained model weight matrix is close to full row-rank, as what has been observed in [20], Spectral Adapter A has nearly double rank capacity compared to LoRA adapter. Furthermore, some prior work explicitly imposes low-rank constraint when training original NNs [50, 43, 66, 22, 68, 24, 9]. Using LoRA adapter to fine-tune such pretrained model weights would destroy their rank constraints while applying spectral adapter preserves the constraints.

Next we proceed to show that top spectral space of pretrained weight matrices is more aligned with ideal neuron direction under a simple setting via subspace decomposition analysis of pretrained model weights. This observation corroborates our choice of tuning top singular vectors in our proposed spectral adaptation mechanism. Empirically, we observe that tuning top directions performs superior to tuning bottom ones, see Appendix F.3 and F.5.1 for related experiments.

3.2 Weight Subspace Alignment

Consider two-layer ReLU network with m hidden nodes and univariate output. For squared loss objective, we can write out the training problem explicitly as

where X ∈ R n × d is the data matrix, ( W ( 1 ) ∈ R d × m , W ( 2 ) ∈ R m ) are first and second layer weights respectively and y ∈ R n is the label vector. For better visualization, we take d = 3 . Consider the case that all data points lie on xy -plane, which mimics the usual observation that data points occupy a low-dimensional manifold. Then we can decompose each first layer neuron W ( 1 ) j ∈ R d into W ( 1 ) j = w j 1 + w j 2 where w j 1 ∈ R( X ) , w j 2 ⊥ R( X ) . With simple algebra, for non-zero weight decay which is often the default setting for current deep learning optimizers, one can derive w j 2 = 0 and thus W ( 1 ) j = w j 1 ∈ R( X ) . Therefore all optimal neurons lie also in xy -plane. However, due to optimization errors, some

of the trained neurons might be slightly deviated from xy -plane, as illustrated in Figure 3, where u i indicates pretrained neuron directions, though most of them lie in xy -plane, some might deviate (i.e., u 4 ). u ⋆ indicates the top singular vector direction of pretrained weight W ( 1 ) which here recognizes the xy -plane orientation, and thus fine-tuning u ⋆ is noiseless and is expected to be more robust.

4 Empirical Results: The Impact of Spectral Information

We experiment our proposed spectral adapter with fine-tuning large language models and diffusion models and compare against various recent PEFT methods. From language model experiments, we observe that Spectral Adapter A performs superior to various PEFT baselines and harnesses higher scores on different benchmarks, which again verifies the effectiveness of incorporating spectral information into the fine-tuning procedure, see Section 4.1 for details. For diffusion model experiments, we will see that the advantage of spectral adapter comes in two-fold: Spectral Adapter A offers a natural solution to existing problems in multi-adapter fusion procedure and Spectral Adapter R manifests finer-grained parameter budgets as well as better parameter efficiency, see Section 4.2 and 4.3 respectively. For a fair comparison with all baselines, we use their official implementation and follow hyperparameter setting in their original reports as long as available. See each individual section for corresponding experimental details. All experiments are done with NVIDIA RTX A6000 GPU.

4.1 Language Model Fine-Tuning: Enhancing Fine-Tuning Results with Spectral Adapter A

For large language model experiments, we present experimental results for fine-tuning DeBERTaV3base model (185M) and Mistral model (7B) on GLUE and GSM8K tasks respectively. Our Spectral Adapter A method achieves superior tuning results compared to various recent PEFT methods in most experiments.

DeBERTaV3-base Experiment. Table 1 shows fine-tuning results of DeBERTaV3-base model on GLUE benchmarks with various PEFT methods. For a fair comparison, we use official implementations for LoRA, DoRA, OFT and AdaLoRA in HuggingFace PEFT library, with hyperparameter setting for LoRA [20] and AdaLoRA [65] following their original reports. We use same hyperparameter setting as LoRA for DoRA and follow the setting used in BOFT [33], a variant of OFT, for OFT experiments. We abbreviate Spectral Adapter A as Spectral A for presentation simplicity and we tune hyperparameters for Spectral Adapter A . See Appendix F.2 for hyperparameter details and F.3 for loss/validation plot comparison. We fine-tune all q, k, v matrices in attention layers. Our Spectral Adapter A achieves highest average score and best scores for most tasks with fewest trainable parameters.

Mistral 7B Experiment. We experiment our Spectral Adapter A with Mistral 7B model [23] fine-tuned for GSM8K task [8]. Since all baseline model reports include no finetuning tasks with the Mistral family, we use official implementations of all baseline methods for comparison and we fix learning rate to be 2 . 5 e -5 for all methods following [51].

We take r = 8 for LoRA, DoRA and Spectral Adapter A to maintain approximately same number of trainable parameters for all methods. Table 2 presents the accuracy comparison where Spectral A stands for Spectral Adapter A . From the result, we observe that our Spectral Adapter A scores higher than both LoRA and DoRA by a large margin and increases the pretrained model baseline significantly, which verifies the effectiveness of the proposed spectral adaptation mechanism. See Appendix F.4 for more about experi-

mental details. Note for a different learning rate, DoRA performs better than LoRA while still worse than our method, see also Appendix F.4 for details.

4.2 Diffusion Model Fusion: Improving Multi-Object Fine-Tuning with Spectral Adapter A

Step A.

Step C.

lecunu

S

Step D.

U

lecunv

V

U+

lecunu

Step B.

bengiou

hintonU

Merged U

MergedUx

S

MergedV“<bengio>,

lecun"

>

lecun

<hinton>standingnear

<lecun>,and

Distributeconceptsalong

differentspectralspace

V

+

>

lecun

bengiov

hinton

=MergedV

high quality"

alake,4K,

bengiou

bengioV

5

hinton

hinton

Addindividualadapters

Collectindividualadapters

Synthesis

Multi-adapter fusion is a current bottleneck in diffusion model fine-tuning tasks with LoRA adapters. Simply adding different LoRA adapters tuned for distinct objects will result in problems involving identity loss and concept binding [12]. To tackle this toughness, different methods emerge such as Gradient Fusion [12] and Orthogonal Adaptation [42]. Specifically, Orthogonal Adaptation method proposes to fix LoRA parameter B to have orthogonal basis and train A solely. Experiments there show that merging LoRA weights with such orthogonal basis helps preserving individual object characteristics compared to its non-orthogonal counterpart. In Orthogonal Adaptation [42], the authors maintain B by manually keeping large orthogonal matrices for different layer sizes and sample r columns from corresponding orthogonal matrix to form B for each LoRA adapter. With knowledge from random matrix theory, such sampled matrices are likely to have orthogonal basis.

Notably, our Spectral Adapter A naturally operates on orthogonal singular vectors and thus introduces an elegant solution to multi-adapter fusion problems by distributing different concept tunings along different columns of singular vector matrices, which maps to wireless communications where the signals are distributed over non-overlapping frequencies. A subtlety here lies in the choice of column space for different fine-tuning tasks: (1) Sample-based methods can be adopted if data privacy is considered and different tuning tasks are done independently. In Appendix F.5, we show that tuning top columns manifests better generation quality compared to both tuning bottom columns and sampling random orthogonal basis as what has been done in Orthogonal Adaptation [42]. Thus there is a trade-off between high-quality generation and concept collapsing, i.e., sampling from top singular vectors is more encouraged while column overlapping between concepts happens more often compared to sampling from the whole set. (2) On the other hand, if fine-tuning tasks are not isolated and can collaborate on the column scheduling, then more deliberate tuning scheduling can be adopted, for example in a two-concept tuning task with r = 4 , the first concept can allocate first to fourth columns and the second concept then claims fifth to eighth columns. Figure 4 demonstrates steps for the same method for three-concept tuning task. Since we expect fine-tuned weights to stay close to original weights, though both row space and column space are tuned in spectral adapter, this adaptation mechanism approximates orthogonal-basis tuning for different objects and thus we expect it helps improving identity preservation for multi-adapter fusion. In this section, we investigate this effect via extensive diffusion model experiments.

