Title: Expanding Sparse Tuning for Low Memory Usage
Venue: NeurIPS 2024
Accepted: yes
Keywords: sparse tuning

parameter-efficient fine-tuning

vision model fine-tuning
Abstract: Parameter-efficient fine-tuning (PEFT) is an effective method for adapting pre-trained vision models to downstream tasks by tuning a small subset of parameters. Among PEFT methods, sparse tuning achieves superior performance by only adjusting the weights most relevant to downstream tasks, rather than densely tuning the whole weight matrix. However, this performance improvement has been accompanied by increases in memory usage, which stems from two factors, i.e., the storage of the whole weight matrix as learnable parameters in the optimizer and the additional storage of tunable weight indexes. In this paper, we propose a method named SNELL (Sparse tuning with kerNELized LoRA) for sparse tuning with low memory usage. To achieve low memory usage, SNELL decomposes the tunable matrix for sparsification into two learnable low-rank matrices, saving from the costly storage of the whole original matrix. A competition-based sparsification mechanism is further proposed to avoid the storage of tunable weight indexes. To maintain the effectiveness of sparse tuning with low-rank matrices, we extend the low-rank decomposition by applying nonlinear kernel functions to the whole-matrix merging. Consequently, we gain an increase in the rank of the merged matrix, enhancing the ability of SNELL in adapting the pre-trained models to downstream tasks. Extensive experiments on multiple downstream tasks show that SNELL achieves state-of-the-art performance with low memory usage, endowing PEFT with sparse tuning to large-scale models. Codes are available at https://github.com/ssfgunner/SNELL.

Expanding Sparse Tuning for Low Memory Usage

Abstract

Parameter-efficient fine-tuning (PEFT) is an effective method for adapting pretrained vision models to downstream tasks by tuning a small subset of parameters. Among PEFT methods, sparse tuning achieves superior performance by only adjusting the weights most relevant to downstream tasks, rather than densely tuning the whole weight matrix. However, this performance improvement has been accompanied by increases in memory usage, which stems from two factors, i.e. , the storage of the whole weight matrix as learnable parameters in the optimizer and the additional storage of tunable weight indexes. In this paper, we propose a method named SNELL ( S parse tuning with ker NEL ized L oRA) for sparse tuning with low memory usage. To achieve low memory usage, SNELL decomposes the tunable matrix for sparsification into two learnable low-rank matrices, saving from the costly storage of the whole original matrix. A competition-based sparsification mechanism is further proposed to avoid the storage of tunable weight indexes. To maintain the effectiveness of sparse tuning with low-rank matrices, we extend the low-rank decomposition by applying nonlinear kernel functions to the whole-matrix merging. Consequently, we gain an increase in the rank of the merged matrix, enhancing the ability of SNELL in adapting the pre-trained models to downstream tasks. Extensive experiments on multiple downstream tasks show that SNELL achieves state-of-the-art performance with low memory usage, endowing PEFT with sparse tuning to large-scale models. Codes are available at https://github.com/ssfgunner/SNELL.

1 Introduction

Fine-tuning has become a predominant way for adapting large pre-trained models to downstream tasks with limited training samples [13, 9, 24, 23]. Nevertheless, fine-tuning all model parameters requires substantial memory usage and is susceptible to over-fitting, making it costly and infeasible for largescale models [58, 2, 11]. To address these limitations, parameter-efficient fine-tuning (PEFT) [64, 27, 62, 30, 8, 22] has been proposed to tune a small subset of parameters while keeping other parameters frozen. PEFT methods can be categorized into addition-based and reparameterization-based methods. The former attaches additional parameters to a frozen pre-trained backbone, while the latter adjusts the original parameters in the pre-trained backbone.

Addition-based methods [53, 62, 30] have achieved remarkable performance on vision tasks. However, adopting additional parameters incurs extra computational costs during the inference process. In contrast, reparameterization-based methods [5, 7, 27] directly fine-tune the original parameters. These methods select specific parameters, involving reduced memory usage compared to full-parameter fine-tuning. Based on the granularity of parameter selection, one primary approach focuses on

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specific parameter matrices. For example, Bitfit [5] only adjusts bias to reduce the volume of tunable parameters while Partialk [30] fine-tunes the last few layers to avoid back-propagation through the entire pre-trained backbone. To further reduce memory usage, LoRA [27] optimizes each selected weight matrix using two low-rank matrices to achieve memory-efficient fine-tuning. Although sufficient in reducing memory usage, these methods usually gain inferior performance compared to addition-based methods [30]. Recently, SPT [22] and GPS [63] found that combining existing PEFT methods with sparse tuning, which only adjusts the most task-related weights in a matrix, can achieve state-of-the-art performance on vision tasks. Concurrently, the effectiveness of sparse tuning has also been observed in NLP tasks [18]. By focusing on individual weights in a matrix, sparse tuning allows for more precise adjustments, thus achieving good performance and mitigated over-fitting risks [18].

However, the performance gained from sparse tuning has been accompanied by high memory usage, as Figure 1(a) shows. Although sparse tuning only updates part of weights in the pre-trained weight matrix, the whole matrix still needs to be stored as learnable parameters in the optimizer and computed for their corresponding gradients in practice. Additionally, sparse tuning necessitates storing the tunable weight indexes, further aggravating the memory demands. The above observation indicates that sparse tuning gains no advantage over full fine-tuning regarding memory usage, especially given the increasing parameter volumes in pre-trained models [58, 2]. A sparse tuning method with low memory usage is urgently required for applications on large-scale pre-trained models.

In this paper, we propose a method that conducts S parse tuning with ker NEL ized L oRA (SNELL) shown in Figure 1(b). SNELL can adapt pre-trained models to downstream tasks with both low memory usage and strong performance. To reduce memory usage, we decompose the tunable matrix for sparsification into low-rank learnable matrices to store fewer parameters in the optimizer and develop a competition-based method to avoid storing the tunable weight indexes. To improve the performance on downstream tasks, we extend LoRA from a kernel perspective and merge low-rank matrices with nonlinear kernel functions to obtain matrices with higher ranks.

