LoRA-Pro: Are Low-Rank Adapters Properly Optimized? (2407.18242v2)

Abstract: Low-rank adaptation, also known as LoRA, has emerged as a prominent method for parameter-efficient fine-tuning of foundation models. Despite its computational efficiency, LoRA still yields inferior performance compared to full fine-tuning. In this paper, we first uncover a fundamental connection between the optimization processes of LoRA and full fine-tuning: using LoRA for optimization is mathematically equivalent to full fine-tuning using a low-rank gradient for parameter updates. And this low-rank gradient can be expressed in terms of the gradients of the two low-rank matrices in LoRA. Leveraging this insight, we introduce LoRA-Pro, a method that enhances LoRA's performance by strategically adjusting the gradients of these low-rank matrices. This adjustment allows the low-rank gradient to more accurately approximate the full fine-tuning gradient, thereby narrowing the performance gap between LoRA and full fine-tuning. Furthermore, we theoretically derive the optimal solutions for adjusting the gradients of the low-rank matrices, applying them during fine-tuning in LoRA-Pro. We conduct extensive experiments across natural language understanding, dialogue generation, mathematical reasoning, code generation, and image classification tasks, demonstrating that LoRA-Pro substantially improves LoRA's performance, effectively narrowing the gap with full fine-tuning. Code is publicly available at \url{https://github.com/mrflogs/LoRA-Pro}.

LoRA-Pro: Are Low-Rank Adapters Properly Optimized?

Abstract

The paper LoRA-Pro: Are Low-Rank Adapters Properly Optimized? introduces a novel approach called LoRA-Pro, aimed at addressing the inherent performance discrepancies between Low-Rank Adaptation (LoRA) and full fine-tuning in the context of parameter-efficient fine-tuning (PEFT) of foundational models. By exploring the optimization dynamics, the authors identify a critical gap in the existing LoRA methodology and propose a solution involving the concept of "equivalent gradient." This innovative approach ensures that the optimization processes of re-parameterized matrices under LoRA more closely mimic those of full fine-tuning, thereby improving performance.

Introduction

Foundational models have revolutionized the field of deep learning, demonstrating remarkable generalization capabilities through extensive pre-training on large datasets. However, the sheer number of parameters in these models presents significant challenges when it comes to fine-tuning for specific downstream tasks. To circumvent the prohibitive computational costs, researchers have increasingly gravitated towards PEFT methods such as LoRA, which notably reduces the number of trainable parameters by re-parameterizing the weight updates as low-rank matrices.

LoRA and Its Limitations

LoRA leverages the insight that weight changes in large models typically reside in a low-dimensional subspace. It re-parameterizes these changes using two low-rank matrices $A$ and $B$, drastically reducing the number of trainable parameters. However, despite its efficiency, LoRA often falls short when compared to full fine-tuning in terms of performance. This paper identifies a key reason for this discrepancy: LoRA's failure to approximate the optimization dynamics of full fine-tuning.

Proposing LoRA-Pro: Methodology

The paper introduces a novel concept termed the "equivalent gradient," which is essential for understanding the optimization nuances in both LoRA and full fine-tuning. By defining the equivalent gradient as a composite of the gradients of the low-rank matrices $A$ and $B$, the authors quantify the differences between the optimization processes of LoRA and full fine-tuning.

To optimize the matrices $A$ and $B$, the paper formulates an objective function that minimizes the discrepancy between the equivalent gradient under LoRA and the gradient obtained from full fine-tuning. The resulting optimization problem yields a closed-form solution that ensures the equivalent gradient follows the optimization trajectory of full fine-tuning.

Theoretical Insights

Two key theorems are presented to justify the effectiveness of the proposed approach:

1. Theorem 1 provides the closed-form solutions for updating matrices $A$ and $B$, showing that these solutions depend on the gradients observed in standard LoRA.
2. Theorem 2 guarantees the convergence of the optimization process, demonstrating that the proposed updates for $A$ and $B$ consistently lead to a reduction in the loss function.

Moreover, Theorem 3 addresses the selection of the matrix $X$ used in the closed-form solutions, ensuring that $X$ is chosen to maintain the gradients of the low-rank matrices as close as possible to those of standard LoRA.

Experimental Results

The paper validates the proposed method through extensive experiments on NLP tasks using the T5-base model. The datasets include a subset of the GLUE benchmark, which provides a comprehensive assessment across various NLP tasks. Compared to standard LoRA and its variants, LoRA-Pro consistently achieves higher average scores, significantly narrowing the performance gap with full fine-tuning. Specifically, LoRA-Pro shows an improvement margin of 6.72 points on average over five datasets compared to standard LoRA.

Implications and Future Work

The implications of this research are multifaceted. Practically, LoRA-Pro offers a more effective fine-tuning strategy for large-scale models, making it feasible to deploy these models in resource-constrained environments without sacrificing performance. Theoretically, the concept of equivalent gradients introduces a new dimension to the understanding of optimization dynamics in re-parameterized models.

Future developments may involve adapting the equivalent gradient concept to other PEFT methods or exploring its potential in different machine learning paradigms. Additionally, further research could investigate the integration of LoRA-Pro with advanced optimization techniques beyond SGD and AdamW, potentially enhancing its robustness and efficacy across various applications.

Conclusion

In conclusion, the paper LoRA-Pro: Are Low-Rank Adapters Properly Optimized? introduces a robust framework that bridges the gap between LoRA and full fine-tuning. By focusing on optimizing the equivalent gradient, LoRA-Pro aligns the optimization processes of low-rank matrices with those of full fine-tuning, resulting in significant performance improvements. Through rigorous theoretical formulations and extensive experimental validations, this research underscores the importance of optimizing not just the approximation of weight updates but the entire optimization trajectory in PEFT methods.