Our experiments follow [42] and build on [12] which studies multi-LoRA fusion. We experiment with multi-object tuning and face generation tasks. Due to space limitation, we present some multi-object tuning results below and we leave the rest to Appendix F.5. For all tasks, we compare against baselines including Gradient Fusion [12], Orthogonal Adaptation [42], and FedAvg [37]. We start with a simple review for these baseline methods.

Baseline Review

To merge different LoRA adapters, say we have a set of LoRA parameters { ∆ θ 1 , . . . , ∆ θ n } where ∆ θ i = A i B T i and pretrained parameter θ 0 , FedAvg [37] proposes to merge them in to a single parameter by taking a weighted average as θ merged = θ 0 + ∑ i λ i ∆ θ i , where λ i is the weight attached to parameter ∆ θ i and is usually taken to satisfy ∑ i λ i = 1 , i.e., θ merged is a convex combination of individual adapters. Gradient Fusion [12] instead considers solving an auxiliary optimization problem of form θ merged = argmin θ ∑ n i = 1 ∥( θ 0 + ∆ θ i ) X i -θX i ∥ 2 F where X i represents the input activation of the i -th concept. Orthogonal Adaptation [42] follows FedAvg method and replaces original LoRA

prompts

a <Vdog2>and a <Vcat >

a <Vdogz> and a <Vcat>

a <Vdogz>and a<Vcat>

and a <Vdog2>

and a <Vdogz>

and a <Vdogz>

reference

method

infront of MountFuji

in a galaxy

onaplayground

5s

Gradient

Fusion

img: 0.8667

txt:0.2583 avg:0.5625

img: 0.8843

txt:0.3523 avg:0.6181

img: 0.8511

txt:0.2981

5s

FedAvg

cat>

img: 0.8745

5txt:0.3464avg:0.6103

img: 0.8755txt:0.2620 avg:0.5688

img:0.8481

txt:0.3039 avg:0.5761

5s

Orthogonal

<Vdog2>

Adaptation

img: 0.8652

txt:0.2810avg:0.5732

img: 0.8540

txt:0.2949avg:0.5742

img: 0.8320 txt:0.3152 avg:0.5737

5s

Spectral AdapterA

condition

sketch

(ours)

img: 0.8828

txt:0.3555 avg:0.6191

img: 0.8784txt:0.3496 avg:0.6142

img: 0.8804txt:0.2974 avg:0.5888

parameters with orthogonal-based LoRA adapters. For our method, to merge different spectral adapters, let θ 0 = U 0 S 0 V T 0 denote the spectral representation of pretrained model weight. Given a set of spectral adapters {( U i , V i ) , . . . , ( U n , V n )} with zero-padding to make the shape the same as ( U 0 , V 0 ) , we follow FedAvg and compute θ merged = ( U 0 + ∑ i λ i U i ) S 0 ( V 0 + ∑ i λ i V i ) T . In the following experiments, we take λ i = 1 / n as in [42] for all FedAvg, Orthogonal Adaptation, and our Spectral Adapter A fusion. Notably, all FedAvg, Orthogonal Adaptation, and our Spectral Adapter A fusion can be done approximately instantly while Gradient Fusion usually takes around 10 ∼ 15 minutes for solving its auxiliary optimization problems for all concept adapters.

Multi-Object Generation

We follow default training setting in [12] and fine-tune the Chilloutmix diffusion model [1] on three custom animal concepts, see original animals in "reference" in Figure 5. For better spatial alignment, we adopt T2I-Adapter [39] with sketch condition and we set guidance equal to one, see also "reference" in Figure 5 for the sketch condition being used. LoRA rank r = 8 is adopted. For baseline comparisons, we use original code for Gradient Fusion [12] and Orthogonal Adaptation [42]. We adapt code of Gradient Fusion for FedAvg method since there is no official implementation available. Custom animal name is replaced with special token < V animal> for fine-tuning. For our Spectral Adapter A , we follow the method depicted in Figure 4 and tune first, second, and third top eighth columns of singular vector matrices for different animal concepts. Figure 5 shows the generation results with different methods for selected prompts. Notably, baseline methods sometimes fail to capture the custom animal concepts while Spectral Adapter A recognizes all custom animals and generates visually satisfactory images. For better measurement, we also compute the alignment scores for each generated image with both reference images and prompt texts. It can be witnessed that our method achieves better alignment scores compared to baselines. See Appendix F.7 for details on alignment score computation.

4.3 Diffusion Model Expressiveness: Improving Parameter Efficiency with Spectral Adapter R

Spectral Adapter R is closely connected to prior Orthogonal Fine-Tuning (OFT ) [45] method which proposes to multiply the pretrained model weights by trainable orthogonal matrices in the finetuning procedure. Motivation behind OFT is to preserve hyperspherical energy which characterizes the pairwise neuron relationship on the unit hypersphere. Unlike OFT which orthogonally rotates neurons, Spectral Adapter R multiplies the topr columns of singular vector space U and V by orthogonal trainable matrices. For our implementation, several options are available for maintaining a trainable orthogonal matrix such as adding an orthogonality penalty in the objective function considered in [65] or via Cayley parameterization considered in [45]. We follow [45] and adopt Cayley parameterization which is supported by Pytorch [44]. Specifically, the orthogonal matrix R is constructed via R = ( I + Q )( I -Q ) -1 with a skew-symmetric matrix Q maintained as ( A -A T )/ 2

where A is our trainable parameter. Compared to adding an auxiliary orthogonality penalty, this parametrization is exact and thus the SVD form is preserved after tuning with Spectral Adapter R and can be adopted directly for subsequent fine-tuning tasks, which we state formally as a lemma below:

Lemma 4.1. With the Cayley parametrization, Spectral Adapter R is an exact rotation operation and thus preserves the structure of the SVD of the fine-tuned weight. Subsequent fine-tunings can be applied consequently without recomputing the SVD each time.

See Appendix C for the proof of above lemma. Unlike LoRA which requires number of trainable parameters to scale with weight size, when tuning topr columns of U an V , Spectral Adapter R only requires two trainable matrices of size r × r and thus can be more parameter-efficient especially for large pretrained weight. For common weight size such as W ∈ R 1024 × 1024 , LoRA with only r = 1 introduces same number of trainable parameters as Spectral Adapter R with r = 32 . For a thorough analysis on parameter efficiency improvement brought by Spectral Adapter R , we here also compare with different variants of LoRA which are proposed for trainable parameter savings. We review all baselines in detail below.

Baseline Review

LoRA

SVDiff

(HyperDreamBooth)

OFT

VeRA

sihgular

dia

go

nal

orthogqnal

diagonal

valuies

We compare our Spectral Adapter R with LoRA [20], SVDiff [15], LiDB [48], OFT [45], and VeRA [25]. Though the other methods are proposed for vision model tuning, VeRA is originally proposed for LLM tuning and we extend it here to diffusion model tuning due to its parameter efficiency. Consider a pretrained weight W ∈ R n × n , SVDiff originally proposes to tune all singular values of flattened CNN weights, here we extend it to tune all singular values of text encoder and U-Net weights for our comparison, thus trainable parameter attached to W will be of size n and is nonadjustable. LiDB stands for Lightweight Dreambooth and proposes to cut down trainable parameter budget by introducing auxiliary frozen matrix A aux ∈ R n × a and B aux ∈ R b × n , then it mimics LoRA but uses A aux AB T B aux in replace of AB T with trainable ( A ∈ R a × r , B ∈ R b × r ) . Thus with a, b < n, LiDB requires ( a + b ) r < 2 nr trainable parameters. In below, we use a = 50 , b = 100 as default in [48]. OFT multiplies the weight matrix by a trainable orthogonal matrix via Cayley parametrization discussed above, thus its complete version requires n 2 trainable parameters. For parameter efficiency, OFT proposes to use block-diagonal trainable matrix with all diagonal blocks being orthogonal. Thus with r diagonal blocks, the number of trainable parameter will be r ×( n / r ) 2 .