Specifically, SNELL updates the pre-trained weight matrix using a sparse low-rank adaptation matrix. This adaptation matrix is first merged with two low-rank learnable matrices and then sparsified toward effective fine-tuning. Compared to storing the whole adaptation matrix, storing low-rank matrices in the optimizer results in lower memory usage. For the sparsification process, we propose a competition-based method inspired by the neuron competition phenomenon in neuroscience [49], avoiding the storage of the tunable weight indexes that incur additional memory usage. The proposed method promotes competition among weights based on their absolute values. Most task-relevant weights are encouraged to have larger absolute values and survive during the fine-tuning process. By setting a sparsity ratio as the hyperparameter and determining tunable weights based on their absolute values in an end-to-end manner, we can eliminate the storage of the tunable weight indexes.

In addition to low memory usage, the performance is also critical for model fine-tuning. However, directly merging two low-rank matrices through the inner product leads to the low-rank structure of the adaptation matrix, which narrows the optimization scope of tunable matrices and further limits the expressiveness of sparse tuning. To overcome this bottleneck, we draw inspiration from DyN [45]

on weight matrix interpretation based on low-dimensional dynamical systems, and reformulate the merging process with nonlinear kernel functions that increase the rank of the merged adaptation matrix. This new formulation enables a more expressive sparse tuning while maintaining a compact representation with low memory.

Extensive experiments are conducted on 24 downstream visual recognition tasks with both plain and hierarchical vision Transformer backbones under supervised and self-supervised pre-training. Results show that SNELL can gain the performance improvement of sparse tuning and the low memory usage of LoRA concurrently. SNELL obtains the state-of-the-art performance on FGVC (91.8% vs. 90.7%) and VTAB-1k (74.6% vs. 74.1%) benchmark with LoRA-level memory usage. Moreover, as Figure 1(c) shows, the low memory-usage advantage of SNELL becomes increasingly apparent as the model size grows, enabling sparse tuning on larger models.

2 Related Work

Parameter Sparsity. In early work, the parameter sparsity usually serves as an optimization objective in model pruning [21, 42]. These pruning methods remove the weights from pre-trained models irrelevant to a specific task, without significantly degrading model performance. The relevance of individual weights can be estimated based on activations [28], redundancy [48], per-layer second derivatives [15], and energy efficiency [57]. Except for the post-training pruning strategy, sparse networks [4, 41, 17] directly introduce parameter sparsity into the training process, removing redundant weights more precisely [17]. Motivated by the advantage of parameter sparsity in model optimization, recent studies introduce sparsity to the fine-tuning of pre-trained models and achieve enhanced model performance on downstream tasks [1, 21, 56]. The parameter sparsity gives rise to a reduced number of trainable parameters and serves as a regularization constraint during fine-tuning [18]. Among sparse tuning, pre-pruning methods adopt model pruning for fine-tuning. These methods sparsify the weight matrix [22, 63] or adapter [1] through pruning metrics [43, 17] to identify learnable parameters for the fine-tuning process. Other methods select trainable parameters during fine-tuning, including learnable mask [64] or diff vectors [20] with sparsity constraints. However, the parameter sparsification methods need to store the indexes of tunable weights, which incurs additional memory usage. For sparse tuning under low memory budget, our competition-based mechanism selects weights relevant to downstream tasks in a learnable manner without storing the tunable weight indexes.

Parameter-efficient Fine-tuning. Fine-tuning is the most predominant approach for adapting a pre-trained model to downstream tasks. However, for large pre-trained models, fine-tuning all parameters is costly and prone to overfit downstream datasets. To tackle these problems, parameterefficient fine-tuning (PEFT) [8, 30, 62], which tunes only a tiny portion of parameters, becomes a desirable choice. Following the taxonomy of SPT [22], PEFT methods can be categorized into addition-based [3, 26, 46, 50, 14, 31, 37, 62] and reparameterization-based [5, 7, 20, 64, 27] methods.

Addition-based methods attach additional trainable parameters to a frozen pre-trained backbone. Adapters [3, 26, 46, 50, 61] adopt a residual pathway and learn a bottleneck layer including two linear projections and a non-linear activation. Prompt-tuning methods [14, 31, 37, 34] add trainable parameters to the input and keep the entire pre-trained model unchanged during training. Recent work [62] attempts to find the optimal configurations to combine multiple addition-based methods. Despite of the popularity and effectiveness of addition-based methods, the additional trainable parameters incur excess computational costs during the inference process [3, 33].

Reparametization-based methods adjust the inherent parameters in the pre-trained backbone to avoid excess computational costs during inference. Early work directly selects parameters with low memory usage for fine-tuning, such as the bias terms [5] and the final few layers of the pre-trained model [7]. To further reduce the memory usage of the selected matrices, LoRA [27] optimizes low-rank matrices that can be reparameterized into the pre-trained weight matrices to reduce memory usage. Exploring finer-grained parameter selection, some studies [20, 64] propose sparse tuning, which involves selecting and tuning individual weights sparsely within the weight matrices. Recently, SPT [22] combines sparse tuning and LoRA in a hybrid framework that achieves state-of-the-art performances on visual PEFT tasks. SPT has revealed that optimizing the weights most relevant to the downstream task through sparse tuning can significantly enhance the performance, which is also supported by SAM [18] and GPS [63]. However, existing sparse tuning framework still faces the challenge of high memory usage brought by sparse tuning. Unlike existing methods, our SNELL

inherits high performance and low memory usage concurrently by sparsifying the adaptation matrix merged with low-rank matrices through nonlinear kernels.

3 Methodology

We first introduce the definitions of sparse tuning [64, 20, 1], LoRA [27] and kernel trick [32] (Section 3.1). Then we propose SNELL, a sparse tuning method including kernelized LoRA that enables high-performance tuning with low-rank learnable matrices (Section 3.2) and a competition-based mechanism that sparsifies weights without additional memory usage (Section 3.3).

