Low-rank adaptation (LoRA) is a widely used parameter-efficient fine-tuning method that places trainable low-rank adapters into frozen pre-trained models. Recent studies show that using fewer LoRA adapters may still maintain or even improve performance, but existing methods still distribute adapters broadly, leaving where to place a limited number of adapters to maximize performance largely open. To investigate this, we introduce PAGE (Projected Adapter Gradient Energy), a gradient-based sensitivity probe that estimates the initial trainable gradient energy available to each candidate LoRA adapter. Surprisingly, we find that PAGE is highly concentrated on a single shallow FFN down-projection across two model families and four downstream tasks. We term this module the dominant adaptation module and show that its layer index is architecture-dependent but task-stable. Motivated by this finding, we propose DomLoRA, a placement method that places a single adapter at the dominant adaptation module. With only ~0.7% of vanilla LoRA's trainable parameters, DomLoRA outperforms it on average across various downstream tasks, including instruction following, mathematical reasoning, code generation, and multi-turn conversation. This method also improves other LoRA variants, supporting the dominant adaptation module perspective as a practical placement guideline.
We study the \textit{parameter placement problem}: given a fixed budget of k trainable entries within the B matrix of a LoRA adapter (A frozen), does the choice of which k matter? Under supervised fine-tuning, random and informed subsets achieve comparable performance. Under GRPO on base models, random placement fails to improve over the base model, while gradient-informed placement recovers standard LoRA accuracy. This regime dependence traces to gradient structure: SFT gradients are low-rank and directionally stable, so any subset accumulates coherent updates; GRPO gradients are high-rank and near-orthogonal across steps, so only elements with consistently signed gradients retain the learning signal. Our scoring procedure identifies these critical parameters in under 10 seconds at less than 0.5% of training cost. Selected parameters concentrate on residual-stream-writing projections (V, O, Down), stable across model families and scales (1.5B - 8B).
Parameter-efficient fine-tuning (PEFT) methods face a tradeoff between adapter size and expressivity: ultra-low-parameter adapters are confined to fixed low-rank subspaces, capping performance even with extended training. We propose BoostLoRA, a gradient-boosting framework that overcomes this limit by iteratively training and merging minimal adapters on the examples the current model gets wrong. A ROTATE SVD basis strategy assigns each round to an orthogonal subspace, so cumulative effective rank grows linearly with the number of rounds while each adapter remains ultra-low-rank. After merging, adapters are discarded, leaving zero inference overhead. On Qwen2.5-3B, BoostLoRA reaches 89.1% on GSM8K and 68.8% on MATH-500, surpassing both the best single-shot ultra-low parameter adapter (TinyLoRA) and full fine-tuning; on code generation it reaches 57.2% on MBPP and 80.4% on HumanEval while full fine-tuning drops below the zero-shot baseline. We also demonstrate cross-architecture transfer on protein binding classification with ESM2-650M and cross-entropy training. BoostLoRA is, to our knowledge, the first PEFT method whose effective rank grows with training, separating per-round parameter cost from total representational capacity.
Low-rank adaptation (LoRA) assigns a uniform rank to every adapted weight matrix - a practical convenience that ignores a fundamental reality: different layers contribute unequally to task adaptation. We address this with a lightweight engineering solution: before fine-tuning begins, run eight calibration backward passes, compute the gradient variance of each LoRA-B matrix as a proxy for layer informativeness, and redistribute the rank budget proportionally. The resulting adapter is a standard LoRA with a per-layer rank pattern - no new parameters, no training overhead, no changes to serving infrastructure. We implement this via an efficient approximation of the empirical Fisher Information Matrix (eFIM) diagonal, restricted to LoRA adapter matrices only, which reduces memory cost by approximately 256x compared to full-model Fisher estimation. On GLUE with DeBERTa-v3-base, FIM-LoRA matches LoRA (88.6 vs. 88.7) at the same parameter budget, and on commonsense reasoning with LLaMA-3-8B reaches 68.5 vs. 68.7 for LoRA. The per-layer rank maps are interpretable: value projections and early-to-middle layers consistently receive higher rank, consistent with established findings on transformer layer roles.