Further reduction of trainable parameter is achieved via sharing the diagonal blocks, which demands only ( n / r ) 2 parameters. In below comparison, we use this shared block-diagonal version for best parameter efficiency of OFT. VeRA proposes to use Λ a A Λ b B T in replace of AB T where Λ a and Λ b are diagonal matrices of size n × n and r × r respectively. Thus the total number of trainable parameters by VeRA is ( n + r ) ∝ n . Table 3 compares dif-

Table 3: Baseline methods comparison for parameter efficiency. Granularity indicates number of trainable parameter budgets available. See Section 4.3 for details.

ferent properties across all methods, where n represents weight size and r represents rank for all methods except for OFT, where r denotes number of diagonal blocks.

Parameter Efficiency

We fine-tune the Chilloumix diffusion model [1] with various PEFT methods on custom vase concept and present the generation results for prompt "a < V vase>" in Figure 6 for various trainable parameter budgets, where grey dash denotes that the corresponding parameter budget is unobtainable with a given adapter no matter how the hyperparameter is chosen and empty entry without grey dash

#param

1K

20K

100K 200K 300K

400K

500K >800K

reference

method

OFT

alignment:0.74alignment:0.80

alignment:0.84

alignment:0.85

(HyperDreamBooth)

alignment:0.71

alignment:0.83

alignment:0.84

SVDiff

alignment:0.79

LoRA

alignment:0.84

alignment:0.84alignment:0.85

VeRA

alignment:0.80

alignment:0.80

alignment:0.83

Spectral AdapterR

(sano)

alignment:0.75alignment:0.82

alignment:0.83 alignment:0.84

alignment:0.85

represents that there is a way to achieve the corresponding parameter budget though the generation result is skipped for better visualization. We follow default LoRA implementation in [12] for LoRA baseline and adjust it for all other methods. From Figure 6, it can be observed that LoRA, OFT, and LiDB start to generate vase close to custom vase with at least 200 k trainable parameters. SVDiff and VeRA are unable to generate ideal vase images even if scaled to large parameter budget. On the contrary, Spectral Adapter R starts to recognize the custom vase concept with only 20 k trainable parameters and has finer-grained parameter choices compared to other methods, i.e., notably Spectral Adapter R can have as few as 1 k parameters while other methods start with at least tens of thousands of trainable parameters. In a word, Spectral Adapter R enjoys finer-grained parameter budget choices and manifests better visual quality with fewer parameters, thus achieves enhanced parameter efficiency compared to various other PEFT methods.

Figure 7 below presents generation results of Chilloutmix diffusion model [1] tuned on custom chair concept with different methods under various parameter budgets. The prompt used is "a yellow < V chair>". See "reference" in Figure 7 for original chair images. From the generation results, it can be observed that LoRA generates reasonable chairs for all rank r = 1 , 2 , 3 though it already induces 273 k parameters even if rank is set to 1 . OFT and VeRA start to recognize custom chair with > 100 k parameters. SVDiff has a single fixed trainable parameter budget of size around 100 k . LiDB forms a

LoRA

273k

546k

819k

Spectral

R

1k

6k

23k

92k

206k

Reference

r=1

r=2

r=3

(ours)

r=2

r=4

r=8

r=16

r=24

OFT

alignment:0.8789alignment:0.8808

29k

118k

474k

alignment:0.8857

1.8m

VeRA

136k 316k

alignment:0.8241

alignment:0.8295

SVDiff

alignment:0.8477

130k

alignment:0.8847

26k

422k

r=64

r=32

r=16

r=8

r=1

r=1024

r=4096

fixed

(HyperDreamBooth)r=1

r=16

r=32

alignment:0.8022 alignment:0.8325 alignment:0.8466 alignment:0.8789 alignment:0.8442 alignment:0.8720 alignment:0.8554 alignment:0.8642

competitive candidate and generates satisfactory images with smallest trainable parameter budget among all baseline methods. However, our Spectral Adapter R still generates images better aligned to

reference images with as few as 20 k trainable parameters and has finer-grained parameter budget choices compared to LiDB. See Appendix F.6 for hyperparameter setting and Appendix F.7 for alignment score computation details.

4.4 Final Note: A Closer Look at SVD Cost

Runtime (/s)

6000

40

30

4000

Diffusion Model

MistralModel

Diffusion Model

MistralModel

(1B)

(7B)

20

(1B)

(7B)

2000

10

0

0

LoRA

SVD

Spectral AdapterA

To alleviate the concerns with respect to online training cost and show that our proposed method is very practical, we provide runtime and GPU storage cost bar plot in Figure 8, which shows runtime and GPU storage cost for LoRA and for our Spectral Adapter A when used for fine-tuning diffusion model in Section 4.2 and Mistral 7B model in Section 4.1. Here we adopt rank r = 8 for both LoRA and Spectral Adapter A . It can be observed that our Spectral Adapter A introduces negligible runtime and storage overhead for current large model size. Modern numerical tools such as randomized SVD [13] can also be exploited for further runtime reduction and the SVD procedure can be parallelized when multiple machines are available. See Appendix E for further investigation.

5 Conclusion and Limitations

In this work, we investigate the incorporation of spectral information of pretrained model weights into current PEFT models by introducing a spectral adaptation mechanism which updates only the top singular vectors of pretrained weights. We investigate the additive and rotational variants of such spectral adaptation mechanism. Theoretically, we show the motivation of tuning top singular vectors by comparing the rank capacity of different fine-tuning models and carrying out weight decomposition of pretrained model layers. Empirically, we verify the superiority of our proposed spectral adaptation method compared to various recent PEFT methods from different aspects via extensive experiments. To our best knowledge, this is the first work considering incorporating spectral information as a practical generic paradigm for fine-tuning tasks and enhances fine-tuning results, parameter efficiency, as well as benefits multi-adapter fusion of existing PEFT methods. For future work, fine-tuning spectral representation of different components, i.e., only the attention layer, of current large models is also worth studying. Other PEFT methods such as AdaLoRA [65] can also be dynamically combined with spectral adaptation.

A limitation of the current work remains in the choice of tuning top spectral space. Though its validity has been theoretically verified under simple settings, further investigation on tuning different columns of singular vector matrices is critical to understanding the role of spectral information in fine-tuning procedure. Besides, fine-tuning spectral representation of different components, i.e., only the attention layer, of current large models is also worth studying. Moreover, the time consumption of singular value decomposition procedure increases as model grows larger and thus faster singular value decomposition method also benefits.

References

Chilloutmix diffusion model. https://civitai.com/models/6424/chilloutmix.

A. Aghajanyan, L. Zettlemoyer, and S. Gupta. Intrinsic dimensionality explains the effectiveness of language model fine-tuning, 2020.

A. Asai, M. Salehi, M. E. Peters, and H. Hajishirzi. Attempt: Parameter-efficient multi-task tuning via attentional mixtures of soft prompts, 2022.

N. Cancedda. Spectral filters, dark signals, and attention sinks, 2024.

A. Chavan, Z. Liu, D. Gupta, E. Xing, and Z. Shen. One-for-all: Generalized lora for parameterefficient fine-tuning, 2023.

Y. Chen, D. Hazarika, M. Namazifar, Y. Liu, D. Jin, and D. Hakkani-Tur. Empowering parameter-efficient transfer learning by recognizing the kernel structure in self-attention. arXiv preprint arXiv:2205.03720 , 2022.

A. Chronopoulou, M. E. Peters, A. Fraser, and J. Dodge. Adaptersoup: Weight averaging to improve generalization of pretrained language models, 2023.

K. Cobbe, V. Kosaraju, M. Bavarian, M. Chen, H. Jun, L. Kaiser, M. Plappert, J. Tworek, J. Hilton, R. Nakano, C. Hesse, and J. Schulman. Training verifiers to solve math word problems, 2021.

M. Denil, B. Shakibi, L. Dinh, M. Ranzato, and N. de Freitas. Predicting parameters in deep learning, 2014.

T. Dettmers, A. Pagnoni, A. Holtzman, and L. Zettlemoyer. Qlora: Efficient finetuning of quantized llms, 2023.

A. Edalati, M. Tahaei, I. Kobyzev, V. P. Nia, J. J. Clark, and M. Rezagholizadeh. Krona: Parameter efficient tuning with kronecker adapter, 2022.