3.1 Preliminaries

Sparse Tuning . Given a downstream training set D = { x ( n ) , y ( n ) } N n =1 , the objective of sparse tuning is to minimize the model's empirical risk on the downstream task, with the sparsity constraints on the volume of tunable weights in weight matrix W ∈ R m × n . The sparsification is usually achieved through a binary mask M ∈ { 0 , 1 } m × n . The objective function can be formulated as

where f ( · ; · ) is a parameterized function over the input ( e.g. , a neural network), L ( · , · ) is a loss function ( e.g. , cross-entropy), and ⊙ denotes element-wise multiplication. The binary mask M can be either a fixed hyperparameter, pre-computed with heuristics such as pre-pruning [22], or a learnable parameter obtained through end-to-end fine-tuning [64]. All these methods require storing M to determine the tunable weights, which results in additional memory usage. More importantly, the tunable parameters W ⊙ M occupy the same amount of memory as the weight matrix W in practice. As a result, the memory usage of sparse tuning is even higher than that of full fine-tuning.

LoRA . Given a pre-trained weight matrix W 0 , LoRA [27] optimizes two low-rank matrices B ∈ R m × r , A ∈ R n × r to reduce the memory usage during fine-tuning. The low-rank matrices A and B can be reparameterized into the pre-trained weight W 0 ,

With r ≪ min( m,n ) , LoRA can achieve high training efficiency and low memory usage by only optimizing the smaller low-rank matrices.

Kernel Trick [32]. In many machine learning tasks, mapping the vectors into higher dimensions is frequently used to achieve linear separability [51]. However, the explicit mapping process incurs significant computational costs. To address this problem, the kernel trick is proposed to efficiently model data relationships in high-dimensional spaces, without the need to explicitly formulate the space. According to Mercer's theorem [6], a kernel function κ : R r × R r → R can express an inner product in some space as κ ( x , x ′ ) = ϕ ( x ) ⊤ ϕ ( x ′ ) , if and only if κ is positive semi-definite (Appendix B). x , x ′ ∈ R r , and ϕ : R r → R d is an implicit feature map. By selecting an appropriate kernel function κ , we can obtain the inner product of two vectors in higher-dimensional space R d ( d ≥ r ) without explicitly formulating the feature map ϕ .

3.2 Kernelized LoRA

We leverage LoRA to reduce the memory usage of sparse tuning in light of its low memory usage. An intuitive solution is to sparsify the adaptation matrix ∆ W composed of the two low-rank matrices. However, the low-rank property of ∆ W can lead to the performance degradation of sparse tuning. For the original sparse tuning, the weight matrix W is free of the rank constraint, and weights are independent of each other. Therefore, we can independently select and optimize weights most relevant to the downstream task. For sparse tuning with LoRA, the adaptation matrix ∆ W with rank r is constrained in R r × ( m + n -r ) , a subspace of R m × n . When r ≪ min( m,n ) , the weight optimization scope of sparse tuning contracts, hindering its performance on downstream tasks.

To achieve sparse tuning with both strong performance and low memory usage, we propose to construct a high-rank matrix using low-rank matrices. Inspired by DyN [45] that fits a high-rank

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matrix using the distance matrix of a low-dimension dynamical system, we extend the distance function to general kernel functions and investigate LoRA in the kernel perspective. Given two vectors x , x ′ ∈ R r , the kernel function κ ( x , x ′ ) can be formulated as an inner product ϕ ( x ) ⊤ ϕ ( x ′ ) with an implicit feature map ϕ : R r → R d . The merging process of LoRA can be seen as applying linear kernel function κ l ( · , · ) on the rows of the learnable parameters A and B ,

where A i, · , B j, · ∈ R r , ϕ l : R r → R r denotes the identity mapping. By replacing κ l ( · , · ) with more complex non-linear kernel functions, we can approximate relations in higher-dimensional spaces R d and obtain matrices with rank larger than r . The merged adaptation matrix in SNELL can be represented by

Note that in practice, explicit computation of A ϕ ∈ R n × d and B ϕ ∈ R m × d is unnecessary. ∆ W can be directly derived based on A and B with the kernel function κ . By extending LoRA in a kernel perspective, SNELL can build high-rank adaptation matrices based on low-rank learnable matrices, empowering strong sparse tuning with low memory usage. We utilize the piecewise linear kernel introduced in Appendix B without a specific statement.

3.3 Competition-based Sparsification Mechanism

Existing methods store tunable weight indexes M ∈ { 0 , 1 } m × n for sparsifying the update of the weight matrix W ∈ R m × n . The storage of M leads to additional memory usage. Inspired by the neuron competition phenomenon in neuroscience [49], we design a competition-based parameter sparsification mechanism to avoid this additional storage. Instead of determining the learnable weights in the optimization process based on M , our objective is to encourage the weights to compete based on their contributions to performance improvement. Weights with stronger contributions survive in the sparsification while the remaining low-contributed weights are zeroed out. The weight contribution is reflected in their absolute values during the end-to-end optimization. During optimization, weights contributing more to the loss reduction are encouraged to have more significant values, while weights contributing less approach zero. By retaining higher importance to significant weights and zeroing out the less impactful weights, we can achieve end-to-end tunable parameter selection by solely relying on the absolute values of weights, avoiding the storage of M .

Specifically, given a merged adaptation matrix ∆ W and a sparsity ratio s ∈ [0 , 1] , we sparsify weights with a soft-threshold function. To induce weight competition during end-to-end fine-tuning, we propose a dynamic threshold ∆ w s , i.e. , the weight having the ⌈ smn ⌉ -th smallest absolute value in ∆ W . This threshold ensures that only a fixed proportion ( s × 100% ) of weights remain non-zero. Therefore, the weights have to compete with each other to be selected instead of just having a larger absolute value than a fixed threshold.

where ∆ W s = (∆ W s ij ) m × n denotes the sparse matrix with sparsity ratio s . In practice, the sparsity ratio s is manually determined regarding specific downstream tasks. Given a sparsity ratio s , the training objective in Equation 1 can be reformulated as

This objective encourages weights that are most relevant to the downstream task to gain more significant values for survival. Adjusting the sparsity ratio s allows us to control the sparsification process precisely and identify the optimal number of tunable parameters for different tasks.

