Same Predictor, Different Learners: How LoRA Factorization Changes Active Learning

Two low-rank adaptation (LoRA) learners can make identical predictions and equivalent updates on common data, yet choose different examples to label next. In rank-one Gaussian task families, exact refactorization under the same posterior-variance rule produces a label-cost ratio for the same prediction risk that grows without bound with pool size. The gap persists with arbitrarily accurate conditional covariance and nearly optimal information about the teacher: learning private coefficients improves few predictions, whereas learning a shared coefficient improves many. We formulate a factorization-independent acquisition Gram from effective updates and effective gradients, retaining the geometry of balanced factor gradients. With matched selection inputs and randomness, it preserves ordered purchases across equivalent factorizations; with corresponding common-data updates, it also preserves learning paths. Controlled multi-round experiments with Qwen3 classifiers isolate acquisition as the source of divergence and recover learner correspondence through common purchases or the invariant Gram. A retrospective analysis of repeated training continuations on MultiNLI finds repeatable per-example correctness changes despite similar mean accuracy. Equivalence on common training data therefore does not extend automatically to active learning: the acquisition geometry must also be specified.