Y. Gu, X. Wang, J. Z. Wu, Y. Shi, C. Yunpeng, Z. Fan, W. Xiao, R. Zhao, S. Chang, W. Wu, Y. Ge, S. Ying, and M. Z. Shou. Mix-of-show: Decentralized low-rank adaptation for multi-concept customization of diffusion models. arXiv preprint arXiv:2305.18292 , 2023.

N. Halko, P.-G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, 2010.

K. Hambardzumyan, H. Khachatrian, and J. May. Warp: Word-level adversarial reprogramming, 2021.

L. Han, Y. Li, H. Zhang, P. Milanfar, D. Metaxas, and F. Yang. Svdiff: Compact parameter space for diffusion fine-tuning, 2023.

S. Hayou, N. Ghosh, and B. Yu. Lora+: Efficient low rank adaptation of large models, 2024.

J. He, C. Zhou, X. Ma, T. Berg-Kirkpatrick, and G. Neubig. Towards a unified view of parameter-efficient transfer learning, 2022.

S. He, R.-Z. Fan, L. Ding, L. Shen, T. Zhou, and D. Tao. Mera: Merging pretrained adapters for few-shot learning. arXiv preprint arXiv:2308.15982 , 2023.

N. Houlsby, A. Giurgiu, S. Jastrzebski, B. Morrone, Q. de Laroussilhe, A. Gesmundo, M. Attariyan, and S. Gelly. Parameter-efficient transfer learning for nlp, 2019.

E. J. Hu, Y. Shen, P. Wallis, Z. Allen-Zhu, Y. Li, S. Wang, L. Wang, and W. Chen. Lora: Low-rank adaptation of large language models, 2021.

C. Huang, Q. Liu, B. Y. Lin, T. Pang, C. Du, and M. Lin. Lorahub: Efficient cross-task generalization via dynamic lora composition, 2024.

M. Jaderberg, A. Vedaldi, and A. Zisserman. Speeding up convolutional neural networks with low rank expansions. arXiv preprint arXiv:1405.3866 , 2014.

A. Q. Jiang, A. Sablayrolles, A. Mensch, C. Bamford, D. S. Chaplot, D. de las Casas, F. Bressand, G. Lengyel, G. Lample, L. Saulnier, L. R. Lavaud, M.-A. Lachaux, P. Stock, T. L. Scao, T. Lavril, T. Wang, T. Lacroix, and W. E. Sayed. Mistral 7b, 2023.

M. Khodak, N. Tenenholtz, L. Mackey, and N. Fusi. Initialization and regularization of factorized neural layers, 2022.

D. J. Kopiczko, T. Blankevoort, and Y. M. Asano. Vera: Vector-based random matrix adaptation, 2024.

T. Lei, J. Bai, S. Brahma, J. Ainslie, K. Lee, Y. Zhou, N. Du, V. Zhao, Y. Wu, B. Li, et al. Conditional adapters: Parameter-efficient transfer learning with fast inference. Advances in Neural Information Processing Systems , 36, 2024.

B. Lester, R. Al-Rfou, and N. Constant. The power of scale for parameter-efficient prompt tuning, 2021.

X. L. Li and P. Liang. Prefix-tuning: Optimizing continuous prompts for generation, 2021.

Y. Li, Y. Yu, C. Liang, P. He, N. Karampatziakis, W. Chen, and T. Zhao. Loftq: Lora-finetuning-aware quantization for large language models, 2023.

Z. Lin, A. Madotto, and P. Fung. Exploring versatile generative language model via parameterefficient transfer learning. arXiv preprint arXiv:2004.03829 , 2020.

Q. Liu, X. Wu, X. Zhao, Y. Zhu, D. Xu, F. Tian, and Y. Zheng. Moelora: An moe-based parameter efficient fine-tuning method for multi-task medical applications, 2023.

S.-Y. Liu, C.-Y. Wang, H. Yin, P. Molchanov, Y.-C. F. Wang, K.-T. Cheng, and M.-H. Chen. Dora: Weight-decomposed low-rank adaptation, 2024.

W. Liu, Z. Qiu, Y. Feng, Y. Xiu, Y. Xue, L. Yu, H. Feng, Z. Liu, J. Heo, S. Peng, Y. Wen, M. J. Black, A. Weller, and B. Schölkopf. Parameter-efficient orthogonal finetuning via butterfly factorization, 2023.

X. Liu, Y. Zheng, Z. Du, M. Ding, Y. Qian, Z. Yang, and J. Tang. Gpt understands, too, 2023.

R. K. Mahabadi, S. Ruder, M. Dehghani, and J. Henderson. Parameter-efficient multi-task fine-tuning for transformers via shared hypernetworks, 2021.

C. H. Martin and M. W. Mahoney. Implicit self-regularization in deep neural networks: Evidence from random matrix theory and implications for learning, 2018.

H. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. Communicationefficient learning of deep networks from decentralized data, 2023.

A. Mitra, H. Khanpour, C. Rosset, and A. Awadallah. Orca-math: Unlocking the potential of slms in grade school math, 2024.

C. Mou, X. Wang, L. Xie, Y. Wu, J. Zhang, Z. Qi, Y. Shan, and X. Qie. T2i-adapter: Learning adapters to dig out more controllable ability for text-to-image diffusion models, 2023.

mrm8488. Lora finetune deberta-v3 huggingface blog, 2021. Available at https://huggingface.co/mrm8488/deberta-v3-small-finetuned-mnli/commits/main.

J. Pfeiffer, A. Kamath, A. Rücklé, K. Cho, and I. Gurevych. Adapterfusion: Non-destructive task composition for transfer learning. arXiv preprint arXiv:2005.00247 , 2020.

R. Po, G. Yang, K. Aberman, and G. Wetzstein. Orthogonal adaptation for modular customization of diffusion models, 2023.

D. Povey, G. Cheng, Y. Wang, K. Li, H. Xu, M. A. Yarmohammadi, and S. Khudanpur. Semiorthogonal low-rank matrix factorization for deep neural networks. In Interspeech , 2018.

pytorch group. Pytorch orthogonal parameterization method implementation, 2023.

Z. Qiu, W. Liu, H. Feng, Y. Xue, Y. Feng, Z. Liu, D. Zhang, A. Weller, and B. Schölkopf. Controlling text-to-image diffusion by orthogonal finetuning, 2023.

A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, G. Krueger, and I. Sutskever. Learning transferable visual models from natural language supervision, 2021.

E. Robb, W.-S. Chu, A. Kumar, and J.-B. Huang. Few-shot adaptation of generative adversarial networks, 2020.

N. Ruiz, Y. Li, V. Jampani, W. Wei, T. Hou, Y. Pritch, N. Wadhwa, M. Rubinstein, and K. Aberman. Hyperdreambooth: Hypernetworks for fast personalization of text-to-image models, 2023.

A. Rücklé, G. Geigle, M. Glockner, T. Beck, J. Pfeiffer, N. Reimers, and I. Gurevych. Adapterdrop: On the efficiency of adapters in transformers, 2021.

T. N. Sainath, B. Kingsbury, V. Sindhwani, E. Arisoy, and B. Ramabhadran. Low-rank matrix factorization for deep neural network training with high-dimensional output targets. In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing , pages 6655-6659, 2013.

H. Skogström. Lora finetune mistral 7b valohai blog, 2024. https://valohai.com/blog/finetunemistral/.

A. Tang, L. Shen, Y. Luo, Y. Zhan, H. Hu, B. Du, Y. Chen, and D. Tao. Parameter efficient multi-task model fusion with partial linearization, 2023.

K. Turgutlu. Answer.ai qdora report, 2024. https://www.answer.ai/posts/2024-04-26-fsdp-qdorallama3.html.

M. Valipour, M. Rezagholizadeh, I. Kobyzev, and A. Ghodsi. Dylora: Parameter efficient tuning of pre-trained models using dynamic search-free low-rank adaptation, 2023.