4 Experiments

4.1 Experimental Setup

Datasets and Metrics. We evaluate our methods on 24 downstream tasks categorized into two groups following SPT [22]. (i) FGVC [30] is a benchmark for fine-grained image classification. This benchmark includes 5 downstream tasks, which are CUB-200-2011 [55], NABirds [25], Oxford Flowers [44], Stanford Dogs [19] and Stanford Cars [12]. We follow the validation splits in [22] if the official validation set is unavailable. (ii) VTAB-1k [59] is a large-scale transfer learning benchmark consisting of 19 visual classification tasks. VTAB-1k can be further divided into three groups, i.e. , natural tasks with natural images, specialized tasks with images captured by specialized equipment, and structured tasks with images mostly generated from synthetic environments. We use top-1 accuracy averaged within each group as our main metric following [22].

Pre-trained Backbones. We conduct experiments on the plain vision Transformer backbone ViTB/16 [16] that is pre-trained on ImageNet [47] with different pre-training strategies following [22], including supervised pre-training and self-supervised pre-training with MAE [23] and MoCo v3 [10]. We also conduct experiments on the representative hierarchical vision Transformer backbone SwinB [38] and CNN backbone ConvNeXt-Base [39] under supervised pre-training. In addition, we fine-tune the supervised pre-trained large-scale models (ViT-L/16 [16], ViT-H/14 [16]) on VTAB-1k to demonstrate the memory-efficiency and high-performance of SNELL.

Competitors. We compare our methods with addition-based methods including MLPk , VPTShallow [30], VPT-Deep [30], Adapterr [26], and SPT-Adapter [22]. For reparameterization-based methods, we compare with Linear, Partial-1, Bias [5], LoRAr [27], SSF [35], and SPT-LoRA [22]. Here r represents the number of bottleneck dimensions in Adapterr and the value of rank in LoRAr and our proposed SNELLr . Details of the competitors are presented in Appendix A.1. We also provide additional comparisons with other approaches [63, 53] in Appendix C.1.

Implementation Details. Following SPT [22], we use the AdamW optimizer [40] with cosine learning rate decay. The batch size, learning rate, and weight decay are 32 , 1 e -3 , and 1 e -4 , respectively. We also follow SPT [22] to implement the standard data augmentation pipeline for VTAB-1K and follow SSF [35] for FGVC as well. SNELL is applied on the pre-trained weight matrix of all linear layers. For each task, we fine-tune the model with different sparsity ratios s to search the optimal volume of tunable parameters for this task. Without specific stating, we adopt the piecewise linear kernel (introduced in Appendix B) as the kernel function for SNELL. Ablation studies on different kernel functions are presented in Figure 4.

4.2 Performance on Downstream Tasks

Performance on Different Benchmarks. Experiments on FGVC and VTAB-1k benchmarks indicate that SNELL achieves the best performance with supervised pre-trained ViT-B/16 backbone as shown in Table 1. SNELL gains large performance improvements over LoRA variants, e.g. , SNELL-8 surpasses LoRA-8 significantly by 5.5% in terms of mean accuracy on the FGVC benchmark. Moreover, SNELL outperforms the state-of-the-art method SPT-LoRA by a clear margin of 0.5% in terms of mean top-1 accuracy on the VTAB-1k benchmark. This stems from the fact that SPT-LoRA only performs sparse tuning on a portion of the weight matrices while employing LoRA for the remaining part. In contrast, the low memory property of SNELL empowers sparse tuning on all the weight matrices, allowing for more precise adjustments and giving rise to superior performance.

Performance on Different Pre-training Strategies. Experimental results on models pre-trained using different strategies are presented in Table 2. SNELL outperforms the state-of-the-art performances on models pre-trained with MAE (71.8% vs. 69.8%) and MoCo v3 (75.5% vs. 75.3%). Furthermore, SNELL consistently outperforms other PEFT methods on every group of downstream datasets. This demonstrates the general effectiveness of SNELL under different pre-training strategies.

Performance on Different Architectures. Following VPT [30] and SPT [22], we apply SNELL to the hierarchal vision transformer Swin-B and the CNN architecture ConvNeXt-Base. Experimental results are shown in Table 3. Results on Swin-B demonstrate that SNELL-8 outperforms existing reparameterization-based PEFT methods by 0.3% and achieves comparable performance to the stateof-the-art addition-based method SPT-Adapter. For ConvNeXt-Base, SNELL achieves a performance improvement of 0.4% compared to the best-reported result. These results obtained on different architectures further validate the versatility and effectiveness of our SNELL approach.

Memory Usage Comparison. We illustrate the effectiveness of SNELL in terms of memory usage by comparing it with various PEFT methods. Figure 3(a) shows the accuracy and memory usage of different methods on ViT-B/16. Although some methods achieve satisfactory performance, their memory usage is excessively large, even surpassing that of full fine-tuning ( e.g. SPT-Adapter and VPT-Deep). In comparison, SNELL achieves superior performance on downstream tasks with memory usage comparable to memory-efficient methods, including LoRA and Adapter.

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Additionally, we present the memory usage evolutions during the fine-tuning process in Figure 3(b) to provide a detailed explanation of how SNELL can save memory. In the model initialization stage, SNELL exhibits a significantly smaller memory usage compared to full fine-tuning. This is because full fine-tuning stores all weight matrices as learnable parameters in the optimizer, whereas SNELL only stores low-rank matrices with smaller parameter volumes. In the feed-forward phase, the memory usage increases with the storage of intermediate variables for backpropagation. Unlike other intermediate variables, the adaptation matrix ∆ W in SNELL solely relies on the low-rank parameter matrices, which are already stored in the optimizer. Therefore, it can be dumped in the feed-forward phase and recovered in backpropagation immediately, saving from a large amount of memory usage (SNELL vs. SNELL storing ∆ W ).

Scaling to Larger Models. To investigate the scalability of SNELL to large models, we apply it to ViT models of varying sizes (ViT-B/16, ViT-L/16, and ViT-H/16 pre-trained on ImageNet21K). We follow the experimental setup presented in Section 4.1, except for modifying the batch size for experiments on ViT-H/14 to 8 and changing the search scope of sparsity ratios s ∈ { 0 , 0 . 9 } .

