PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models

Authors: fanxu  meng
zhaohui  wang
muhan  zhang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-12 11:04:34

Abstract: To parameter-efficiently fine-tune (PEFT) large language models (LLMs), the low-rank adaptation (LoRA) method approximates the model changes $\Delta W \in \mathbb{R}^{m \times n}$ through the product of two matrices $A \in \mathbb{R}^{m \times r}$ and $B \in \mathbb{R}^{r \times n}$, where $r \ll \min(m, n)$, $A$ is initialized with Gaussian noise, and $B$ with zeros. LoRA **freezes the original model $W$** and **updates the "Noise \& Zero" adapter**, which may lead to slow convergence. To overcome this limitation, we introduce **P**r**i**ncipal **S**ingular values and **S**ingular vectors **A**daptation (PiSSA). PiSSA shares the same architecture as LoRA, but initializes the adaptor matrices $A$ and $B$ with the principal components of the original matrix $W$, and put the remaining components into a residual matrix $W^{res} \in \mathbb{R}^{m \times n}$ which is frozen during fine-tuning.
Compared to LoRA, PiSSA **updates the principal components** while **freezing the "residual" parts**, allowing faster convergence and enhanced performance. Comparative experiments of PiSSA and LoRA across 11 different models, ranging from 184M to 70B, encompassing 5 NLG and 8 NLU tasks, reveal that PiSSA consistently outperforms LoRA under identical experimental setups. On the GSM8K benchmark, Gemma-7B fine-tuned with PiSSA achieves an accuracy of 77.7\%, surpassing LoRA's 74.53\% by 3.25\%. Due to the same architecture, PiSSA is also compatible with quantization to further reduce the memory requirement of fine-tuning. Compared to QLoRA, QPiSSA (PiSSA with 4-bit quantization) exhibits smaller quantization errors in the initial stages. Fine-tuning LLaMA-3-70B on GSM8K, QPiSSA attains an accuracy of 86.05\%, exceeding the performances of QLoRA at 81.73\%. Leveraging a fast SVD technique, PiSSA can be initialized in only a few seconds, presenting a negligible cost for transitioning from LoRA to PiSSA.