T. Vu, B. Lester, N. Constant, R. Al-Rfou, and D. Cer. Spot: Better frozen model adaptation through soft prompt transfer. arXiv preprint arXiv:2110.07904 , 2021.

Z. Wang, R. Panda, L. Karlinsky, R. Feris, H. Sun, and Y. Kim. Multitask prompt tuning enables parameter-efficient transfer learning, 2023.

G. Xiao, Y. Tian, B. Chen, S. Han, and M. Lewis. Efficient streaming language models with attention sinks, 2024.

L. Xu, H. Xie, S.-Z. J. Qin, X. Tao, and F. L. Wang. Parameter-efficient fine-tuning methods for pretrained language models: A critical review and assessment, 2023.

Y. Xu, L. Xie, X. Gu, X. Chen, H. Chang, H. Zhang, Z. Chen, X. Zhang, and Q. Tian. Qa-lora: Quantization-aware low-rank adaptation of large language models, 2023.

A. X. Yang, M. Robeyns, X. Wang, and L. Aitchison. Bayesian low-rank adaptation for large language models, 2024.

F. Zhang and M. Pilanci. Riemannian preconditioned lora for fine-tuning foundation models, 2024.

F. F. Zhang, L. Li, J.-C. Chen, Z. Jiang, B. Wang, and Y. Qian. Increlora: Incremental parameter allocation method for parameter-efficient fine-tuning. ArXiv , abs/2308.12043, 2023.

L. Zhang, L. Zhang, S. Shi, X. Chu, and B. Li. Lora-fa: Memory-efficient low-rank adaptation for large language models fine-tuning, 2023.

M. Zhang, H. Chen, C. Shen, Z. Yang, L. Ou, X. Yu, and B. Zhuang. Loraprune: Pruning meets low-rank parameter-efficient fine-tuning, 2023.

Q. Zhang, M. Chen, A. Bukharin, N. Karampatziakis, P. He, Y. Cheng, W. Chen, and T. Zhao. Adalora: Adaptive budget allocation for parameter-efficient fine-tuning, 2023.

Y. Zhang, E. Chuangsuwanich, and J. Glass. Extracting deep neural network bottleneck features using low-rank matrix factorization. In 2014 IEEE international conference on acoustics, speech and signal processing (ICASSP) , pages 185-189. IEEE, 2014.

H. Zhao, H. Tan, and H. Mei. Tiny-attention adapter: Contexts are more important than the number of parameters, 2022.

Y. Zhao, J. Li, and Y. Gong. Low-rank plus diagonal adaptation for deep neural networks. In 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , pages 5005-5009. IEEE, 2016.

Y. Zhu, J. Feng, C. Zhao, M. Wang, and L. Li. Counter-interference adapter for multilingual machine translation, 2021.

B. Zi, X. Qi, L. Wang, J. Wang, K.-F. Wong, and L. Zhang. Delta-lora: Fine-tuning high-rank parameters with the delta of low-rank matrices, 2023.

Appendix

A Prior Work

Here we provide an overview of recent PEFT methods. Dating back to 2019, Houlsby et al. [19] develop the idea of parameter-efficient fine-tuning and introduce Adapter model, which injects trainable components between pretrained model layers, though the number of trainable parameters has been reduced due to the small size of adapters, this method incurs inference latency and is thus not desirable. Later improvement of Adapter fine-tuning focuses on improving inference latency [49, 26], fusing multiple adapters [7, 41, 18], modifying adapter model architecture [67], introducing parallelism [17, 69], and creating task-specific and layer-specific adapter [35, 30]. Another line of fine-tuning is prompt-tuning [27] which usually adds the trainable components into the prompt. Variants of prompt-tuning involve WARP [14], prefix-tuning [28], P-tuning [34], and ATTEMPT [3] which consider injecting different forms of trainable components. Multitask prompt-tuning is considered in [55, 56].

The more relevant PEFT methods to our spectral adaptation mechanism involves LoRA [20] and OFT [45], which inspires our Spectral Adapter A and Spectral Adapter R respectively. LoRA originates from the observation that model fine-tuning is intrinsically low-rank [2]. Variants of LoRA involve different methods proposing dynamic allocation of LoRA rank budgets [54, 62, 65, 6]. LoRA has been combined with model pruning [64] and quantization [10, 59, 29]. Some other variants further cut down the trainable parameter budget or activation storage by modifying LoRA model [25, 11, 63]. DoRA [32] fixes LoRA's low-rank limitation by decomposing pretrained model weights and isolating their magnitudes. Laplace-LoRA [60] incorporates Bayesian inference into LoRA parameters to improve calibration. LoRAHub [21], MOELoRA [31], and L-LoRA [52] consider multitask LoRA. Delta-LoRA [70] updates pretrained weights simultaneously from information of LoRA parameters. GLoRA [5] generalizes LoRA by introducing a prompt module. Another line of variants focuses on analyzing the optimization scheme of LoRA model [61, 16]. OFT studies the multiplicative fine-tuning and its variant BOFT [33] improves OFT by utilizing butterfly parametrization for better information delivery efficiency. [58] offers a comprehensive review of recent development of PEFT methods.

B Rank Capacity Proof

Proof. Consider weight matrix W ∈ R n × m with n ≤ m of full row rank. For LoRA parameter A ∈ R m × r , B ∈ R n × r with n ≥ r, final weight matrix W + AB T has rank in [ n -r, n ] . With Spectral Adapter A parameters A S ∈ R m × r , B S ∈ R n × r where n ≥ 2 r . Let X r denote the first r columns of any matrix X and X -r denote the rest columns, final weight matrix (( U r + A S ) S r ( V r + B S ) T ) + U -r S -r V T -r has rank in [ n -2 r, n ] . Therefore, R( LoRA; W ) = r and R( Spectral Adapter A ; W ) = 2 r can be derived trivially.

C Cayley Parameterization Proof

Proof. With any trainable square matrix A, we set Q = ( A -A T )/ 2 and thus Q = -Q T and Q is skew-symmetric thereby. Now we show that for any skew-symmetric Q , ( I + Q )( I -Q ) -1 is orthogonal. Let O = ( I + Q )( I -Q ) -1 , then

which shows that the Cayley parametrization is exact and no re-SVD is needed for orthogonality preservation.

D Connection to DoRA

In DoRA [32], the authors observe that plain LoRA method tends to either increase or decrease the magnitude and direction updates proportionally and thus lacks ability to make slight direction change together with large magnitude change, to come across this limitation, the authors propose to decompose pretrained model weights into magnitude and direction and update them separately. The magnitude is replaced with a trainable scalar and the direction is updated with original LoRA method. Experiments in [32] show that such decomposition helps improve effectiveness of LoRA significantly. Here we show that our Spectral Adapter A is closely connected to the weight decomposition trick used in DoRA when pretrained model weight is of vector form. We note that in DoRA, after the weight decomposition, each column becomes unit-length while in Spectral Adapter A , we also operates on matrices with unit-length columns. Specifically, consider a pretrained model weight w 0 ∈ R n × 1 , then DoRA becomes

where w is a trainable scalar initialized at ∥ w 0 ∥ 2 . b and a are trainable parameters of size n × 1 and 1 × 1 respectively, with ba = 0 at initialization. Comparably, Spectral Adapter A becomes

with trainable vector a ′ ∈ R n × 1 and trainable scalar b ′ both initialized at zero. We can thus equivalently view ∥ w 0 ∥ 2 ( 1 + b ′ ) as a single trainable scalar initialized at ∥ w 0 ∥ 2 , which then plays the role of magnitude adapter as w in DoRA. a ′ is adopted for directional adaptation since it directly operates on the normalized base vector.

