Compressing Large Language Models using Low Rank and Low Precision Decomposition

Summary

This paper proposes a framework that combines quantization and low-rank approximation. Given a neural net weight matrix $W\in \mathbb{R}^{n\times d}$, it considers the following decomposition: $W=Q+LR$, where $Q$ is a quantized sketch of $W$ with very few bits (2 or 4 in their experiments), and $L\in \mathbb{R}^{n\times k}, R\in \mathbb{R}^{k\times d}$ are low-rank factors that serve as a "correction" to the quantization sketch $Q$. They are usually quantized with more bits (4 or 16). To compute respective $Q, L, R$, the authors propose an alternating minimization framework: starts with $L, R=0$, update $Q$ as $Q_t=Q_Q(W-L_{t-1}R_{t-1})$, then solve for $L_t, R_t$ by solving the rank-constrained regression against $W-Q_t$ and $X\in \mathbb{R}^{m\times d}$, where $X$ is the calibration dataset. They further show their algorithm could be used in conjunction with LoRA and fine-tuning the randomized Hadamard transform. Extensive experiments show that their proposed algorithm gives low perplexities and high accuracies when compared against earlier algorithms.

Strengths

This paper proposes a method that is a good mixture of quantization and low-rank approximation, as simply using only one of quantization sketch or low-rank approximation intuitively "misses" an important part. The algorithm proposed in this paper is a natural and simple alternating minimization framework. This paper also contains a theoretical analysis for approximation error under assumptions on the target rank. Empirically, it is also shown the proposed algorithm has good performance under small rank and bit precision. The paper is also well-written.

Weaknesses

Overall, I don't see big weaknesses of this paper. One potential direction for improvement is theoretically, the alternating minimization framework could possibly be sped up given $k\ll n, d$, especially the rank-constrained regression, see [1]. On the other hand, it is not clear how to integrate these possible algorithmic pieces into post-processing a model weight, so I think the approach and runtime analysis of this paper are fair. [1] Gu, Song, Yin and Zhang. Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time. ICLR'24.

Questions

If we allow $L$ and $R$ to be computed to full bit-precision, could the statement / result of Theorem 4.1 be simplified?

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

Yes.

Summary

This paper introduces CALDERA, a new post-training compression algorithm for large language models (LLMs). CALDERA uses the inherent low-rank structure of LLM weight matrices by approximating them via a low-rank, low-precision decomposition $W \approx Q + LR$, where $L$ and $R$ are low-rank factors with quantized entries. CALDERA provides an effective way to compress large language models by exploiting their low-rank structure, enabling efficient distribution and deployment of LLMs on resource-constrained hardware while maintaining strong performance.

Strengths

1. CALDERA introduces a new post-training compression technique for LLMs that exploits their inherent low-rank structure, setting it apart from existing methods. Also, the paper provides theoretical upper bounds on the approximation error of CALDERA using a rank-constrained regression framework, lending credibility to the approach. 2. Experimental results demonstrate that CALDERA outperforms existing post-training LLM compression techniques in the low-bit regime (less than 2.5 bits per parameter), highlighting its effectiveness. By enabling efficient distribution and deployment of LLMs on resource-constrained hardware, CALDERA can make LLMs more accessible and promote their broader adoption.

Weaknesses

This paper is generally well-written. Here are some minor weaknesses: 1. The performance of CALDERA depends on the choice of target rank and quantization bit budget. The paper does not provide a systematic way to determine the optimal values for these hyperparameters. Is it possible to apply meta-learning here to decide the optimal values? 2. The experiments focus on compressing LLaMa models. It would be valuable to see how CALDERA performs on a wider range of LLMs to assess its generalizability. 3. As mentioned by the authors, the iterative nature of CALDERA's optimization process may require more computational resources compared to simpler compression methods, although this is a one-time cost.

Questions

How does CALDERA perform compared to other compression approaches? Is it possible to combine CALDERA with these techniques?

Rating

7

Confidence

2

Soundness

4

Presentation

3

Contribution

3

Limitations

Please see weaknesses.

Summary

This paper introduces CALDERA, a novel post-training method that combines quantization and low-rank decomposition techniques for compressing large language models. The primary contribution lies in the design of a combined pipeline and the application of low-rank decomposition. Experimental results demonstrate that integrating the designed low-rank decomposition technique enhances compression performance compared to using quantization alone.

Strengths

1. This paper addresses a highly significant and widely studied research topic: compressing large language models. 2. The author not only conducts experiments but also provides numerical evidence to demonstrate the effectiveness of the designed compression method. 3. Experimental results indicate that the combined methods designed in this study effectively enhance the performance of large language models compressed using quantization-only methods.

Weaknesses

1. **Missing Experimental Results:** The experiments conducted in this paper are inadequate. Many critical experiments conducted in previous works have not been included. For instance: - Evaluation of throughput for LLaMA models compressed by CALDERA and QuIP# under the same average bits with the same batch size and sequence length. It's crucial to assess whether combining low-rank decomposition with quantization compromises efficiency compared to uncompressed LLMs. - Evaluation results on LLMs with different architectures from LLaMA and on generation datasets. - Evaluation of time consumption between CALDERA and QuIP# for compressing LLMs. - Evaluation of compression performance using different calibration sets and varying numbers of calibration data. 2. **Lack of Novelty:** The novelty of this paper is limited. The primary contributions are the design of a combination framework and a low-rank decomposition algorithm, both of which appear simplistic and resemble iterative weight compression methods with minor losses. The use of the existing quantization method LDLQ for combination and the effectiveness of the designed low-rank decomposition method in achieving minimal compression loss are questionable. Replacing it with established work like data whitening method in SVD-LLM[1] might yield better performance, given SVD-LLM's theoretical proof of achieving global optimality through data whitening. *[1] Wang, X., Zheng, Y., Wan, Z., & Zhang, M. (2024). SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compression. https://arxiv.org/abs/2403.07378* 3. **Intuitive Experimental Results:** While it's intuitive that combining low-rank decomposition with quantization can improve performance over quantization-only methods, existing works such as SVD-LLM already demonstrate this. To add value to the proposed methods, it's crucial to include experimental results where the average bits are equal to or less than 1-bit, demonstrating capabilities that quantization-only methods cannot achieve.