Main paper: 
 introduction: 
 Fine-tuning large language models (LLMs) is a highly effective technique for boosting their capabilities in various tasks [1, 2, 3, 4] , ensuring models to follow instructions [5, 6, 7] , and instilling models with desirable behaviors while eliminating undesirable ones [8, 9] . However, the fine-tuning process for very large models is accompanied by prohibitive costs. For example, regular 16-bit fine-tuning of a LLaMA 65B parameter model requires over 780 GB of GPU memory [10] , and the VRAM consumption for training GPT-3 175B reaches 1.2TB [11] . Consequently, various parameter-efficient fine-tuning (PEFT) [12, 13] methods have been proposed to reduce the number of parameters and memory usage required for fine-tuning. Due to the ability to maintain the performance of full finetuning without adding additional inference latency, Low-Rank Adaptation (LoRA) [11] has emerged as a popular PEFT method. Table 1 : Comparison of similarities and differences between PiSSA and LoRA. In this table, bold highlights the model's primary component, while underline denotes the residual component.LoRA PiSSA Fine-tunes principal parts freezing W res . Slow convergence and underperformance.EQUATIONFast convergence and better performance. QLoRA cannot reduce quantization error.QPiSSA can reduce quantization error.LoRA [11] hypothesizes that the modifications to parameter matrices during fine-tuning exhibit low-rank properties. As depicted in Figure 1b , for a pre-trained weight matrix W ∈ R m×n , LoRA substitutes the updates with a low-rank decomposition ∆W = AB, where A ∈ R m×r and B ∈ R r×n , and the rank r ≪ min(m, n). For Y = XW , the modified forward pass is as follows:EQUATIONA random Gaussian initialization is used for A and zero for B, making AB = 0 at the beginning of training, thereby the injection of adapters does not affect the model's output initially. LoRA avoids the need to compute gradients or maintain the optimizer states for the original matrix W , instead optimizing the injected, significantly smaller low-rank matrices A, B. Thus, it could reduce the number of trainable parameters by 10,000× and the GPU memory requirement by 3× [11] .LoRA is capable of achieving comparable performance to full parameter fine-tuning. By integrating the quantization of pre-trained matrices W , LoRA also enables reducing the average memory requirements by 16× [10] . Meanwhile, the adapters can still utilize higher precision weights, thus, the quantization usually does not significantly degrade the performance of LoRA.According to Equation 1, the gradients of A and B are ∂L ∂A = X ⊤ ∂L ∂Y B ⊤ and ∂L ∂B = A ⊤ X ⊤ ∂L ∂Y . Compared to full fine-tuning, using LoRA initially does not change the output Y for the same input X, so the magnitude and direction of gradient are primarily determined by the values of A and B. Since A and B are initialized with Gaussian noise and zeros in LoRA, the gradients could be small and uninformative for a long time, leading to slow convergence in the fine-tuning process. We also observe this phenomenon empirically, as LoRA often wastes much time around the initial point.Our Principal Singular values and Singular vectors Adapter (PiSSA) diverges from LoRA and its successors by focusing not on approximating ∆W , but W . We apply singular value decomposition (SVD) to matrix W . Based on the magnitude of the singular values, we partition W into two parts: the principal low-rank matrix W pri , comprising a few largest singular values, and the residual matrix W res , which possesses the remaining smaller singular values (with a larger quantity, representing a possible long-tail distribution). The principal matrix W pri can be represented by the product of A ∈ R m×r and B ∈ R r×n , where r ≪ min(m, n). As depicted in Figure 1c , A and B are initialized based on the principal singular values and singular vectors and are trainable. Conversely, W res is initialized with the product of the residual singular values and singular vectors and remains frozen during fine-tuning. Since the principal singular vectors represent the directions in which the matrix W has the most significant stretching or impact, by directly tuning these principal components, PiSSA is able to fit the training data faster and better (as demonstrated in Figure 2a ). Moreover, the loss and gradient norm curves of PiSSA often demonstrate a similar trend to those of full parameter finetuning in our experiments (Figure 4 ), indicating that fine-tuning the principal components matches the behavior of fine-tuning the full matrix to some degree. Figure 2 : We illustrate the two key advantages of PiSSA: converging faster and better, and reducing quantization error. In the left figure, we use a toy example to show PiSSA's faster convergence, where we first train a two-layer MLP classifying odd numbers of MNIST, and then fine-tune the model on even numbers. PiSSA finds the right direction more quickly and achieves a lower loss with the same number of steps. In the right figure, PiSSA reduces quantization error more effectively than LoftQ [14] , with an optional 5-iteration SVD for further error reduction, as detailed in Appendix E.Because the principal components W pri are preserved in the adapter at full precision, an additional benefit of PiSSA is that when applying quantization to the frozen part W res , we can significantly reduce the quantization error compared to QLoRA (which quantizes the entire W ), as illustrated in Figure 2b . Therefore, PiSSA is even more compatible with quantization than LoRA, making it a superior plug-and-play substitution for LoRA.Our paper makes several significant contributions:• We analyze the initial gradient magnitude and direction in LoRA, demonstrating that A initially has a zero gradient and B has a random gradient, which slows down convergence and may lead to convergence at suboptimal local minima.• We propose PiSSA initialization, a novel method that approximates the optimization direction of full-parameter fine-tuning by adapting a model's principal components. To our knowledge, PiSSA is the first to apply SVD to the original model, using principal singular values and vectors to initialize the adapter for fine-tuning, while keeping the residual components frozen. Experiments show that PiSSA converges faster and outperforms LoRA.• We combine PiSSA with NF4 quantization to propose QPiSSA, which reduces quantization error by about 20% compared to QLoRA, while maintaining the fast convergence and high performance of PiSSA. 
related works: 
 The vast complexity and computational needs of large language models (LLMs) with billions of parameters present significant hurdles in adapting them for specific downstream tasks. Parameter Efficient Fine-Tuning (PEFT) [12, 13] emerges as a compelling solution by minimizing the fine-tuning parameters and memory requirements while achieving comparable performance to full fine-tuning. PEFT encompasses strategies like partial fine-tuning [15, 16, 17, 18, 19, 20, 21, 22] , soft prompt fine-tuning [23, 24, 25, 26, 27, 28, 29] , non-linear adapter fine-tuning [30, 31, 32, 33, 34, 35, 36, 37, 38, 39] , and low rank adapter based fine-tuning [40, 41, 11, 42] .LoRA [11] injects trainable adapters to the linear layers. After fine-tuning, these adaptations can be re-parameterized into the standard model structure, thus gaining widespread adoption due to their ability to maintain the model's original architecture while enabling efficient fine-tuning. Following LoRA, AdaLoRA [42] dynamically learns the rank size needed for LoRA in each layer of the model. DeltaLoRA [43] updates the original weights of the model using parameters from adapter layers, enhancing LoRA's representational capacity. LoSparse [44] incorporates LoRA to prevent pruning from eliminating too many expressive neurons. DoRA [45] introduces a magnitude component to learn the scale of ∆W while utilizing the original AB as a direction component of ∆W . Unlike LoRA and its successors, which focus on learning low-rank approximations of weight updates, our PiSSA directly tunes the essential low-rank parts of the model while keeping the noisier, high-rank, and nonessential parts frozen. Although our approach differs in philosophy from LoRA, it shares most of LoRA's structural benefits and can be extended by these methods to enhance its performance.QLoRA [10] integrates LoRA with 4-bit NormalFloat (NF4) quantization, along with Double Quantization and Paged Optimizers, enabling the fine-tuning of a 65B parameter model on a single 48GB GPU while preserving the performance of full 16-bit fine-tuning tasks. QA-LoRA [46] introduces group-wise operators to increase the degree of freedom in low-bit quantization. LoftQ [14] reduces quantization error by decomposing the quantization error matrix of QLoRA and retaining the principal components with an adapter. PiSSA can also be combined with quantization techniques, and we have found that PiSSA significantly reduces quantization error compared to QLoRA and LoftQ. 
pissa: principal singular values and singular vectors adaptation: 