E Cost Investigation (More Detailed)

Here we address the potential concern about the overhead of our proposed spectral adaptation mechanism. Firstly, we note that spectral adapter introduces similar number of trainable parameters and can be merged into original model weights, thus it is lightweight for sharing and introduces no additional inference latency, which preserves the strengths of additive fine-tuning methods. Therefore, the major overhead concern exists in the runtime and GPU storage overhead during online training. Note our method involves only matrix multiplication in the forward procedure and thus should run as quick as LoRA. Though the SVD procedure can bring additional runtime overhead, it needs to be done only once for a single model and can be reused for later fine-tuning on various downstream tasks. Besides, modern numerical tools such as randomized SVD [13] can also be exploited and the SVD procedure can be parallelized when multiple machines are available. As for GPU storage, unlike SVDiff [15] where all SVD components are required for training procedure thus introducing significant GPU storage burden, our method requires only the top spectral space to be stored additionally and consumes similar GPU storage to LoRA for relatively small tuning ranks (which is usually the case).

F Supplemental Materials for Experiments

F.1 Experimental Setup for Figure 1

For Figure 1 experiments, we follow QDoRA [53] experimental setup for fine-tuning Llama3 8B model, where all k_proj, q_proj, v_proj, up_proj, down_proj, and gate_proj weights are tuned. We adopt the same data processing method and train on 10 K Orca Math data (shuffled) as in [53]. We fix learning rate as 1 e -5 for all methods as in QDoRA and train for one epoch with batch size 8 . r = 8 is adopted for LoRA, DoRA, AdaLoRA, and Spectral Adapter A while for OFT, we set number of diagonal blocks to be 800 to maintain similar amount of trainable parameters. LoRA alpha is set to be 16 following DoRA [32] convention and AdaLoRA hyperparameter is set following what has been used for MNLI benchmark in the original AdaLoRA report [65] with regularization set to 1 e -3 which we find works better. For evaluation, we test on GSM8K [8] benchmark for exact matching. For more comparisons, Figure 9 provides training loss for smaller rank r = 4 (oft_ r = 1600 ) and larger rank r = 64 (oft_ r = 95 ). All settings are the same except that LoRA alpha is always kept as

Full

LoRA

DoRA

OFT

AdaLoRA

Full

LoRA

DoRA

OFT

AdaLoRA

Training Loss

Spectral Adapter^ (ours)

Training Loss

Spectral Adapter^ (ours)

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

Global Steps

Global Steps

200

400

600

800

1000

1200

200

400

600

800

1000

1200

twice as rank number. From Figure 9 we can observe that though increasing trainable parameters closes the gap between different tuning methods, our spectral adapter method is always superior to other PEFT methods and stays closest to full fine-tuning.

F.2 Hyperparameter Setting for DeBERTaV3-base Experiment (Section 4.1)

Table 4 shows the hyperparameter setting for our Spectral Adapter A used for fine-tuning DeBERTaV3base model in Section 4.1. We set number of diagonal blocks to be 4 and enable block sharing for OFT to maintain similar amount of trainable parameters.

F.3 More About DeBERTaV3-base Experiment

Left plot in Figure 10 presents the training loss and validation score comparisons of LoRA, SVDiff and our Spectral Adapter A for fine-tuning DeBERTaV3-base model on CoLA benchmark. We set learning rates for both LoRA and Spectral Adapter A as what has been used in popular public blog [40] for LoRA fine-tuning with DeBERTaV3-base model, which is not tuned in favor of our method. For SVDiff, since it is originally proposed for vision model tuning, we extend it to this experiment by tuning all singular values of pretrained weights. We find the same learning rate leads to poor fine-tuning results with SVDiff, we thus pick the best learning rate among [ 1 e -3 , 1 e -4 , 1 e -5 ] according to validation performance and set learning rate to be 1 e -3 . We use r = 8 for LoRA and Spectral Adapter A . From Figure 10, it can be observed that Spectral Adapter A achieves better training and validation performance compared to both LoRA and SVDiff.

Interestingly, in LoRA [20], the authors provide a correlation analysis between the LoRA additive component △ W = AB T and original pretrained weight matrix W (see Section H.3 in [20]), and they find that the additive component does not contain the top singular directions of W . The authors therefore conclude that the learned LoRA component amplifies "task-specific" directions which are not emphasized in the pretrained weight matrix. Naively, this seems to suggest that tuning top singular subspace of pretrained weights is not ideal and one should identify the desired "task-specific" directions to improve LoRA. Here we show that this is not the case and fine-tuning top directions provides a significant improvement to LoRA. In the right plot of Figure 10 above, we experiment

0.7

Loss

Test Accuracy

6 ×10-1

0.5

Training

4 ×10-1

0.3

3 ×10-1

0.1

Q

4

8

12

LoRA

SVDiff

Spectral Adapter (Ours)

0.7

Training Loss

6 ×10-1

Test Accuracy

0.5

4 ×10-1

0.3

3 × 10-1

0.1

8

12

LoRA

Spectral Bottom

SpectralTop

tuning the top eighth rank and the bottom eighth rank of singular vector space in our Spectral Adapter A , which we present as "Spectral Top" and "Spectral Bottom" respectively. Remarkably, "Spectral Top" converges faster and scores higher than LoRA, which is then superior to "Spectral Bottom". This result unravels the fact that tuning different part of spectral space brings different tuning effect and tuning the top columns of singular vector space improves LoRA tuning significantly. See Section 3 for more theoretic insights.

F.4 Hyperparameter Setting for Mistral 7B Experiment (Section 4.1)

Table 5 shows training hyperparameter setting for fine-tuning Mistral 7B model in Section 4.1. We train with bfloat16 precision and fine-tune all q_proj, k_proj, v_proj, o_proj, and gate_proj weights. We evaluate with lm-evaluation-harness [47]. Table 6 shows accuracy comparison of different tuning methods with learning rate 1 e -5 . Our Spectral Adapter A still exceeds both LoRA and DoRA.

F.5 Supplemental Materials for Multi-Adapter Fusion Experiment (Section 4.2)

F.5.1 Comparison of Single Object Generation

We present more experimental results to show that Spectral Adapter A with top ranks tuned behaves at least as good as LoRA with same parameter budget and is better than Orthogonal Adaptation [42], which is likely due to that Orthogonal Adaptation fixes LoRA parameter B and thus has limited expressiveness. We also show that tuning bottom ranks in spectral adapter behaves worse than all other methods. Figure 11 shows generation results for custom toy concept tuning, where Orthogonal Adaptation and Spectral Adapter A (bottom) generate inaccurate happy-face octopus, sad-face octopus, and green tortoise. Figure 12 shows generation results for custom animal concept tuning, where Orthogonal Adaptation and Spectral Adapter A (bottom) sometimes miss first dog concept.

Orthogonal

Spectral Adapter^

Spectral Adapter^

Prompts

LoRA

Adaptation

(bottom)

(top)

wearing

aredhat

a<Voyg>

swimming

in the pool

a<Voy> sit

on the chair

<Vtoy2>

a<Viog

in front of

eiffel tower

a<Vioy>on

atable

a<Viog>

swimming

in the pool

a<Voy>

swimming

in the pool

a<Vo>sit

on the chair

a<Voy?

swimming

in the pool

<Voy?

a<Voy sit

on the chair

reference

Orthogonal

Spectral Adapter A

Spectral Adapter^

Prompts

LoRA

Adaptation

(bottom)

(top)

a<Vo>

walking on

the street

<Vdog>

wearing

a headphone

a<Vdo>

in front of

eiffel tower

<Vcat>

a<Veat>

walking on

the street

a<Vcat>

swimming

in the pool

on the chair

<Vdog2>

a<Va>

in front of

eiffel tower

a<Vdog>

wearing

aheadphone

a<Vd>

swimming

in the pool

reference

<V

cat >

<Vdog2?

F.5.2 More Multi-Adapter Fusion Generation Results

Here we present more results for multi-adapter fusion generation. Figure 13 shows generation results for multi-object generation for custom toy concepts and Figure 14 presents generation results for multi-character generation for three computer scientists. See below for experimental details.

Multi-Object Generation. As in Section 4.2, we fine-tune Chilloutmix diffusion model [1] on four custom toy concepts, see "reference" in Figure 13 for original toy images. We use r = 8 for all methods and tune first, second, third, and fourth top eighth columns of singular vector space of pretrained weights for first, second, third, and fourth toys in our Spectral Adapter A . We follow all default experimental settings in [12] and tune all embedding layer, U-Net, and text-encoder. For better spatial alignment, we employ T2I-Adapter with sketch condition listed in "reference" in Figure 13. We randomly select three scenes and prompt fused-adapters for the results, see "prompts" in Figure 13 for individual prompt being used. From Figure 13, it can be observed that FedAvg and Orthogonal Adaptation generate unsatisfactory happy-face octopus and green tortoise toys. On the contrary, our spectral adapter generates high-quality images similar to Gradient Fusion while saving much more time.

