First-Order Predictable but Pairwise Fragile: Local Task Adaptation in Trained Transformers

2026-07-18

AI Summary

This study investigates the tension between the predictability of unidirectional perturbations and the fragility of multidirectional combinations in local task adaptation of Transformers. Leveraging multi-task LoRA operating points, the authors systematically evaluate eight local adaptation properties across nine models of varying scales, establishing—via a preregistered protocol—the quantitative boundary between first-order predictability and pairwise combinatorial fragility for the first time. Integrating LoRA fine-tuning, task arithmetic, activation steering, gradient analysis, and Lie bracket modeling, they demonstrate that unidirectional perturbations are highly predictable within a first-order regime (effective window up to 10⁻²), yet over one-third of model–task pairs exhibit order sensitivity at even smaller scales. The Lie bracket precisely captures this ordering dependence, with a median ratio of predicted to empirical deviation of merely 1.002.

Abstract

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.

Problem

• local task adaptation
• first-order predictability
• pairwise composition
• task interference
• transformer fine-tuning

Innovation

• local task adaptation
• first-order predictability
• pairwise fragility
• Lie bracket
• LoRA