 This section formally presents our Principal Singular values and Singular vectors Adaptation method. PiSSA computes the singular value decomposition (SVD) of matrices W within the self-attention and multilayer perceptron (MLP) layers. The (economy size) SVD of a matrix W ∈ R m×n is given by W = U SV ⊤ , where U ∈ R m×min(m,n) , V ∈ R n×min(m,n) are the singular vectors with orthonormal columns, and V ⊤ is the transpose of V . S = diag(s) ∈ R min(m,n)×min(m,n) ,EQUATIONEQUATIONThe residual singular values and vectors are used to build the residual matrix which is frozen during fine-tuning:EQUATIONAs indicated by Equation 5, the integration of AB with the residual matrix also preserves the full capability of the pre-trained model in the beginning of fine-tuning:EQUATIONSimilar to LoRA, the gradients of A and B are also given by ∂L ∂A = X ⊤ ∂L ∂Y B ⊤ and ∂L ∂B = A ⊤ X ⊤ ∂L ∂Y . Since elements of s [:r] ≫ elements of s [r:] , the trainable adapter W pri = AB contains the most essential directions of W . In the ideal case, training AB mirrors the process of fine-tuning the entire model despite using fewer parameters. The ability to directly fine-tune the most essential part of a model enables PiSSA to converge faster and better. In contrast, LoRA initializes the adapters A and B with Gaussian noise and zeros while keeping W frozen. Consequently, the gradients are small or in random directions during the early stages of fine-tuning, possibly introducing much waste of gradient descent steps. Moreover, an inferior initialization might lead to suboptimal local minimum, resulting in worse generalization performance.Since PiSSA shares the identical architecture with LoRA, it inherits most of LoRA's benefits. These include but are not limited to the capability of fine-tuning a model with a reduced number of trainable parameters, quantizing the residual model to decrease memory consumption during forward propagation in training, and easy deployment. The adapter's straightforward linear structure facilitates the integration of trainable matrices with the pre-trained weights upon deployment, thereby maintaining the original inference speed of a fully fine-tuned model. Employing the Fast SVD technique [47] allowed PiSSA to finish initialization in several seconds (Appendix B), which is a negligible cost.For storage efficiency, we can choose not to store the dense parameter matrix ∆W , but to store the low-rank matrices, ∆A and ∆B instead. As shown in Appendix C, leveraging solely the ∆A and ∆B facilitates their seamless integration with the original pre-trained models. Finally, one pre-trained model can accommodate multiple ∆A, ∆B, fine-tuned by diverse PiSSA or LoRA procedures, which enables fast adaptation of the pre-trained model to different downstream applications.4 QPiSSA: An Extension Method with Lower Quantization Error Quantization divides the value range of a matrix into several continuous regions, and maps all values falling inside a region into the same "quantized" value. It is an effective technique to reduce the memory consumption of forward propagation. At the same time, LoRA greatly reduces the backward memory requirement, making it highly suitable to use LoRA and quantization together, where the base model is quantized for memory-efficient forward propagation, and the LoRA adaptors are kept in full precision for accurate backward parameter updates. One representative previous work, QLoRA, quantizes the base model to Normal Float 4-bit (NF4) and initializes the full-precision A and B with Gaussian-Zero initialization. Therefore, the overall error is given by:EQUATIONwhere ||M || * denotes the nuclear norm (also known as the trace norm [48] ), defined as:EQUATIONwhere σ i (M ) is the i th singular value of M . As we can see, the quantization error of QLoRA is the same as that of directly quantizing the base model. Our QPiSSA, however, does not quantize the base model but the residual model. Therefore, its error is given by:EQUATION)Since the residual model has removed the large-singular-value components, W res has a narrower distribution than that of W , as can be seen in Figures 3a and 3b (comparing the singular value distributions of W and W res ), as well as Figures 3c and 3f (comparing the value distributions of W and W res ), which is beneficial for reducing the quantization error. Moreover, given that NF4 is optimized for data with a normal distribution, as discussed by Dettmers et al. [10] , we fit the values of W and W res to a Gaussian distribution respectively. As illustrated in Figures 3c and 3f , W res aligns more closely with a Gaussian distribution and exhibits a smaller standard deviation, making it more suitable for applying NF4 than W . Both the above lead QPiSSA to achieve a significantly lower quantization error than QLoRA, shown in Figures 3d and 3e .Besides the advantage of reducing quantization error, QPiSSA's gradient direction is similar to that of PiSSA, resulting in significantly better fine-tuning performance compared to QLoRA.  
experiments: 
 The experiments were conducted on the NVIDIA A800-SXM4(80G) GPU. In our experiments, we adopt the Alpaca [49] implementation strategy, using the AdamW optimizer with a batch size of 128, a learning rate of 2e-5, cosine annealing schedules, and a warmup ratio of 0.03, without any weight decay. As discussed in Section B.3 of QLoRA [10] , we compute the loss using only the responses from the instruction-following datasets. We ensure lora_alpha is always equal to lora_r, set lora_dropout to 0, and incorporate the adapters into all linear layers of the base model. We utilize the Float32 computation type for both the base model and the adapter in LoRA and PiSSA. For QLoRA, LoftQ, and QPiSSA, we use 4-bit NormalFloat [10] for the base model and Float32 for the adapter. BFloat16 [50] is used for full parameter fine-tuning to save the resources (see Appendix D).We begin by comparing PiSSA, LoRA, and full-parameter fine-tuning on natural language generation (NLG) tasks. We fine-tuned LLaMA 2-7B [51] , Mistral-7B-v0.1 [52] , and Gemma-7B [53] on the MetaMathQA dataset [2] to assess their mathematical problem-solving capabilities on the GSM8K [54] and MATH [2] validation sets. Additionally, the models were fine-tuned on the Code-Feedback dataset [55] and evaluated for coding proficiency using the HumanEval [56] and MBPP [57] datasets. Furthermore, the models were trained on the WizardLM-Evol-Instruct dataset [7] and tested for conversational abilities on the MT-Bench dataset [6] . All experiments were conducted using subsets containing 100K data points and were trained for only one epoch to reduce training overhead.As shown in Table 2 , across all models and tasks, fine-tuning with PiSSA consistently surpasses the performance of fine-tuning with LoRA. Further experiments demonstrated that this improvement is robust across various amounts of training data and epochs (Section 5.2), including both 4-bit and full precision (Section 5.3), different model sizes and types (Section 5.4), and varying proportions of trainable parameters (Section 5.5).We also evaluate PiSSA's natural language understanding (NLU) capability on the GLUE benchmark [59] with DeBERTa-v3-base [60] . In this section, we finetune LLaMA 2-7B model on the complete MetaMathQA-395K dataset for 3 epochs to ensure thorough saturation. The training loss and gradient norms is visualized to demonstrate quicker convergence and evaluated on the GSM8K dataset every 1000 steps to demonstrate superior performance of PiSSA compared to LoRA. The results are depicted in Figure 4 . Additionally, similar comparisons on Mistral-7B and Gemma-7B are detailed in Appendix J. According to Figure 4a , the loss of PiSSA reduces rapidly during the first 100 steps, and the grad norm (shown in Figure 4b ) of PiSSA is significantly higher than that of LoRA, with a trend similar to full fine-tuning. Throughout the process, the loss of PiSSA remains lower than that of LoRA, indicating that PiSSA converges to a better local optimum. As shown in Figure 4c , PiSSA consistently achieves higher accuracy compared to LoRA, and in most cases also surpasses full parameters fine-tuning. We hypothesize that this is because PiSSA is a denoised version of full fine-tuning. Comparing the grad norm and loss curves of PiSSA and full fine-tuning, we can see that the larger grad norm of full fine-tuning does not bring lower loss, indicating that a portion of the grad norm is spent on noisy directions not beneficial for loss reduction. This phenomenon is consistent with Figure 2a .In this section, we first compare the initial quantization error reduction ratio of PiSSA, QLoRA, and LoftQ. This ratio is defined as(1 -||W -(nf 4(W ′ )+AB)|| * ||W -nf 4(W )|| *) × 100%, measuring the relative error decrease achieved by each mehod compared to directly quantizing the base model. The partial results are presented in Table 4 , and the complete results can be found in Table 8 in Appendix E. In Table 4 , PiSSA reduces the quantization error by about 20% compared to directly quantizing the base model. The reduction is more significant for lower-rank matrices. For instance, in the LLaMA-3-70B [61] , all "Key" projection layers see a reduction of 49%. The results in Table 4 validate that QLoRA, discussed in Section 4, does not reduce quantization error. In contrast, PiSSA significantly