Multi-Character Generation. We also experiment fine-tuning Chilloutmix diffusion model [1]

a<Voyand a<Vyz>anda<Voy

prompts

a<Voyu>and a <Voyz> and a<Voya?

a<Voy? and a<Voyz and a<Voy?

reference

method

and a<Voy? on the sofa

aforest

15min

Gradient

Fusion

5s

FedAvg

5s

Orthogonal

Adaptation

5s

Spectral AdapterA

(Ours)

condition

sketck

with photos of three computer scientists Yoshua Bengio, Yann LeCun, and Geoffrey Hinton. As in multi-object generation, we use r = 8 for all methods and tune first, second, and third top eighth columns of singular vector space of pretrained weights for Bengio, Lecun, and Hinton in our Spectral Adapter A . We use T2I-Adapter [39] with keypose condition. See "reference" in Figure 14 for scientists' photos and keypose condition being used. Figure 14 shows generation results for prompt "< V bengio> and < V lecun> and < V hinton>, standing near a lake, 4K, high quality, high resolution" with different fused adapters, from which it can be observed that our spectral adapter generates picture of most consistent styles across characters and renders all scientists' faces clearly.

FedAvg

5s

OrthogonalAdaptation

5s

reference

<Venge> and <Vecw? and <Vhanto?. standingneara lake,4K, high quality, high resolution.

o：

<Vuto?

keypose

condition

GradientFusion

Spectral Adapter^ (Ours)

5s

15min

F.6 Supplemental Materials for Parameter Efficiency Experiment (Section 4.3)

In this section, we present more tuning results with various parameter budgets for parameter efficiency experiment studied in Section 4.3, see Section 4.3 for baseline method explanation. Table 7 shows the learning rates used for each baseline method and Table 8 shows learning rates used for our method, the rest experimental settings are default as in [12].

LoRA

273k

546k

819k

Spectral

R

1k

6k

23k

92k

206k

Reference

r=1

r=2

r=3

(ours)

r=2

r=4

r=8

r=16

r=24

OFT

alignment:0.8974alignment:0.9208

29k

r=64

118k

r=32

474k

r=16

alignment:0.9301

1.8m

r=8

VeRA

136k

r=1

alignment:0.8100

316k

r=1024

alignment:0.8266

r=4096

SVDiff

alignment:0.8764

130k

fixed (HyperDreamBooth)r=1

alignment:0.8925

26k

422k

alignment:0.9140

r=16

r=32

alignment:0.8222 alignment:0.8564  alignment:0.9057 alignment:0.9267

allignment:0.8554

alignment:0.8339

alignment:0.8916

alignment0.8999

Figure 15 shows generation results of Chilloutmix diffusion model [1] fine-tuned on custom table concept with different methods under various parameter budgets. The prompt used is 'a < V table>'. LoRA generates acceptable images for all rank r = 1 , 2 , 3 though it starts with 273 k parameters even if rank is set to 1 . OFT generates desirable images only for parameter budget > 400 k . VeRA and LiDB start to generate reasonable images with > 300 k trainable parameters and SVDiff has only a single fixed parameter budget. Meanwhile, our Spectral Adapter R recognizes the shape of custom table with as few as 6 k parameters and produces ideal images since 100 k parameters. See Appendix F.7 for alignment score computation details.

F.7 Alignment Score Computation

For better quantitative measurement, we compute alignment scores for our Figure 5,6,7,15 results. Specifically, we first compute CLIP [46] embedding for all generated/reference images and prompt texts, then we compute the cosine similarity between generated images' embedding and reference images' embedding to serve as their alignment score. Likewise, text score stands for cosine similarity between generated images' embeddings and their corresponding prompt texts' embeddings. Intuition here is that if an image is close to another image (or text), their CLIP vectors are expected to stay close as well. For Figure 5 alignment score computation, we crop each generated image vertically into three columns, then we compute their alignment scores to each corresponding reference animal, we finally take the mean of these three scores. For Figure 6, 7, 15 scores, we compute average score over three random trials, with each trial consisting of 8 generated images.

NeurIPS Paper Checklist

1. Claims

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

Answer: [Yes]

Justification: The introduction and abstract state that we study spectral fine-tuning mechanism which helps current PEFT from various perspectives. Our model in Section 2 and experimental results in Section 4 properly address all the points.

Guidelines:

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

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

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

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

2. Limitations

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

Answer: [Yes]

Justification: Section 5 discusses limitation of the current work.

Guidelines:

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

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

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

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

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

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

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

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

3. Theory Assumptions and Proofs

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

Answer: [Yes]

Justification: See Appendix B and C for proof of Lemma 3.1 and 4.1 respectively.

Guidelines:

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

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

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

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

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

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

4. Experimental Result Reproducibility

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

Answer: [Yes]

Justification: Experimental details are included in each subsection (or appendix) of corresponding experiments. For example, Table 4 provides all hyperparameter settings for DeBERTaV3-base experiment in Section 4.1.

Guidelines:

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

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

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

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

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

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

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

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

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

5. Open access to data and code

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

Answer: [Yes]

Justification: Code has been made public at https://github.com/pilancilab/ spectral_adapter .

Guidelines:

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

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

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

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

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

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

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

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

6. Experimental Setting/Details

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

Answer: [Yes]

Justification: See Section 4 and corresponding appendix sections in Appendix F for experimental details.

Guidelines:

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

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

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

7. Experiment Statistical Significance

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

Answer: [Yes]

Justification: See Table 2 and 6.

Guidelines:

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

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

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

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

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

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

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

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

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

8. Experiments Compute Resources

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

Answer: [Yes]

Justification: We mention "all experiments are done with NVIDIA RTX A6000 GPU" at the end of first paragraph in Section 4. Moreover, in Section 4.4, we specifically plot the memory and time of execution of some experiments.

Guidelines:

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

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

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

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

9. Code Of Ethics

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

Answer: [Yes]

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

Guidelines:

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

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

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

10. Broader Impacts

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

Answer: [NA]

Justification: Though large language/diffusion models might be used to generate fake profiles/images, our proposed fine-tuning model does not introduce additional societal impacts.

Guidelines:

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

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

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

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

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

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

11. Safeguards

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

Answer: [NA]

Justification: The paper poses no such risks.

Guidelines:

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

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

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

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

12. Licenses for existing assets

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

Answer: [Yes]

Justification: All related work has been cited.

Guidelines:

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

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

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

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

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

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

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

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

13. New Assets

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

Answer: [Yes]

Justification: See Section 2 for model architecture details. Supplemental code is submitted in an anonymous way.

