Localized Adaptation Reveals Distinct Learning Signatures in Transformers

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.

Task arithmetic, sequential fine-tuning, activation steering, and first-order random search all operate through relatively small perturbations around an already trained checkpoint, and they rely on different local approximations: individual perturbations should be first-order predictable, task updates should compose with controlled interference, useful tangent structure should be stable and possible to estimate, and weight edits should have counterparts in representation space. We measure 8 such properties with the same harness around a multitask LoRA operating point, on 9 transformers (82M-7B), with a prospectively registered property list, thresholds, and test split. We find a shared one-direction validity window up to the tested scale $10^{-2}$, but no universal radius for pairwise composition or update ordering. Along individual directions, changes of the probe loss remain first-order predictable throughout the grid: a perturbation's effect on the loss is essentially its projection onto the gradient, which is also what makes local random search work. Pairwise structure, however, proves to be far more fragile: on over a third of the measured (model, task pair) combinations, two-update order sensitivity sets in strictly inside that window; task-gradient subspaces rotate within tens of steps; additivity under our fixed activation probe fails at full task-vector scale on several models, including both held-out 7B models; and no model median passes the registered global mean-vector weight-to-steering correspondence bar. For two sequential task-gradient steps, the leading order-dependent term is the Lie bracket $H_B\textbf{g}_A-H_A\textbf{g}_B$; its normalized prediction $c(η)=ηκ+O(η^2)$ tracks the measured defect at median ratio 1.002, while the onset scale $η^\dagger\approx0.10/κ$ spans three orders of magnitude across models and task pairs.

Reinforcement learning (RL) has become a central component of post-training large language models (LLMs), yet little is understood about how RL adaptation is distributed across transformer layers. Existing approaches typically update all model parameters uniformly, implicitly assuming that every layer contributes similarly to the gains obtained during RL post-training. In this work, we challenge this assumption through a systematic layer-wise study of RL training. Surprisingly, we find that training a single transformer layer can recover most of the gains achieved by full-parameter RL training, and in some cases even surpass it. To quantify this phenomenon, we introduce the quantity layer contribution, which measures the fraction of full RL improvement recovered by training a layer in isolation. Across seven models spanning two model families (Qwen3, Qwen2.5), three RL algorithms (GRPO, GiGPO, Dr. GRPO), and multiple task domains including mathematical reasoning, code generation, and agentic decision-making, we observe a remarkably stable pattern: RL gains are highly concentrated in a small subset of, and in many cases even a single, transformer layers. More strikingly, the same structural pattern consistently emerges: high-contribution layers concentrate in the middle of the transformer stack, while layers near the input and output ends contribute substantially less. The resulting layer rankings remain strongly correlated across datasets, tasks, model families, and RL algorithms.