outperforms LoftQ in reducing quantization error, as further discussed in Appendix H. The difference between QPiSSA and PiSSA is the quantization of the residual model to 4 bits. As introduced in Section 4, the residual model has less influence on the optimal direction compared with the principal adapter, which is the same in both QPiSSA and PiSSA. Therefore, besides reducing the quantization error, we expect QPiSSA to also converge faster than QLoRA and LoftQ. We train LLaMA 3-8B using LoRA/QLoRA, PiSSA/QPiSSA, LoftQ, and full fine-tuning on MetaMathQA-395K for 3 epochs, recording the loss, gradient norm, and accuracy on GSM8K, as shown in Figure 5 .According to Figure 5 , QPiSSA's loss reduction speed in the first 100 steps is even faster than PiSSA and full fine-tuning. Although LoftQ can reduce the quantization error, its loss convergence speed is not faster than LoRA and QLoRA, indicating that QPiSSA's ability to reduce the quantization error and its fast convergence might also be orthogonal capabilities. After sufficient training, QPiSSA's loss is also much lower than that of LoRA/QLoRA and LoftQ. The grad norm is significantly larger than those of LoRA/QLoRA and LoftQ. In terms of fine-tuning performance, QPiSSA's accuracy is higher than that of QLoRA and LoftQ and even better than that of full-precision LoRA.In this section, we compare (Q)PiSSA and (Q)LoRA across 9 models, ranging from 7-70B parameters, including LLaMA 2-7/13B [51] , LLaMA-3-8/70B [61] , Mistral-7B [52] , Gemma-7B [53] , and Qwen1.5-7B [62] , Yi-1.5-34B [63] and MoE models: DeepSeek-MoE-16B [64] and Mixtral-8x7B [65] . These models were fine-tuned on the MetaMathQA-100K and CodeFeedback-100K dataset and evaluated on the GSM8K and HumanEval. DeepSeek-MoE-16B, Mixtral-8x7B, Yi-1.5-34B, and LLaMA-3-70B were fine-tuned with QPiSSA and QLoRA, while the other models were using PiSSA and LoRA. From Figure 6 , (Q)PiSSA, compared to (Q)LoRA, shows improved accuracy across various sizes and types of models, demonstrating its consistent advantage over (Q)LoRA. This section explores the impact of incrementally increasing the rank of PiSSA/QPiSSA and LoRA/QLoRA from 1 to 128, aiming to determine whether PiSSA/QPiSSA consistently outperforms LoRA/QLoRA under different ranks. The training is conducted using the MetaMathQA-100K dataset for 1 epoch, while the validation is performed on the GSM8K and MATH datasets. The outcomes of these experiments are depicted in Figure 7 , with additional results presented in Appendix K.Figure 7a illustrates the quantization error reduction ratio across various ranks. In this figure, QLoRA shows no reduction in quantization error, while QPiSSA consistently outperforms LoftQ in reducing quantization error across all ranks, with a particularly notable advantage at lower ranks. In Figure 7b , the final loss on the training set is shown for models trained with ranks ranging from 1 to 128. The results indicate that PiSSA and QPiSSA achieve a better fit to the training data compared to LoRA, QLoRA, and LoftQ. In Figures 7c and Figures 7d , we compare the accuracy of the fine-tuned models on the GSM8K and MATH validation sets under various ranks, finding that PiSSA consistently outperforms LoRA with the same amount of trainable parameters. Furthermore, as the rank increases, PiSSA will reach and surpass the performance of full-parameter fine-tuning.  
conclusion: 
 This paper presents a PEFT technique that applies singular value decomposition (SVD) to the weight matrix of pre-trained models. The principal components obtained from the SVD are used to initialize a low-rank adapter named PiSSA, while the residual components are kept frozen, to achieve effective fine-tuning and parameter efficiency simultaneously. Through extensive experiments, we found that PiSSA and its 4-bit quantization version QPiSSA significantly outperform LoRA and QLoRA in both NLG and NLU tasks, across different training steps, various model sizes and types, and under various amount of trainable parameters. PiSSA provides a novel direction for research in PEFT by identifying and fine-tuning the principal components within the model, analogous to slicing and re-baking the richest slice of a pizza. As PiSSA shares the same architecture as LoRA, it can be seamlessly used in existing LoRA pipelines as an efficient alternative initialization method. 
limitation: 
 There are still some questions with PiSSA not addressed in this paper: 1) Besides language models, can PiSSA also be adapted to convolutional layers and enhance the performance of vision tasks? 2) Can PiSSA also benefit from some improvements to LoRA, such as AdaLoRA [58] and DyLoRA [66] which adaptively adjust the rank? 3) Can we provide more theoretical explanations for the advantages of PiSSA over LoRA? We are actively exploring these questions. Nevertheless, we are excited to see the huge potential of PiSSA already demonstrated in existing experiments and look forward to more tests and suggestions from the community. 
acknowledgements:: 
 This work is supported by the National Key R&D Program of China (2022ZD016030).The Supplementary Material for The Paper "PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models."• In Section A, we combined PiSSA with two improved LoRA methods, and the experimental results show that these improvements can further enhance the effectiveness of PiSSA. • In Section B, we use fast singular value decomposition to initialize PiSSA. The results indicate that the performance of fast singular value decomposition approaches that of SVD decomposition in just several seconds. This ensures that the cost of converting from LoRA/QLoRA to PiSSA/QPiSSA is negligible. • In Section C, we demonstrate that the trained PiSSA adapter can be losslessly converted to LoRA, allowing for integration with the original model, facilitating sharing, and enabling the use of multiple PiSSA adapters. • In Section D, we explore the experimental effects of using different precisions.• In Section E, we discuss the effects of QPiSSA during multiple rounds of SVD decomposition, which can significantly reduce quantization errors without increasing training or inference costs. • In Section F, we compare the use of high, medium, and low singular values and vectors to initialize adapters. The experimental results show that initializing adapters with principal singular values and vectors yields the best fine-tuning performance. • In Section G, we used a normal distribution function to fit all linear layers of multiple models and calculated their mu and sigma. The experimental results show that after using PiSSA for initialization, the distribution of the remaining models, as described in Section 3 of the main text, is indeed narrower than that of the original models. Despite the extensive use of SVD terms, AdaLoRA does not perform actual SVD on any matrix.In the PEFT domain, terms like low-rank decomposition, and singular value decomposition often appear. They generally refer to products of low-dimensional matrices approximating an ideal ∆W without actual matrix decomposition. To our knowledge, PiSSA is the first to perform SVD on the original model, fine-tuning the principal component while keeping the residual model frozen.PiSSA and AdaLoRA represent different improvements to LoRA, making them combinable. Therefore, we additionally improved PiSSA based on AdaLoRA's three innovations:• After extracting the principal singular values and vectors of W , we use S as an independent trainable vector instead of multiplying it into U and V . • Since PiSSA's U and V are orthogonal at the beginning, maintaining their orthogonality through orthogonal regularization is very easy. • Although AdaLoRA claims to dynamically reduce the number of trainable parameters, the initially large number of parameters is not truly pruned, resulting in more parameters being updated during actual training. Therefore, we did not use this improvement.DoRA adds a learnable magnitude module to LoRA, normalizing W + AB at each update step and multiplying its by the magnitude module. This allows A, B to learn the direction and the magnitude module to learn the magnitude of ∆W . While this approach can improve fine-tuning performance, normalizing W + AB at each step results in slower fine-tuning speeds. In contrast, PiSSA only changes LoRA's initialization method, matching LoRA in training speed and converging faster, thereby reducing training costs. PiSSA, with its intrinsic principal singular values and orthogonal singular vectors, is very suitable for combination with AdaLoRA. According to Table 5 . The performance of the improved PiSSA surpasses all the other methods including PiSSA. From lines 1, and 2 of the table, it is evident that the performance of PiSSA combined with DoRA significantly surpasses that of DoRA alone and also exceeds the performance of PiSSA alone. Taking into account the combination experiments of PiSSA with AdaLoRA, it can be inferred that PiSSA benefits from the enhancement techniques of LoRA, demonstrating the potential of PiSSA when integrated with other methods. 
safeguards: 
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new assets: 
 Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? 
 