As depicted in Figure 3(c), the memory usage of full fine-tuning increases rapidly as the model size grows. This observation highlights that existing PEFT methods like VPT and SPT, despite their advanced performances, incur substantial memory costs when applied to large-scale models due to even higher memory usage than full fine-tuning. In contrast, SNELL exhibits a notable advantage in terms of memory usage for larger models (similar to LoRA-8). When applied to ViTH/14, the memory usage of SNELL is only approximately 50% of that required for full fine-tuning, exemplifying its significant memory-saving capability on large models.

Regarding the performance, as shown in Table 4, SNELL-8 outperforms LoRA-8 on all dataset groups (Natural, Specialized, and Structured) as well as the mean accuracy for both ViT-L and ViT-H on the VTAB-1k benchmark. This demonstrates the effectiveness of SNELL for adapting large pre-trained models to downstream tasks.

4.3 Ablation Studies

Effect of Kernelized LoRA. We explore the effectiveness of kernelized LoRA by comparing the performance of sparsifying a full-rank matrix, the merged adaptation matrix of LoRA, and the merged adaptation matrix of kernelized LoRA. Experimental results are presented in Table 5(a). We can see that sparsifying the merged adaptation matrix of LoRA significantly underperforms a full-rank matrix. This reveals that the low-rank property of the merged adaptation matrix in LoRA greatly compromises the weight selection scope, leading to performance degradation for sparse tuning. However, when we replace LoRA with kernelized LoRA, the performance becomes notably comparable to that of the full-rank matrix under the strong sparsity constraint ( s = 0 . 9 ). This indicates that kernelized LoRA can effectively leverage sparse tuning while maintaining a low memory usage.

Effect of Sparse Tuning. Table 5(b) shows the performance comparison between SNELL and kernelized LoRA to explore the effectiveness of sparse tuning. For kernelized LoRA with different ranks, applying sparse tuning can consistently improve their performance. Moreover, as the rank of the learnable matrix increases, the performance of kernelized LoRA decreases while that of SNELL increases. This difference stems from the model regularization. Similar to sparse regularization, the low-rank property of LoRA that constrains the dependence between individual weights, can also be taken as a form of regularization. As the rank of the learnable matrix increases, the effect of low-rank regularization diminishes. Consequently, kernelized LoRA becomes more susceptible to over-fitting and encounters performance degradation. In contrast, SNELL employs both low-rank and sparse regularization. Higher ranks enable better sparsification towards downstream tasks, boosting sparse regularization that counteracts the diminished low-rank regularization. Therefore, a higher rank may lead to over-fitting in kernelized LoRA, but it can further enhance performance with sparse tuning.

Effect of Different Kernel Function. We investigate the effectiveness of different kernel functions in kernelized LoRA. First, we explore the ability of different kernel functions to fit randomly generated full-rank matrices based on low-rank matrices using the gradient descent algorithm (introduced in Appendix A.3). As shown in Figure 4(a), we explored four kinds of kernel functions. Compared with the linear kernel function, nonlinear kernel functions can reconstruct the full-rank target matrix more accurately based on the low-rank matrices. Subsequently, we explore the performance of kernelized LoRA with different kernel functions for pre-trained model fine-tuning in Figure 4(b). We find that using piecewise linear distance as the kernel function can achieve better results compared to linear kernel function (LoRA), while using Sigmoid and RBF kernels leads to severe performance degradation. This is because the complex non-linear kernel functions such as the exponential function increase the optimization difficulty in deep networks shown in Figure 4(c). More comparisons between LoRA and kernelized LoRA (with piecewise linear kernel) are presented in Appendix C.3.2.

Optimal Sparsity Ratio for Different Downstream Tasks. We provide the optimal sparsity ratio of SNELL-8 on tasks from VTAB-1k benchmarks in Figure 5. The optimal sparsity ratio varies

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significantly across different downstream tasks within the same group ( e.g. , Cifar vs. Sun397, dSprloc vs. Clevr-Dist). Furthermore, we can observe that the Natural task group exhibits a higher average optimal sparsity ratio compared to the Specialized group, while the Structured group demonstrates the lowest ratio. This observation aligns with the example illustrated in Figure A6, where cross-domain adaptation from a model pre-trained on natural images (ImageNet) to images of Specialized and Structured groups necessitates a larger number of tunable parameters.

5 Conclusion

In this work, we proposed a PEFT method named SNELL ( S parse tuning with ker NEL ized L oRA) to conduct high-performance sparse tuning with low memory usage. To reduce memory usage, we sparsified the adaptation matrix merged with low-rank matrices rather than the pre-trained weight matrix to reduce the volume of learnable parameters stored in the optimizer. Then we designed a competition-based sparsification mechanism to avoid the additional memory usage of storing the tunable weight indexes. To reveal the effectiveness of sparse tuning, we utilize nonlinear kernel functions to merge the adaptation matrix, increasing the rank of the merged matrix to maintain a compact representation suitable for sparse tuning with low memory usage. Extensive experiments demonstrated the ability of SNELL to leverage the high performance of sparse tuning and the low memory usage of LoRA. For future work, we will apply SNELL on larger models such as LLMs and improve its training efficiency. For limitations discussion, please refer to Appendix E.2.

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A More Details of Experimental Setup

A.1 More Details of Contenders.

Full: fully tunes all the model parameters (including backbone and classification head).

Linear: freezes all the backbone parameters and only tunes the linear classification head.

Bias [5]: freezes all model parameters except for the bias term and the linear classification head.

Partial-1: freezes the backbone except for the last 1 layer and also tunes the classification head as described in [30].

MLP-3: freezes the backbone and tunes the classification head implemented by a trainable 3-layer multi-layer percepton as described in [30].

VPT-Shallow [30]: freezes all the backbone parameters while introducing additional trainable prompts to the input space of the pretrained ViT.

VPT-Deep [30]: freezes the backbone while appending additional trainable prompts to the sequence in the multi-head self-attention layer of each ViT block.

Adapterr [26]: freezes all the backbone parameters while adding a down projection, a ReLU non-linearity, and an up projection layer sequentially in the feed-forward network (FFN) of each visual Transformer block. We report the performance implemented by [22] for comparison.

