Parameter-efficient fine-tuning (PEFT), particularly Low-Rank Adaptation (LoRA), adapts large language models (LLMs) by training only a small fraction of parameters. However, as the rank of the low-rank matrices used for adaptation increases, LoRA often exhibits an unstable “double descent” phenomenon, characterized by transient divergence in the training loss, which delays convergence and impairs generalization by causing instability due to the attraction to sharp local minima. To address this, we introduce **LoRA-MGPO**, a framework that incorporates Momentum-Guided Perturbation Optimization (MGPO). MGPO stabilizes training dynamics by mitigating the double descent phenomenon and guiding weight perturbations using momentum vectors from the optimizer’s state, thus avoiding dual gradient computations. Additionally, an adaptive normalization scheme scales the magnitude of perturbations based on an exponential moving average (EMA) of gradient norms, further enhancing stability. While EMA controls the magnitude of the perturbations, MGPO guides their direction, ensuring a more stable optimization trajectory. Experiments on a suite of natural language understanding and generation benchmarks show that LoRA-MGPO consistently achieves superior performance over LoRA and other PEFT methods. The analysis indicates that LoRA-MGPO leads to smoother loss curves, faster convergence, and improved generalization by stabilizing the training process and mitigating the attraction to sharp minima. The code is publicly available at [https://github.com/llm172/LoRA-MGPO](https://github.com/llm172/LoRA-MGPO).

Parameter-efficient fine-tuning (PEFT) is essential for adapting large language models (LLMs), with low rank adaptation (LoRA) being the most popular approach. However, LoRA suffers from slow convergence, and some recent LoRA variants, such as PiSSA, primarily rely on Singular Value Decomposition (SVD) for initialization, leading to expensive computation. To mitigate these problems, we resort to Nyström method, which follows a three-matrix manipulation. Therefore, we first introduce StructuredLoRA (SLoRA), investigating to introduce a small intermediate matrix between the low-rank matrices (A) and (B). Secondly, we propose NyströmLoRA (NLoRA), which leverages Nyström-based initialization for SLoRA to improve its effectiveness and efficiency. Finally, we propose IntermediateTune (IntTune) to explore fine-tuning exclusively the intermediate matrix of NLoRA to furthermore boost LLMs’ efficiency. We evaluate our methods on 5 natural language generation (NLG) tasks and 8 natural language understanding (NLU) tasks. On GSM8K, SLoRA and NLoRA achieve accuracies of 56.48% and 57.70%, surpassing LoRA by 33.52% and 36.41% with only 3.67M additional trainable parameters. IntTune boosts average NLG performance over LoRA by 7.45% while using only 1.25% of its parameters. These results demonstrate the efficiency and effectiveness of our approach in enhancing model performance with minimal parameter overhead.