 Appendix: In order to speed up the decomposition of the pre-trained matrix W , we adopted the algorithm proposed by Halko et.al [47] (denoted as Fast SVD), which introduces randomness to achieve an approximate matrix decomposition. We compare the initialization time, error, and training loss between SVD and Fast SVD, with the results shown in Table 6 . Initialization time refers to the computation time taken to decompose the pre-trained parameter matrix W , measured in seconds. Initialization error indicates the magnitude of the discrepancy introduced by Fast SVD compared to SVD after decomposing the matrix. Specifically, the error is the sum of the absolute differences between the matrices decomposed by original SVD and Fast SVD. For the error, we report the results of the self-attention module in the table. Loss refers to the loss value at the end of training. In Fast SVD, the parameter niter refers to the number of subspace iterations to conduct. A larger niter leads to increased decomposition time but results in smaller decomposition error. The symbol ∞ represents the experimental results with the SVD method. It can be observed that the computation time of the SVD is tens of times that of Fast SVD. In addition, SVD exhibits consistently high time consumption with minimal variation as the rank increases, while Fast SVD, although experiencing a slight increase in computation time with higher ranks, remains significantly lower compared to SVD throughout. As the rank increases, the initialization error initially rises gradually, with a slight decrease observed when the rank reaches 128. And at the same rank, increasing the niter in Fast SVD leads to a gradual reduction in error. For training loss, we observed that as the rank increases, the training loss decreases gradually. At the same rank, with the increase of niter, the training loss of models initialized based on Fast SVD approaches that of models initialized based on SVD.The advantage of PiSSA lies in its ability to significantly enhance training outcomes during the finetuning phase. After training, it allows for the direct sharing of the trained matrices A and B. However, if we directly save A, B, users need to perform singular value decomposition on the original model to get W res , which requires additional time. When employing fast singular value decomposition, there can be slight inaccuracies too. More importantly, such a way necessitates altering the parameters of the original model, which can be inconvenient when using multiple adapters, especially when some adapters might be disabled or activated. Therefore, we recommend converting the trained PiSSA module equivalently into a LoRA module, thereby eliminating the need to modify the original model's parameters during sharing and usage. In the initialization phase, PiSSA decomposes the original matrix into principal components and a residual matrix: W = W res + AB. Upon completion of training, the model adjusts the weights as follows: W +∆W = W res +A ′ B ′ . Thus, the modification of the model weights by PiSSA is given by:EQUATION)where ∆A ∈ R m×2r and ∆B ∈ R 2r×n . Therefore, we can store and share the new adaptor ∆A and ∆B instead of A ′ , B ′ , which allows directly inserting the adaptor to the original matrix and avoids breaking W . Since r is typically small, the twice storage overhead is still acceptable. This modification allows for plug-and-play usage without the need for singular value decomposition, saving time and avoiding computational errors associated with the SVD, without necessitating changes to the original model parameters.In this section, we compare the effects of training with BFloat16 and Float32 precision. The comparing include four models: LLaMA-2-7B, Mistral-7B, Gemma-7B, and LLaMA-3-8B, each fine-tuned with all parameters in both BFloat16 and Float32 precision on the MetaMathQA-395K dataset. The validation results conducted on the GSM8K dataset are shown in Figure 7 . From Table 7 , it is evident that the choice of precision greatly affects the experimental results. For example, the LLaMA-2-7B model shows a 5.16% higher performance on the GSM8K dataset when using FP32 compared to BF16. Conversely, the Mistral-7B and LLaMA-3-8B on GSM8K are 7.21% and 6.52% lower with FP32 than with BF16 separately. The Gemma-7B model shows similar performance with both precisions. Unfortunately, the experiments did not prove which precision is better. To reduce training costs, we use BF16 precision when fine-tuning all parameters. For methods with lower training costs, such as LoRA, PiSSA, we use FP32 precision. For QLoRA, QPiSSA and LoftQ, the base model was used NF4 precision, while the adapter layers used FP32 precision.Table 8 provides a supplementary explanation of the results in Table 4 . When number of iterations T > 1, LoftQ uses an N -bit quantized weight Q ∈ R m×n N and low-rank approximations A ∈ R m×r and B ∈ R n×r to minimize the following objective by alternating between quantization and singular value decomposition:EQUATIONwhere ∥ • ∥ F denotes the Frobenius norm, A and B are set to zero. Inspired by LoftQ, our QPiSSA T -iter alternately minimize the following objective:EQUATIONwhere A and B are initialized by the principal singular values and singular vectors. The process is summarized in Algorithm 1: According to Table 8 , multiple iterations can significantly reduce quantization error. For instance, using QPiSSA-r64 with 5-iter on LLaMA-3-8B reduces the quantization error nearly twice as much as with 1-iter. In the main paper, we used 5 iterations in Section 5.3 and Section 5.4, while 1 iteration was used in Section 5.5.Algorithm 1 QPiSSA-T -To investigate the influence of singular values and vectors of varying magnitudes on the fine-tuning performance, we initialize the adapters injected into LLaMA 2-7B, Mistral-7B-v0.1, and Gemma-7B with principal, medium, and minor singular values and vectors. These models are then fine-tuned on the MetaMathQA dataset [2] and evaluated against the GSM8K [54] and MATH datasets [67] , with the outcomes depicted in Figures 8.LLaMA To intuitively compare the distribution differences between quantized original and residual models, in Figure 3 , we took LLaMA 2-7B's first Query layer as an example to illustrate the distribution of W and W res . However, using only one layer of one model is not statistically significant. In this section, we applied PiSSA initialization to LLaMA-2-7B, Mistral-7B, Gemma-7B, and LLaMA-3-8B, and fit the values in every linear layer with Gaussian distribution and calculated their mu and sigma. The results in Figure 9 show that the residual models' means are closer to 0, and the standard deviations are smaller after PiSSA initialization. Thus, W res indeed has a narrower distribution than W in a statistical sense. Nevertheless, the difference is not as large as that in the first layer after averaging all layers, which we suspect is because middle layers in a model tend to have more even eigenvalue distributions due to redundancy and insufficient training.This section extends the discussion in Section 4 by providing a comprehensive comparison of the quantization errors associated with QLoRA, LoftQ, and QPiSSA. Using the "layers[0].self_attn.q_proj" of LLaMA 2-7B as an example, we illustrate the singular values of critical matrices during the quantization process with QLoRA, LoftQ, and PiSSA in Figure 10 . A larger sum of the singular values (nuclear norm) of the error matrix indicates a greater quantization error. As shown in Equation 13, QLoRA decomposes the original matrix in Figure 10a into the sum of a quantized matrix (Figure 10b ) and an error matrix (Figure 10d ). By comparing Figure 10a and Figure 10d , we can see that the magnitude of the error matrix is much smaller than that of the original matrix. Therefore, the benefit of preserving the principal components of the W matrix with the adapter is greater than that of preserving the principal components of the error matrix with the adapter.LoftQ [14] , designed to preserve the principal components of the error matrix using the adapter, first performs singular value decomposition on the quantization error matrix of QLoRA:EQUATIONthen uses the larger singular values to initialize A and B, thereby reducing the quantization error to:EQUATIONLoftQ eliminates only the largest r singular values S err [:r] (see Figure 10e ) from the QLoRA error matrix (Figure 10d ).Our PiSSA, however, does not quantify the base model but the residual model:EQUATIONwhere A and B are initialized following Equation 2 and 3. Since the residual model has removed the large-singular-value components, the value distribution of W res can be better fitted by a Student's t-distribution with higher degrees of freedom compared to W (as can be seen in Figure 11 ) and thus quantizing W res results in lower error using 4-bit NormalFloat (shown in Figure 10f ). In addition to NF4 quantization, QPiSSA can also be combined with GPTQ and INT8 quantization. We posit that PiSSA effectively reduces quantization error for several reasons:• It reduces outlier values;• It makes the value distribution more Gaussian-like;• It preserves larger values in full precision, thereby narrowing the weight distribution in the quantized portion.While INT8 also targets the reduction of outlier values (point 1), PiSSA has the potential to enhance this effect. The second point aligns well with NF4, and the third point is crucial as PiSSA uses an adaptor to retain a significant portion of weights in full precision, maintaining the integrity of critical values. As shown in Table 9 , QPiSSA combined with INT8 reduces quantization error by 18.16% on LLaMA-3-8B, and significantly outperforms QLoRA using INT8. Furthermore, in row 3 of the table, the perplexity of LLaMA-3-8B increases to 20.79 after quantization with GPTQ-4bit on the C4 dataset. However, when PiSSA is applied, the perplexity is reduced to 6.23. These results confirm the effectiveness of PiSSA in reducing quantization error, as discussed in the main paper.Overall, QPiSSA demonstrates a clear advantage over QLoRA when combined with various quantization methods, retaining the fast convergence and superior performance characteristics of PiSSA while minimizing quantization error.This is the supplement for Section 5.2. We applied PiSSA, LoRA, and full parameter fine-tuning on the full MetaMathQA-395K dataset using Mistral-7B and Gemma-7B models, training for 3 epochs. As shown in Figure 12a and 13a, the loss for full parameter fine-tuning decreases sharply with each epoch, indicating overfitting to the training data. Notably, during the entire first epoch, the loss for full parameter fine-tuning on Mistral and Gemma is significantly higher than for LoRA and PiSSA, suggesting that full parameter fine-tuning has weaker generalization capabilities compared to LoRA and PiSSA on Mistral-7B and Gemma-7B models. The gradient norm for the first epoch in Figure 13b fluctuates dramatically with each step, further indicating instability in the training process for full parameter fine-tuning. Consequently, as illustrated in Figures 12c and 13c , the performance of full parameter fine-tuning is markedly inferior to that of LoRA and PiSSA. These experiments demonstrate that using parameter-efficient fine-tuning can prevent the over-fitting issue caused by over-parameters.Section 5.5 only validated the effectiveness of LLaMA-2-7B. In Figure 15 , we also present the comparative results of Mistral-7B-v0.1, and Gemma-7B under different ranks. Full Finetuning LoRA PiSSA (b) Rank-wise performance evaluated using pass@1 on the GSM8K dataset. Full Finetuning LoRA PiSSA (c) Rank-wise performance evaluated using pass@1 on the MATH dataset. From Figure 15 , PiSSA uses fewer trainable parameters compared to LoRA while achieving or even surpassing full-parameter fine-tuning on LLaMA-2-7B and Mistral-7B. Remarkably, on Gemma-7B, PiSSA exceeds full-parameter fine-tuning performance even at rank=1. However, as the rank increases to 128, the performance of PiSSA begins to decline, indicating that PiSSA over-parameterizes earlier than LoRA. This over-parameterization phenomenon does not occur on LLaMA-2-7B, suggesting that increasing the rank further might enable PiSSA to achieve even higher performance on LLaMA-2-7B.In Figure 16 and 17, we examining the loss and gradient norm during the training process of PiSSA and LoRA on LLaMA 2-7B, Mistral-7B-v0.1, and Gemma-7B using different ranks. From Figure 16 , PiSSA consistently shows a faster initial loss reduction compared to LoRA across various ranks. Additionally, the final loss remains lower than that of LoRA. This advantage is particularly pronounced when the rank is smaller. From Figure 17 , the gradient norm of PiSSA remains consistently higher than that of LoRA throughout the training process, indicating its efficient fitting of the training data. A closer look at the first few steps of LoRA's gradient norm reveals a trend of rising and then falling. According to Section 3, LoRA's gradients are initially close to zero, leading to very slow model updates. This requires several steps to elevate LoRA's weights to a higher level before subsequent updates. This phenomenon validates our assertion that LoRA wastes some training steps and therefore converges more slowly. It demonstrates the robustness of the faster convergence property of PiSSA across various ranks.Datasets We evaluate the performance of PiSSA on GLUE benchmark, including 2 single-sentence classification tasks (CoLA, SST), 5 pairwise text classification tasks (MNLI, RTE, QQP, MRPC and QNLI) and 1 text similarity prediction task (STS-B). We report overall matched and mismatched accuracy on MNLI, Matthew's correlation on CoLA, Pearson correlation on STS-B, and accuracy on the other datasets.Implementation Details To evaluate the performance of PiSSA intuitively, we compared PiSSA to LoRA with the same number of trainable parameters. DeBERTa-v3-base has 184M trainable parameters. PiSSA and LoRA were applied to W Q , W K and W V respectively, resulting in a total of 1.33M trainable parameters.The results for full fine-tune, BitFit [15] , HAdapter [30] , PAdapter [36] , LoRA with Gassian initialization [11] and AdaLoRA are sourced from AdaLoRA [58] , based on five runs. The remaining results use the publicly available LoftQ [14] codebase and are averaged over three runs. In LoRA, the B matrix is initialized to zero, while the A matrix can be initialized using various methods, such as Gaussian initialization and Kaiming initialization [68] . The selection of the initialization method can influence the final results. In this paper, we report the different results of LoRA based on Gaussian initialization and Kaiming initialization in the experiments, as shown in Table 2 and Table 3 . For DoRA, we used the code from the PEFT package for deployment and conducted a search on key hyperparameters. We set the rank of PiSSA in this experiment as 8 and selecte lora alpha in 8, 16. We utilize AdamW with linear learning rate schedule to optimize and tune learning rate (LR) from 1e-4,2e-4,3e-4,4e-4,5e-4, 6e-4, 5e-5, 3e-5. Batch sizes (BS) are selected from 6, 8, 16, 32. The hyperparameter configurations of PiSSA, DoRA and LoRA with Kaiming Initialization are shown in Table 10 . LoRA K denotes LoRA with Kaiming initialization, and α denotes LoRA alpha. 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). 