Lorar [27]: freezes all the backbone parameters while adding a concurrent branch including two low-rank matrices to the weight matrices in the multi-head self-attention layers to approximate efficiently updating them. The low-rank matrices can be merged into the backbone weights after fine-tuning. We report the performance implemented by [22] for comparison.

SPT [22]: identifies the tunable parameters for a given task in a data-dependent way, and utilizes LoRA (SPT-LoRA) or Adapter (SPT-Adapter) for weight matrices with a large number of tunable parameters and sparse tuning for weight matrices with a small number of tunable parameters.

VQT [53]: introduces a handful of learnable query tokens to each layer for adaptation.

DoRA [36]: decomposes the pre-trained weight into two components, i.e. , magnitude and direction, for fine-tuning. It specifically employs LoRA for directional updates to efficiently minimize the number of trainable parameters.

GPS [63]: identifies task-dependent tunable weights and applies sparse tuning to these weights.

A.2 Dataset Samples for the Downstream Tasks

We visualize some sampled images from different downstream tasks of VTAB-1k [59]) in Figure A6. The VTAB-1k benchmark encompasses a diverse range of tasks, including natural images, remote sensing, medical images, etc. Notably, our SNELL has achieved state-of-the-art (SOTA) performance on these datasets, demonstrating its general effectiveness.

A.3 Implementation details of Figure 4(a)

Given a random matrix W ( gt ) ∈ R m × n , we fit this matrix by merging two low-rank learnable matrices B ∈ R m × r , A ∈ R n × r with different kernel functions κ ,

We use gradient descent for 1 e 5 optimization steps, employing the Adam optimizer with a learning rate of 1 e -4 . We fit 10 randomly generated matrices for each kernel function presented in Table A6 and report the average MSE Loss in Figure 4(a).

B Introduction of Utilized Kernel Functions

Kernel Function Definition (positive semi-definite). Consider a vector space R r , a kernel function κ : R r × R r → R is called a positive semi-definite kernel on R r if

holds for all x 1 , ..., x n ∈ R r , c 1 , ..., c n ∈ R , n ∈ N .

Given two vectors x , x ′ ∈ R r , we show the utilized kernel functions in Table A6. We introduce additional learnable parameters (the e.g. α for Sigmoid and RBF kernel, α p for piecewise linear kernel) that enable the merged adaptation matrix ∆ W to accommodate both positive and negative values. The additional parameters select certain elements in the matrix and assign them negative values, without compromising the high-rank property of the merged adaptation matrix ∆ W . We set P = 2 for the piecewise linear kernel.

(a)

(b)

C Additional Experiments

C.1 More Comparisons with Existing Methods

Given that some methods do not provide performance or implementation details on both FGVC and VTAB benchmarks, we present a comparison between SNELL and these methods in the appendix rather than in Table 1. First, we provide comparisons with GPS [63] on the FGVC benchmark in terms of performance and memory usage in Table A8. With comparable performance, SNELL has a significant advantage over GPS in terms of memory usage. Then, we compare SNELL with VQT [53] on VTAB-1k dataset in Table A9. SNELL significantly outperforms VQT (76.9% vs. 68.8%).

C.2 Per-task results on the VTAB-1k benchmark

We provide the per-tasks results on the VTAB-1k benchmark using ViT-B/16 supervised pre-trained on ImageNet21K in Table A7. Our SNELL has demonstrated superior performance by achieving SOTA performance on 13 downstream tasks. Additionally, SNELL achieves SOTA performance on the mean accuracy across all tasks (74.6% vs. 74.1%), indicating its effectiveness in various domains.

C.3 More ablation studies.

C.3.1 Comparison between Competition-based Sparsification and Pre-defined Weight Mask

To verify the effectiveness of the proposed competition-based sparsification mechanism, we compare the performance on FGVC datasets between kernelized LoRA (KLoRA-8-Fixed) with pre-defined fixed masks, and SNELL in Table A10. The weight masks are generated by SPT [22]. For a fair comparison, we utilize the same data augmentation as SPT. Compared to our dynamic masking strategy, pre-defined fixed masking can hardly identify and adjust the most task-relevant weights in an end-to-end fashion, which leads to performance degradation (89.4 vs. 90.3).

C.3.2 Comparison between Kernelized LoRA and LoRA

We compare the performance of LoRA and kernelized LoRA on the VTAB-1k benchmark, where all weight matrices of the pre-trained models are fine-tuned to ensure a fair comparison. The experimental results are presented in Table A11. Through experiments with different ranks, we observed that kernelized consistently outperforms LoRA across various task groups. The replacement of the inner product with nonlinear kernel functions leads to stronger expressive ability, which in turn contributes to improved performance on downstream tasks.

C.3.3 Additional Memory Usage from Nonlinear Kernel Functions

In Figure 3(a), we observe that SNELL requires additional memory usage compared to LoRA due to the incorporation of nonlinear kernel functions. To explore whether the impact of this additional usage hinders the usability of SNELL on large models, we compare the memory usage between SNELL and LoRA on models as the model size grows (in Table A12). As the model size expands, the incremental memory usage of SNELL becomes negligible.

C.4 Experiments on Large Language Models

We apply SNELL on LLaMA2-7B [52] to adapt to the commonsense reasoning benchmark. As Table A13 shows, SNELL achieves a better performance than LoRA. This shows the applicability of SNELL to NLP tasks. Many other vision PEFT approaches lack this capability, as they necessitate a full level of memory usage for fine-tuning as Figure 3(a) shows.

C.5 Training Time Analysis

Table A14 provides a comparison of training time costs between SNELL and other PEFT methods using NVIDIA GeForce RTX 4090 GPU. The training time of SNELL-8 is slightly higher than LoRA-8 (0.557 vs. 0.443). By further comparing the training time of SNELL-8 and SNELL-8 (saving ∆ W ), it becomes apparent that the increase in time cost primarily stems from the recomputation

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of the merged adaptation matrix ∆ W . Conversely, the time cost associated with sparsification (KLoRA-8 vs. SNELL-8) and kernelization (LoRA-8 vs. KLoRA-8 (saving ∆ W )) is relatively small. Indeed, despite the slight increase in time cost due to the recomputation, one significant advantage is the significant performance improvement and memory efficiency shown in Figure 3.

