LoRA-GA$^2$: Low Rank Adaptation with Multi-step Gradient Adaptive Alignment

Low-Rank Adaptation (LoRA) is a prominent fine-tuning method for large models, achieving competitive performance with reduced memory overhead. However, a persistent performance gap remains between LoRA and full fine-tuning. Recent studies have sought to narrow this gap by employing one-step gradient approximations of pretrained weights to align LoRA updates with the principal directions or intrinsic dimensionalities of full fine-tuning updates. Nevertheless, these approaches fail to capture the full dynamics of the gradients. In this paper, we propose LoRA-GA$^2$, an effective fine-tuning algorithm that fully leverages multi-step gradient information. Specifically, we introduce a lightweight probe for multi-step gradients of pretrained weights that incurs no additional GPU memory cost and only marginal time overhead. We further employ a spectrum-aware, importance-based rank allocation and optimal initialization derived from multi-step gradients. Extensive experimental results demonstrate that LoRA-GA$^2$ consistently outperforms existing LoRA variants while preserving the efficiency advantages of vanilla LoRA. For instance, LoRA-GA$^2$ surpasses the leading baseline by an average of 0.66 points on the GLUE benchmark, and outperforms the strongest baseline by 1.03 points on GSM8K and 0.87 points on HumanEval, respectively.

SAD-LoRA: Spectral Alignment for Low-Rank Knowledge Distillation

Distilling a fine-tuned teacher into a LoRA-adapted student is a standard recipe for parameter-efficient compression, but output-level KD does not explicitly control which rank-$r$ weight subspace the adapter occupies. We propose \textbf{SAD-LoRA} (\textbf{S}pectral \textbf{A}lignment \textbf{D}istillation), which selects this subspace from the data-weighted student-space reference update $\DWT\Sigx^{1/2}$ and maintains it during training via a differentiable principal-angle loss on $\colspan(B)$. We show that the data-weighted distillation error decomposes exactly into subspace misalignment, within-subspace coefficient mismatch, and irreducible rank residual; standard KD can affect the first term only indirectly through output gradients. On controlled synthetic problems with a flat teacher spectrum, SAD-LoRA reduces the subspace-misalignment term from $51\%$ to nearly zero and lifts final subspace alignment from $0.49$ to $1.00$. On RoBERTa-large to RoBERTa-base distillation across six GLUE tasks, SAD-LoRA improves rank efficiency: at $r{=}4$, it matches or beats the strongest included spectral baseline on five of six tasks, and at $r{=}8$ it gives the best result on SST-2 and CoLA. Ablations identify subspace alignment as the load-bearing component, while coefficient matching is auxiliary.

LAARA: Layer-Aware Adaptive Rank Allocation for Parameter-Efficient Fine-Tuning

Low-Rank Adaptation is widely used for parameter-efficient fine-tuning, yet existing methods typically assign the same adapter rank to every transformer layer despite their heterogeneous adaptation requirements. In this work, we show theoretically and empirically that uniform rank allocation is fundamentally suboptimal. Motivated by this observation, we propose LAARA (Layer Aware Adaptive Rank Allocation framework), a search-free framework that dynamically allocates ranks using lightweight diagonal Fisher estimates computed during training. LAARA combines projection-wise normalization, logarithmic compression, blended adapter importance estimation, and a vote-to-change dampening mechanism to produce stable and efficient rank adaptation. Experiments on GLUE and MathInstruct benchmark demonstrate that LAARA consistently matches or outperforms popular state of the art approaches such as LoRA, AdaLoRA, DyLoRA, and Bitfit while using significantly fewer trainable parameters. Our results show that Fisher-guided rank allocation provides a principled and effective foundation for adaptive parameter-efficient fine-tuning. The code is publicly available at: https://anonymous.4open.science/r/LAARA-D305/LAARA.py