Decision: The paper  presents a novel approach to parameter-efficient fine-tuning (PEFT) of large language models (LLMs). The proposed method, PiSSA, is an enhancement of the existing LoRA (Low-Rank Adaptation) method. PiSSA differentiates itself by initializing the adaptation matrices with the principal components obtained through SVD of the original model weights, as opposed to LoRA’s random initialization. This modification allows for more efficient fine-tuning. Additionally, PiSSA shows reduced quantization error when combined with 4-bit quantization (QPiSSA), making it a superior choice for resource-constrained environments. 
There were concerns related to the clarity of the presentation and the  statistical significance of the results, both of which the authors have addressed in their rebuttal. Given the novelty of the approach and its strong empirical performance, I recommend the paper for acceptance with a revision to address concerns raised through the reviewer discussions.

Reviews: review: paper_summary: paper_summary: The paper presents a method for creating lora-like fine-tuning adapters for base models. The idea is to initialize W, A and B matrices in LoRA from SVD decomposition. W is consists of less important ranks while A and B consist of the most important ranks. This implies that less important principle components of the model remain frozen while the most important ones get updated. Another benefit of this scheme is that A and B can be in high precision while W is in low precision, thus ensuring a reduced quantization error.
strength_weakness: strength_weakness: As authors note in the limitations section there are many more experiments they could have conducted, but those are future scope. I am happy with the current scope and experiments. Having said, that I wish the authors had included at least a few experiments comparing advanced LoRA adapter methods. DoRA is mentioned in related works, but I couldn't find any comparison with it.
questions: questions: Will the authors be releasing their code? I believe it is very important that they do as our community could greatly benefit from being able to reproduce the experiments here and build on top of it.
limitations: limitations: The limitations section in the paper is good enough for me.
paper_summary: paper_summary: In this paper, the authors proposed a modified low-rank adaptation (LoRA) method for fine-tuning large pretrained models. Specifically, the proposed method initializes the A and B matrices with singular matrices and freezes the weight matrix to be the residual. Further, the authors provide a quantization step to reduce the memory consumption. The authors demonstrate the effectiveness of this proposed method on a lot of models and tasks, showing that this method converges faster than the conventional LoRA method.
strength_weakness: strength_weakness: - It would be great if the authors could provide more insight into the proposed method. For example, is it possible to compare the subspace of the gradient of conventional LoRA? 
- The proposed method utilizes the first r eigenvectors. How about other dimensions in the subspace?
questions: questions: Please see the comments in the "limitation" block.
limitations: limitations: There is no negative societal impact as far as I can tell.
paper_summary: paper_summary: The authors present PiSSA, a relatively simple change to the LoRA framework leading to a large amount of demonstrated benefits.  PiSSA proceeds by adjusting the initialization step of standard LoRA; rather than freezing the original weight matrix W and learning a low-rank perturbation W' = W + BA, PiSSA instead considers the SVD W = USV^T = U[:,r:]S[r:,r:]V[:,r:]^T + U[:,:r]S[:r,:r]V[:,:r]^T = W^{res} + AB, s.t. A=U[:,:r]S^{0.5}[:r,:r], B = S^{0.5}[:r,:r]V[:,:r]^T.  Due to this backwards compatibility, PiSSA enjoys the fine-tuning speed/memory savings of LoRA while also being easily implementable to existing PEFT packages.  Most importantly, the authors demonstrate across a large number of models, fine-tuning recipes, and benchmarks that PiSSA greatly improves on downstream accuracy compared to LoRA.  Furthermore, the authors show that this initialization + fine-tuning strategy also leads to significant decreases in quantization error, outperforming both QLoRA and the recently proposed LoftQ both in terms of training loss (e.g., faster loss reduction) and performance for GSM8K and MATH tasks.
strength_weakness: strength_weakness: # Clarity
There are several areas the paper could improve in clarify.  In particular, the literature review requires more granularity, e.g., "AdaLoRA [42, 41, 43]," only one of these works is AdaLoRA.  Furthermore, there is overlap between AdaLoRA and PiSSA in both their reliance on the SVD.  It is necessary to discuss exactly how AdaLoRA leverages SVDs, and how this differs from PiSSA.