D Additional Visualization

D.1 Performance of Different Sparsity Ratio

Figure A7 depicts the accuracy of different sparsity ratios on datasets in VTAB-1k. Different downstream tasks exhibit diverse preferences for the sparsity ratio. For instance, CIFAR-100 tends to favor a smaller sparsity ratio (0.2), while DTD prefers a larger sparsity ratio (0.8). Both Sun397 and Retinopathy tasks also lean towards a larger sparsity ratio (0.99). This highlights the need to consider the specific characteristics of each task when determining the optimal sparsity ratio.

D.2 Analysis of Tunable Parameters

We analyze the tunable weights of SNELL for different downstream tasks. In Figure A8, we compute the number of weights in the weight matrix W Q of self-attention [54] selected by multiple tasks. We find that most of the weights are only selected by a single downstream task (Tuned Times=1). Moreover, we find that in blocks of different depths, there will be a small part of weights that are selected by multiple downstream tasks, which indicates that there exists a small number of crucial parameters to improve the model's performance on downstream tasks.

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E Discussion

E.1 Tunable Parameter Volume Computing

We justify our choice to omit to report the volume of learnable parameters. First, computing the volume of tunable parameters in SNELL is difficult. In the case of LoRA, the volume corresponds to the shape of the learnable low-rank matrices. Conversely, for sparse tuning, the volume is determined by the number of updated weights. However, SNELL employs low-rank matrices as learnable parameters and achieves additional updated weight reduction by sparsifying the merged matrices. When using the parameter volume computation method of LoRA, calculating the reduction in parameters due to sparsification becomes challenging. Conversely, applying the parameter volume computation method of sparse tuning would be inherently unfair, given that SNELL is specifically optimized using low-rank matrices. Second, the parameter efficiency is a pathway to achieve high performance and low memory usage rather than an objective for model improvement, because performance improvement and memory usage reduction hold practical value. In our experiments, SNELL has demonstrated its advantages in terms of high performance and low memory usage, which we consider more valuable than the pursuit of fewer learnable parameters.