Guidelines:

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

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

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

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

14. Crowdsourcing and Research with Human Subjects

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

Answer: [NA]

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

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

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

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

15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

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

Answer: [NA]

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

Guidelines:

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

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

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

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

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn summary, this paper investigates advancements in Parameter-Efficient Fine-Tuning (PEFT) for pre-trained neural networks by integrating spectral information from pretrained weights, aiming to enhance the classic LoRA approach. By employing Singular Value Decomposition (SVD), the authors introduce two spectral adaptation techniques: additive tuning and orthogonal rotation of the top singular vectors.\n\nSTRENGTHS:\nThe topic of LoRA adaptation addressed in this paper is valuable due to its applicability to a wide range of high-level and low-level tasks. The authors introduce two variants of the proposed enhanced LoRA method. In specific datasets and tasks, these variants demonstrate improvements over state-of-the-art (SOTA) methods.\n\nWEAKNESSES:\nThere are some typos that need to be corrected. For example, a spelling mistake in the word \"Apppendix\" in line 32, it should be corrected to \"Appendix.\" Additionally, the word \"digged\" in line 20 should be corrected to \"dug\" to use the proper past tense of the verb \"dig.\".\n\nIt is not common to put the whole literature review in Appendix.\n\nIn line 53, the phrase \"orthogonal rotating the top singular vector space.\" This phrase should be corrected to \"orthogonally rotating the top singular vector space\" to properly use the adverbial form \"orthogonally,\" which modifies the verb \"rotating.\"\n\nAdditionally, if the spectral space is modified, the rank will also change. This implies that the optimal ranks of the two spectral spaces differ, making it unfair to compare different LoRA absed methods using the same rank.\n\nThere is a logic error of the claim stated in line 82-93: Specifically, the statement that \"these methods require storing all U, S and V during training while only the diagonal vector of S is tuned, which nearly doubles the storage requirement compared to pretraining when fine-tuning on downstream tasks\" is misleading. Storing all components (U, S, and V) does indeed increase storage, but it's not clear why this would \"nearly double\" the storage requirement. The increase in storage would depend on the specifics of the matrix dimensions and the storage format. The phrase needs to clarify how the storage requirement nearly doubles to avoid logical inconsistency.\n\nQUESTIONS:\nLiterature [36, 4, 56] already studied the  spectral space of model weights; it is not clear what is new in this paper, and highlighting the difference would make the contribution of this paper much clearer. \n\nThe inclusion of the revised Singular Value Decomposition (SVD) based fine-tuning, as compared to classic LoRA, is time-consuming and the computational burden increases with the number of fine-tuned blocks or layers. How does the paper address this issue?\n\nLIMITATIONS:\nInstead of fine-tuning the singular values of the weights, this paper proposes fine-tuning the singular vectors of the weights. However, the motivation behind this approach is not convincingly presented. For example, The motivation for fine-tuning the singular value vectors is not clearly articulated. As we know, U is an orthogonal matrix representing the left singular vectors, S is a diagonal matrix of singular values, and V is an orthogonal matrix representing the right singular vectors. A critical question that arises is whether U and V maintain their orthogonality after fine-tuning. Orthogonal U and V matrices provide an optimal basis for representing the weight matrices. Losing orthogonality results in a suboptimal basis, which can lead to less efficient representations of the neural network weights. Also, without orthogonality, the interpretability and distribution of the singular values will also be affected. Furthermore, orthogonal matrices are numerically stable and well-conditioned, meaning small changes in the data lead to small changes in the results. Without orthogonality, the resulting matrices may become ill-conditioned, causing numerical instability and issues in the training and optimization algorithms.\n\nAdditionally, the key contribution of the paper lacks clarity and contains logical errors. It would be helpful if the authors could address these concerns and provide further clarification.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes fine-tuning pretrained model weights in the spectral space for parameter efficiency. It explores two spectral adaptation mechanisms: additive tuning and orthogonal rotation of top singular vectors. The authors introduce these methods, providing theoretical analyses on rank capacity and robustness to support their approach. Experiments on language and diffusion model fine-tuning demonstrate the proposed method's superiority over previous parameter-efficient fine-tuning techniques.\n\nSTRENGTHS:\n1. The proposed spectral adapters, which introduce spectral adaptation, are interesting.\n\n2. The theoretical analysis showing that spectral adapters have a larger rank capacity than LoRA is reasonable.\n\n3. Experiments on language and diffusion model fine-tuning demonstrate that the proposed method outperforms other parameter-efficient fine-tuning techniques while maintaining efficiency.\n\nWEAKNESSES:\n1. The analysis of spectral adaptation robustness in Section 3.2 could be clearer. It would be helpful to provide what $\\mathcal{R}(X)$ denotes and more clearly explain why fine-tuning $u^*$ is considered noiseless.\n\n2. The paper's organization might benefit from some restructuring. Consider moving certain content, such as Lemma 4.1 and Table 3, from the experiments section to the methods section. Additionally, restructuring the experiments section could improve clarity.\n\n3. While spectral adaptation is proposed, it would be valuable to more clearly demonstrate in the methods section how this approach is superior to prior works like SVDiff and OFT.\n\n4. In Figures 1 and 8, using validation loss instead of training loss only for comparing PEFT methods could provide more meaningful insights.\n\n5. It might be beneficial to discuss limitations in a separate section rather than within the Experiments section.\n\n6. Including quantitative measures alongside the qualitative comparisons in the image generation results could strengthen the analysis.\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nThe authors mention some limitations in the checklist part.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors proposed to modulate top-r singular vectors after performing SVD on the pretrained weights. Both theoretical analysis and experiments have shown that the proposed two types of spectral fine-tuning methods can improve the representation capacity of low-rank adapters.\n\nSTRENGTHS:\n1. The introduction of the proposed method is pretty clear, and the theoretical analysis and review of other PEFT methods help the readers quickly and comprehensively understand the core of the proposed method.\n2. The advantages of the proposed method under low-rank conditions are very obvious (Figure 1 and Figure 8).\n\nWEAKNESSES:\n1. The proposed method have two versions, including spectral adapter^A and spectral adapter^R. However, the application boundaries of these two methods are not clear. Some experiments are applied by adapter^A, but others are applied by adapter^R. It's better to further discuss the difference or relationship between these two types of versions.\n2. It's better to study the selection of columns of U and V. For example, bottom-r selection and random-r selection.\n3. If we can combine addition and rotation?\n\nQUESTIONS:\nMy main concerns are listed in the weaknesses.\n\nAdditional question: as illustrated in the paper, the spectral adapters are initialized once at the beginning. This is done by carrying out SVD on pre-trained model weights, identifying the most significant components to finetune on. However, analysis from GaLoRE (GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection) shows that the primary components could shift from one to another during the training. So, it’s better to examine the possibility of re-parametrizing the spectral adapters back into model weights, doing SVD again, and creating new spectral adapters to catch new primary components periodically during the training. By doing this we might come up with a better result.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a new low rank adapter for large models. The idea is to apply the adapter in the SVD decomposition of a weight matrix. Two methods are proposed. First, train parameters that get added to top r columns of U and V matrices. Second, train parameters that rotate top r columns of U and V matrices. The resulting methods are shown to perform well on LLMs and diffusion models.\n\nSTRENGTHS:\nI like the idea presented in the paper. It is intuitive and simple. A similar idea was presented recently in the PiSSA paper [1]. I think the experiments are sufficient and interesting. Moreover, compared to PiSSA that directly tunes the top r eigenvectors, additively tuning them has the benefit of providing a clear intuition when merging different adapters. I specially enjoyed reading the discussion on adapter merging as this is a topic that I have been thinking for a while now.\n\n\n[1]: https://arxiv.org/abs/2404.02948\n\nWEAKNESSES:\nEvaluation is a big problem in this paper.\n\n- LLM evaluations are few while diffusion evaluations seem to be mostly quantitative. I want to caveat this by saying that while few, the GSM-8k experiments with Mistral-7B are good enough to convince me that the method works. However, a comprehensive evaluation would have been more convincing and could shed light on cases where the method fails. \n- I wish the authors had focused more on LLM evaluations. I may be biased as I work on LLMs.\n- Figure 8 of supplementary (above like 586) is interesting and the authors should consider moving it to the main body of the paper.\n\nQUESTIONS:\n- Does training for more epochs change the results?\n- Why did you only pick GSM8K?\n- How do you accommodate that different methods may require different LRs to learn optimally?\n\nLIMITATIONS:\nThe limitations section is missing. One limitation that I can think of is: is it possible for a model to overfit since we are always tuning the most important eigenvectors. What if the fine-tuning dataset is structured so that only a select non-top eigenvectors need to be updated. Datasets like GSM-8k restrict the model output to a narrow pattern that is significantly different from what the model outputs normally. Hence, updating the top eigenvectors makes sense. However, what if the fine-tuning dataset is supposed to modify only a few things that the model has learnt. For instance, there's a lora that switches the model output from Joe Biden to whoever is the next US president. I believe that updating the top eigenvectors is actually a problem here, and, as the authors note in lora merging section (lines 210-214), there  may need to be some eigenvector scheduling.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-15"}