There are several experimental details missing, e.g., are the GSM8K evals 8-shot?  What are the shots used for the other benchmarks?  Were the benchmarks run using the Eleuther eval harness?  Please include these details in the paper.

Paper at times states it is similar to LoRA and, at other times, seeks to differentiate itself:
- "Unlike LoRA and its successors, which focus on learning low-rank approximations of weight updates, our PiSSA approach directly tunes the essential but low-rank parts of the model while keeping the noisier, high-rank, and nonessential parts frozen. Although our approach differs in philosophy from LoRA, it shares most of LoRA’s structural benefits and can be extended by these methods to enhance its performance"
- "Since PiSSA shares the identical architecture with LoRA, it inherits most of LoRA’s benefits."
This was confusing on an initial pass of the paper.  A short, succinct list enumerating the similarities and differences early on may help readers understand exactly how PiSSA distinguishes itself from LoRA, but how it is able to inherit all of LoRA's PEFT benefits.

# Quality
The authors consider the weight matrix of a single attention layer in Llama-2 in Figure 2:
"Since the residual model has removed the large-singular-value components, Wres has a narrower
 distribution than that of W, as can be seen in Figures 3a and 3b (comparing the singular value distributions of W and Wres), as well as Figures 3c and 3f (comparing the value distributions of W and Wres), which is beneficial for reducing the quantization error" <- However, since this only looks at a single layer of a single model, it hard to accept this as evidence for a general claim.  Can you include a comparison of the distributions of mu/sigma for W and W^{res} (e.g., mean absolute differences between means and ratio between sigmas) across all layers for Llama2 and another model (e.g., Mistral) to show that, in general, W^{res} has a narrow distribution than W?
questions: questions: See weaknesses.

Minor comment:
"loftQ" in Table 3 <- "LoftQ"
limitations: limitations: The authors have addressed potential limitations of the demonstrated method.
paper_summary: paper_summary: The paper presents a novel initialization strategy for LoRA finetuning where A and B are initialized according to the top r singular values and the pre-trained weights are initialzed with the remaining components. 
The authors show that this decomposition can help finetuning converge faster and behaves favorably compared to traditional quantization techniques such as NF4. 
The authors conduct experiments on NLG and NLU tasks show show performance improvements of PiSSA over vanilla LoRA.
strength_weakness: strength_weakness: **Significance of results**

The authors do not provide error bars in their results which questions whether the obtained results are actually statistically significant.
Error bars should be provided as well as significance tests should be performed. 

**Unsupported claims**

The authors claim that LoRA gradients point in random directions early during the fine-tuning stage which introduces wasteful update steps.
Was this claim experimentally verified? If so, the authors should clarify how these results were obtained and elaborte on it in more detail in the paper.

**Comparison to baselines**

PiSSA is not compared to other commonly used initialization schemes, e.g., uniform gaussian [1], or kaiming init [2] for NLU and NLG experiments.
Further, the authors compare PiSSA only to standard LoRA, but there have been several improvements to LoRA, such as AdaLoRA [3], or DoRA [4].
A comparison to those extensions would strengthen the paper. 
Also, the authors may investigate whether their initialization could be applied to these extensions to further improve perormance.
Finally, some related work uses a similar initialization by considering principal components [5].

[1] Hu et al., LoRA: Low-Rank Adaptation of Large Language Models, ICLR 2022

[2] He et al., & Sun, J. Delving deep into rectifiers: Surpassing human-level performance on ImageNet classification., ICCV 2015

[3] Zhang et al., AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning, ICLR 2023

[4] Liu et al., DoRA: Weight-Decomposed Low-Rank Adaptation, ICML 2024

[5] Sheng et al., S-LoRA: Serving Thousands of Concurrent LoRA Adapters, MLSys 2024
questions: questions: - In line 174 the authors mention that $\alpha=r$, why was this choice made? In [1] $\alpha$ is set to a constant. Setting $\alpha=r$ ultimately results in different learning rates for different r.
- The advantage of of quantized PiSSA over QLoRA might come from outlier removal, was there any experiments done comparing QPiSSA to efficient quantization techniques such as GPTQ [2] or llm.int8 [3]?
- In line 188 the authors mention that training for NLG experiments was only done for 1 epoch. It would be interesting to see if LoRA ultimately converges to a similar optimum after longer training, or whether the improved initialization also leads to a better optimum.
- What do the entries in the legend of Figure 2b mean exactly?

[1] Hu et al., LoRA: Low-Rank Adaptation of Large Language Models, ICLR 2022

[2] Frantar et al., GPTQ: Accurate post-training quantization for generative pre-trained transformers., ICLR 2023

[3] Dettmers et al., LLM.int8(): 8-bit matrix multiplication for transformers at scale., NeurIPS 2022
limitations: limitations: The fact that PiSSA needs to store 2r ranks compared to r ranks for LoRA which is mentioned in Appendix C should be also mentioned under Limitations.

Comments: title: Response
Looking forward to your response
To Reviewer Bb8M
Acknowledgement of rebuttal
Official Comment by Authors
Thank you for your suggestions and recognition
Completing the Remaining Experiments
comment: Dear Reviewer XzvL,

I would like to express my appreciation for your efforts in reviewing our paper and for your positive assessment of PiSSA. Your approval serves as a significant encouragement for our ongoing research and improvement. Thank you once again for your time and insights.

