Which and When to Admit: Gradient Admission for Data-Centric Small Language Model Finetuning

Recent literature on fine-tuning Large Language Models highlights a fundamental debate. While Full Fine-Tuning (FFT) provides greater representational plasticity, Low-Rank Adaptation (LoRA) can match or surpass FFT performance while constraining updates to a low-rank space and potentially benefiting from additional regularization. Through empirical evaluation across diverse tasks (SQL, Medical QA, and Counterfactual Knowledge) and varying language models (Gemma-3-1B, Qwen2.5-1.5B, and Qwen2.5-3B), we observe both trends and find that the better static architecture depends on the task and model. Spectral and truncation analyses further show that endpoint compressibility alone does not explain these task differences, suggesting task-score sensitivity and constrained optimization trajectories as possible explanations. To address this challenge, we propose a Mixture of LoRA and Full (MoLF) Fine-Tuning, a unified framework that enables continuous navigation between both training regimes. MoLF dynamically routes updates between FFT and LoRA at the optimizer level to ensure that exact gradient signals are available to both experts throughout training, while only selected experts update their weights. For memory-constrained environments, we also introduce MoLF-Efficient, which freezes base weights and only routes updates among a pair of LoRA experts of potentially varying rank. Our evaluations show that MoLF either improves on or stays within $1.5$ percentage points of the better of FFT and LoRA across the nine tested settings, while MoLF-Efficient outperforms both AdaLoRA and AdaMix in eight of nine settings, with gains over the stronger baseline of up to $11.70$ percentage points on Fact, $3.13$ on Med, and $2.98$ on SQL.

Data attribution methods using gradient similarity are widely used to analyze and select training data for large language models, but what gradient similarity actually measures is debated. Some interpret it as identifying task-relevant skills, while other work reports that surface form is the main factor. We resolve this debate for supervised fine-tuning examples by varying task and answer format independently. Specifically, we render benchmarks in different answer formats, such that datasets can share a task without a format or a format without a task. We find that gradient alignment follows the answer format, as benchmark pairs sharing an answer format align strongly (disattenuated cosine near 0.4), while same benchmarks rendered with different answer format classes show no alignment (near 0.0). We demonstrate that this ordering holds from the earliest pretraining checkpoints through post-training, and across model scales and families. We then analyze the released selections of LESS, a gradient-based data selection method for instruction tuning, and find that each target's selections over-represent the target's own answer format. Hence, we demonstrate that gradient-based attribution methods track format similarity more than task semantics, meaning that such methods, as well as the semantic interpretation of the gradient, should be tested on data where answer format and task vary independently for greater robustness and reliability.

Online batch selection fine-tunes a language model on the most useful part of each candidate batch. Selectors that match the gradient of the candidate batch are attractive because they need no held-out data, yet they rarely beat training on the whole batch. We show why. In-sample gradient matching uses every example as part of its own target, so its objective credits each example with its own gradient noise. This is the covariance penalty that makes training error optimistic, now sitting on the diagonal of the gradient Gram matrix: it steers selection toward the noisiest examples and makes the full batch the best solution the objective can reach. The fix costs nothing. For each example, the other candidates form an independent sample of the data distribution, so removing the diagonal turns the matching objective into an unbiased estimate of the update's error with respect to the population gradient. The minimizer of this leave-one-out objective weights examples by their gradient signal-to-noise ratio (SNR), and whenever per-example SNR is heterogeneous enough, half of a batch yields a lower-error update than the whole batch; we give the exact condition. We build \method{} on this principle. It computes the Gram matrix in the metric of the Adam preconditioner during the ordinary backward pass, selects a weighted subset greedily with a $(1-e^{-γ})$ guarantee, and uses no held-out data. Across four fine-tuning tasks and seven backbones from 1.5B to 8B parameters, LOOM improves on full-batch training by 2.3 and 2.4 points on Llama-3.1-8B and Qwen2.5-7B, exceeds every in-sample gradient matcher by 2.4 points and the validation-guided GREATS and OPUS by 1.6--2.0, and selects injected label noise at under a fifth of its base rate.