E.2 Limitation Discussion

Despite achieving state-of-the-art performance with low memory usage, SNELL requires more training time than LoRA. The additional training time cost comes from the recomputation of the merged matrix ∆ W in the backpropagation process presented in Appendix C.5. However, it is crucial to note that this limitation can be solved. Firstly, given the unique characteristics of the kernel matrix, more efficient methods [29] can be employed to calculate the merged adaptation matrix ∆ W . Secondly, by designing appropriate GPU operators, it is possible to avoid explicitly calculating ∆ W [45] during the fine-tuning process like LoRA and reparameterize the learnable low-rank matrices into the pre-trained weight matrices after the fine-tuning process.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a method called SNELL for vision model fine-tuning. It extends matrix decomposition from a kernel perspective and designs a novel sparsification mechanism for end-to-end parameter selection. Experimental results show that the proposed SNELL achieves low memory requirements and high performance on downstream tasks simultaneously.\n\nSTRENGTHS:\n1.\t The proposed method is novel and can achieve low memory requirements and high performance concurrently, which is valuable for practical applications. This is particularly noteworthy as many existing methods primarily focus on reducing tunable parameter volumes rather than memory usage.\n2.\t The proposed method can perform the data-dependent parameter selection and tuning in low-resource scenarios. I think this approach has the potential to advance the study of other areas, for example, model editing.\n3.\t This paper conducts extensive experiments, especially on large-scale vision models including ViT-L and ViT-H, demonstrating the high performance and memory efficiency of the proposed SNELL.\n4.\t This paper extends LoRA with a kernel perspective for sparse tuning. The kernel perspective contributes to new insights into LoRA and promotes its further improvements.\n5.\t This paper is well-written and easy to follow.\n\nWEAKNESSES:\n1.\t It would be better to introduce the utilized kernel function in the method section for the first time rather than in the experiment section.\n2.\t The authors should mention in the paper that the implementation details of Figure 4(a) are located in Appendix A.3, for readers to easily access the specific details of this figure.\n3.\t For Figure 3(b), authors should explain why the memory savings of SNELL are more significant for larger models.\n4.\t Although the paper includes comprehensive experiments, I still want to know the comparison between the proposed SNELL and other visual PEFT methods, such as MOSA [1] and VQT [2].\n\n\n[1] Zhang et. al., “MoSA: Mixture of Sparse Adapters for Visual Efficient Tuning”, Arxiv:2312.02923.\n\n[2] Tu et. al., “Visual Query Tuning: Towards Effective Usage of Intermediate Representations for Parameter and Memory Efficient Transfer Learning”, CVPR 2023.\n\nQUESTIONS:\nPlease check detailed comments in the Weaknesses part.\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on the sparse tuning task of the PEFT methods, by introducing the kernel function modeling applied into the LoRA two low-rank matrices or adapters, which costs low memory usage and leads to better performances. To achieve this target, a Kernelization for LoRA is proposed to map two low-rank matrices dot-production from a lower rank r to a implicit higher rank d, such that leading to stronger model capacity based on relatively marginal memory cost. Further, a soft mask adaptive strategy is used to implement the sparsification for fine-tuning so as to reduce the memory usage. Extensive experiments are presented to validate both the memory efficacy and performances improvements.\n\nSTRENGTHS:\n1. Mapping the low-rank matrices dot-production into an implicit space using conventional kernel function sounds like a promising direction, since some typical machine learning methods help with the representation ability of the model while reducing marginally small computation cost.\n\n2. The experimental results are extensive and the ablative studies showcase convincing result.\n\n3. The performances look superior to most of previous works, including both the adapter methods and weights decomposition methods.\n\n4. The presentation and writing of this paper are easy to follow.\n\nWEAKNESSES:\n1. The Competition-based Sparsification Mechanism is kind of overclaim by this paper, because the primary idea of this soft-thresholding is widely utilized to re-activate or de-activate some neurons or bits so that leading to sparsification. But competition-based is not really demonstrated on this point, neither showing competition among each bits of the mask nor clear explanation of the ∆W computation. Can you make a better clarity?\n\n2.  For the memory usage comparisons among LoRA, Adapter and SNELL, it seems SNELL displays little advantages over these works. I suppose that the improvements comes from the kernel function modeling as this implementation enable the model with stronger capacity to adapt for downstream task, such as the non-linear modeling ability, etc. Can the author support more comprehensive explanation and experimental comparisons?\n\n3. Some mistaken typos are presented in the paper writing. In Line6, 'accompanied by increases in' should be 'accompanied by increasing in'.\n\nQUESTIONS:\n1. Among Line47-48, I am confused about fine-tuning implementation that storing only the tunable weight into the optimizer using such as Pytorch is still practical. Hence, we may not need to put all the pretrained and frozen weights into the specific optimizer during the training phase. Can the authors make a clear explanation about this part?\n\nThe other questions please refer to the weaknesses above.\n\nLIMITATIONS:\nOverall, the presentation of this paper and the extensive experiments of this paper sounds promising. However, I still have the confusions above in terms of the motivation or the actual contribution of the Sparsification part.\n\nI will keep watching the responses from the authors to evaluate if my concerns are well addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a method called SNELL for achieving sparse tuning of pre-trained models with low memory usage. The authors employ LoRA to reduce the number of learnable parameters in the optimizer and utilize kernel tricks to ensure the merged matrix maintains a high rank. Additionally, during the sparse training phase, the authors introduce a competition-based sparsification mechanism to avoid the storage overhead of weight indices. Experiments on two benchmarks demonstrate that this method achieves leading performance while reducing memory usage.\n\nSTRENGTHS:\n1. This work introduces SNELL, a sparse tuning method based on LoRA, which combines the memory efficiency of LoRA with the performance of sparse tuning.\n2. Experiments conducted on two benchmarks show that SNELL consistently outperforms LoRA and compares memory usage with the baseline. The authors also conduct comprehensive experiments on different pre-training methods, model architectures, and component ablations.\n3. The paper is well-organized and well-written, making it easy to follow.\n\nWEAKNESSES:\n1. The idea of this paper isn't that novel. The overall approach of this work is a combination of existing fine-tuning techniques, LoRA and sparse tuning, with improvements to LoRA relying on existing kernel tricks. The competition-based sparsification mechanism merely sets a soft threshold.\n2. As a work on sparse tuning, it’s really a pity that this paper does not compare with the latest pure sparse tuning work, GPS[1], nor is it mentioned in the related work section. This severely limits the contribution of this work. In fact, comparing with the state-of-the-art sparse tuning work is essential. If the sole aim is to reduce memory usage at the expense of performance, it may not be very meaningful.\n3. As a PEFT method, the paper does not report the number of trainable parameters, citing the difficulty of calculation due to the sparse tuning method. This is unreasonable to me, as other sparse tuning methods (e.g., GPS) have reported this metric. This metric is also an important evaluation standard for PEFT.\n4. Generally, reparameterization-based methods refer to those where the additional modules can be reparameterized into the original model after training, such as LoRA and SSF[2] (which is also not compared with SNELL). In contrast, methods like Bitfit, Partial, and sparse tuning are not in this category and are referred to as specification-based methods in reviews like Delta-Tuning[3].\n\n[1] Zhang Z, Zhang Q, Gao Z, et al. Gradient-based Parameter Selection for Efficient Fine-Tuning[C]. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2024: 28566-28577.\n\n[2] Lian D, Zhou D, Feng J, et al. Scaling & shifting your features: A new baseline for efficient model tuning[J]. Advances in Neural Information Processing Systems, 2022, 35: 109-123.\n\n[3] Ding N, Qin Y, Yang G, et al. Delta tuning: A comprehensive study of parameter efficient methods for pre-trained language models[J]. arXiv preprint arXiv:2203.06904, 2022.\n\nQUESTIONS:\n1. During the experiments, was there a performance comparison with sparse tuning applied to the entire pre-trained model?\n2. The paper mentions that the competition-based sparsification mechanism can save memory usage for weight indices. What proportion of the total learnable parameters does this memory usage represent?\n3. In the sparse tuning part, have you tried fixing the sparse matrix during training to evaluate its effect?\n\nLIMITATIONS:\nThe paper mentions that to achieve sparsification, the re-computation of the LoRA merged adaptation matrix ΔW can lead to reduced computational efficiency. Although the authors explain that designing appropriate GPU operators can solve this problem, this may not be feasible in practice or may incur higher costs. If this approach can solve the problem, then the high memory usage issue of sparse tuning itself could also be addressed similarly.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-13"}

comments: SUMMARY:
The paper introduces SNELL (Sparse tuning with kerNELized LoRA), a method aimed at reducing memory usage during PEFT of large pre-trained models. SNELL achieves this by decomposing the tunable weight matrices into low-rank matrices and utilizing a competition-based sparsification mechanism, thereby avoiding the storage of large weight matrices and tunable weight indexes. This approach is demonstrated to achieve SOTA performance on multiple visual recognition tasks while maintaining low memory usage, making it viable for large-scale models.

STRENGTHS:
1. The introduction of kernelized LoRA for sparse tuning is a novel approach that effectively combines the advantages of LoRA's low memory usage with the performance benefits of sparse tuning.
2. SNELL significantly reduces memory usage compared to traditional fine-tuning and other PEFT methods, which is critical for scaling to larger models.
3. Extensive experiments demonstrate that SNELL achieves state-of-the-art performance on various benchmarks, including FGVC and VTAB-1k, while maintaining low memory usage.

WEAKNESSES:
1. While the focus is on visual recognition tasks, it would be beneficial to explore the applicability of SNELL to other domains such as natural language processing. I will expect the authors can further show some results on the large language models.
2. While LoRA offers benefits such as reduced memory usage, quick task switching, and the ability to serve multiple adapters, SNELL's use of dynamic masking during fine-tuning negates these advantages. I'm curious if the authors could provide results using pre-defined fixed masks instead, which might help retain some of LoRA's beneficial properties.

QUESTIONS:
See weaknesses.

LIMITATIONS:
See weaknesses.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-16