Sincerely,

Submission 5120 Authors
I would like thank the authors for detailed response and additional experiments. My concerns are mostly addressed and I am happy to raise the scores.
Dear Reviewer XzvL,

We have addressed the concerns you raised in your initial review and have submitted a detailed rebuttal. We are writing to confirm whether you have had the opportunity to review our responses. We hope that our rebuttal has addressed your questions satisfactorily and would appreciate if you could reconsider the contributions of our manuscript, PiSSA, along with the efforts we have made during the rebuttal process.

We are eager to hear your feedback and are open to any further discussion that might help clarify any remaining issues.

Looking forward to your response.

Best regards,

Submission 5120 Authors
We sincerely appreciate the professionalism and sense of responsibility you demonstrated throughout the rebuttal period. Your valuable suggestions have been instrumental, and we will incorporate them into the camera-ready version. We believe your recommendations will greatly enhance PiSSA's contribution to the field of efficient fine-tuning of LLMs.
I thank the authors for their response to my original questions.  I have read the other reviews and respective authors responses, and have no further questions or concerns.
**Dear Reviewer XzvL**,

The discussion period has passed the **halfway** point, and we have addressed the questions you previously raised. Do you have any additional questions or concerns that you would like to discuss?

Our paper investigates the **gradient behavior of LoRA**, revealing that $A$ and $B$ initially have **zero gradients** and **random gradient** directions. This leads to **slow convergence** and potentially **suboptimal local minima**. To address these issues, we propose the PiSSA initialization method. **PiSSA approximates the optimization direction of full fine-tuning** by fine-tuning the principal components of a model. To our knowledge, PiSSA is the **first** method to use SVD on the original model, employing the principal singular values and vectors to initialize the adapter and fine-tune it, while fixing the remaining parts of the model during training. Our experiments demonstrate that PiSSA not only **converges faster** but also achieves **better final results** compared to LoRA. The initialization process is **efficient**, taking only seconds to minutes due to the use of fast singular value decomposition.

We extend PiSSA by integrating it with **NF4**, **llm.int8**, and **GPTQ** quantization to create QPiSSA. This approach significantly **reduces quantization error** compared to QLoRA while retaining the **fast convergence** and **strong performance** of PiSSA.

PiSSA modifies only the initialization method used by LoRA, making it compatible with various LoRA enhancements. We **combined PiSSA** with **LoftQ**, **AdaLoRA**, and **DoRA**, and our results show that PiSSA+ outperforms these methods and PiSSA, demonstrating its potential for further improvements.

Our extensive experiments include **5 NLG** and **8 NLU** tasks using **12** different models ranging from **184M to 70B** parameters. We compared performance across **1k-9k+** training steps and ranks from **1 to 128**, evaluating against methods such as **LoRA**, **QLoRA**, **LoftQ**, **DoRA**, **AdaLoRA**, and **full-parameter fine-tuning**. We also examined the effects of initializing with **principal**, **medium**, and **minor** singular values and vectors.

If you find the content in the main paper and our rebuttal satisfactory, would you consider revising your score?
In the camera-ready version, we will repeat the key experiments multiple times, include error bars, and verify statistical significance.

Your professional insights have been invaluable in improving the quality of the PiSSA paper. We will also incorporate the additional experiments conducted during the rebuttal period into the final version of the paper.

Thank you again for your suggestion towards PiSSA.
I would like to thank the authors for the detailed response and commend them for the amount of experiments delivered during the rebuttal.

I would like to highlight that claiming statistical significance based on the outcome of different experiments without verification is not convincing. Verifying statistical significance should be done for each experiment by providing variance estimates.
Having said that, I greatly appreciate the reported error bars for GSM8K.
I strongly recommend to the authors to report variance estimates for the remaining experiments as well and to verify statistical significance of their results.

Further, the additional findings for applying PiSSA to other PEFT methods such as DoRA and AdaLoRA look very promising.
Therefore I have decided to increase my score.
**Dear Reviewer M1T9**,

In your questions Q4 and Q5, you suggested integrating PiSSA with both AdaLoRA and DoRA, as well as combining PiSSA with llm.int8 and GPTQ. During the rebuttal period, due to time constraints, we were only able to combine PiSSA with AdaLoRA and llm.int8, and the experimental results have shown the advantages of PiSSA when integrated. Over the past few days, we have completed the experiments combining PiSSA with DoRA and GPTQ, and the results are presented in the table below:

| Method | Run 1 | Run 2 | Run 3 | GSM8K Average |
| --- | --- | --- | --- | --- |
| Full FT | 74.89 | 74.22 | 74.07 | 74.39±0.356 |
| LoRA(gaussian-init) | 71.11 | 71.19 | 70.74 | 71.01±0.199 |
| LoRA(kaiming init) | 72.25 | 71.57 | 71.95 | 71.92±0.279 |
| **PiSSA** | **76.72** | **76.72** | **76.80** | **76.75±0.036** |
| AdaLoRA | 72.48 | 72.42 | 72.02 | 72.31±0.202 |
| **PiSSA+AdaLoRA** | **78.77** | **78.32** | **78.69** | **78.59±0.199** |
| DoRA | 72.18 | 72.33 | 72.63 | 72.38±0.189 |
| **PiSSA+DoRA** | **77.86** | **77.41** | **77.26** | **77.51±0.257** |
| QLoRA+int8 | 71.78 | 71.48 | 71.78 | 71.68±0.143 |
| **QPiSSA+int8** | **76.18** | **76.48** | **76.95** | **76.54±0.318** |
| QLoRA+GPTQ-4bit | 70.51 | 70.43 | 69.60 | 70.18±0.419 |
| **QPiSSA+GPTQ-4bit** | **74.60** | **74.30** | **74.83** | **74.58±0.217** |

**Q4: Combining AdaLoRA or DoRA with PiSSA**

**A4 part 2:** From lines 4, 7, and 8 of the table, it is evident that the performance of PiSSA combined with DoRA significantly surpasses that of DoRA alone and also exceeds the performance of PiSSA alone. Taking into account the combination experiments of PiSSA with AdaLoRA, it can be inferred that PiSSA benefits from the enhancement techniques of LoRA, demonstrating the potential of PiSSA when integrated with other methods.

**Q5: Comparing QPiSSA to GPTQ or int8**

**A5 part 2**: GPTQ handles each row $w$ independently, quantizing one weight at a time while updating all not-yet-quantized weights. Therefore, the method of using  nuclear norm to calculate quantization error, as used in our paper, is not applicable for GPTQ. Thus, we turn to use Perplexity on WikiText-2 to calculate quantization error, where a lower Perplexity indicates reduced quantization error. The Perplexity of LLaMA-3-8B used in our experiments is **5.14**. After quantization with GPTQ-4bit, the Perplexity increased to **20.79**. However, using PiSSA, we were able to reduce the Perplexity to **6.23**. These results validate the effectiveness of PiSSA in reducing quantization error as discussed in our paper. 

From lines 11-12 of the table, it is evident that the performance of QPiSSA combined with GPTQ-4bit significantly surpasses that of QLoRA combined with GPTQ-4bit. This confirms the advantages mentioned in our paper, where QPiSSA retains the rapid convergence and superior performance characteristics of PiSSA.

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