Questions

See above.

Rating

2

Confidence

5

Soundness

3

Presentation

1

Contribution

1

Limitations

N.A.

Summary

The article introduces CALDERA, a post-training compression algorithm for large language models (LLMs) that leverages the low-rank structure of weight matrices to achieve significant compression. CALDERA approximates a weight matrix $W$ using a low-rank, low-precision decomposition $W \approx Q + LR$, where $Q$, $L$ and $R$ are quantized. The method aims to reduce the memory and computational footprint of LLMs, facilitating their deployment on memory-constrained edge devices.

Strengths

Innovative Compression Technique: CALDERA combines low-rank approximation with low-precision quantization, addressing both the redundancy and precision issues in LLM weight matrices. Theoretical Foundations: The algorithm is backed by rigorous theoretical guarantees on the approximation error, which enhances its reliability. Performance: Empirical results show that CALDERA outperforms existing post-training compression techniques in terms of zero-shot performance, particularly when compressed to less than 2.5 bits per parameter. Adaptability: The method supports low-rank adaptation, which can further enhance the performance of the compressed models on specific tasks.

Weaknesses

Complexity: The algorithm involves a nested optimization process, which may introduce significant computational overhead during the compression phase. Dependency on Calibration Data: The performance of CALDERA relies on the availability and quality of calibration data, which may not always be accessible or representative. Limited Empirical Comparisons: While the article claims superiority over existing methods, the empirical comparisons might be limited in scope, potentially omitting some relevant state-of-the-art techniques. Potential Issues Quantization Artifacts: Aggressive quantization, especially at very low bit budgets, might introduce artifacts that could degrade model performance in specific scenarios. Generalization: The effectiveness of the method across a diverse range of LLMs and tasks needs thorough validation, as it might not generalize well beyond the tested models and datasets. Optimization Stability: The iterative nature of the optimization process might lead to stability issues, particularly for large models with highly non-convex loss landscapes.

Questions

See Weaknesses

Rating

4

Confidence

2

Soundness

2

Presentation

3

Contribution

2

Limitations

Yes

Summary

This work proposes a new decomposition procedure to achieve low-precision and low-rank compression of large language model (LLM) weight matrices. Such a new method could capture high singular components accurately while compressing less significant ones. An efficient algorithm is proposed to optimize the quantized backbone and low-rank factors, which can be fine-tuned to enhance the model performance. The work also provides a tighter approximation error bound and is validated by compressing the LlaMA LLMs to below 2.5 bits per parameter.

Strengths

1. The proposed CALDERA method can effectively compress LLMs without degrading the performance too much. And it is demonstrated through a series of experiments that the method is capable of model compression and low-rank adaptive fine-tuning at the same time. 2. The paper also provides a rigorous approximation error analysis of the proposed CALDERA method.

Weaknesses

I think the presented experiment study is limited. I mainly have the following suggestions. 1. It would be better to supplement the experiments with LoftQ and LQ-LoRA, and add a set of experiments with full parameter fine-tuning as a baseline. 2. Please provide an experimental comparison of model size. 3. It would be better to add some experiments on evaluating the computation cost of the proposed CALDERA.

Questions

1. Since the CALDERA method requires many time-consuming operations like SVD and matrix inversion, can this be improved in future work? 2. Are there any efficient ways to improve the selection strategy of the hyperparameter rank?

Rating

7

Confidence

4

Soundness

3

Presentation

3

Contribution

3

Limitations

In my opinion, how to efficiently tune the hyperparameter rank is the main limitation of the proposed method.

Further clarifications

Dear Reviewer 32tN, Please let us know if your queries have been addressed satisfactorily. As mentioned in our response, we've thoroughly incorporated your feedback, along with suggestions from the other reviewers. We hope that our response has positively influenced your perception of our work. If you require further clarifications to potentially reconsider your score, we are enthusiastic about engaging in further discussion. Please do not hesitate to contact us. We highly value the generous contribution of your time to review our paper.

Further clarifications

Dear Reviewer ZG6L, Please let us know if your queries have been addressed satisfactorily. As mentioned in our response, we've thoroughly incorporated your feedback, along with suggestions from the other reviewers. We hope that our response has positively influenced your perception of our work. If you require further clarifications to potentially reconsider your score, we are enthusiastic about engaging in further discussion. Please do not hesitate to contact us. We highly value the generous contribution of your time to review our paper.

Further clarifications

Dear Reviewer xRtR, Please let us know if your queries have been addressed satisfactorily. As mentioned in our response, we've thoroughly incorporated your feedback, along with suggestions from the other reviewers. We hope that our response has positively influenced your perception of our work. If you require further clarifications to potentially reconsider your score, we are enthusiastic about engaging in further discussion. Please do not hesitate to contact us. We highly value the generous contribution of your time to review our paper.