When Can First-Order Models of Fine-Tuning Bound Forgetting?

Fine-tuning a language model on new data can make it forget facts that it should keep. We ask whether measurements taken at the start of a fine-tuning run can bound, for each protected fact, the probability that the run makes the model forget it. In LoRA fine-tuning with stochastic gradient descent on models from 0.6B to 14B parameters, a first-order response model estimated by finite-difference probes predicts changes of per-fact margins with correlation 0.974–0.998. Predictions of forgetting built on this model nevertheless failed; in Qwen3-0.6B, forgetting requires parameter changes far outside the region in which the model was validated. The probes can, however, bound the probability that a margin first falls below a boundary near zero: we derive Freedman and Azuma first-passage bounds for a linear surrogate of the margin and test on new runs whether they hold for the model. The bounds contain a term R that measures how much the response coefficients change during the run. The simplified Freedman bound, which sets R = 0, certified most facts but was violated in 14 of 112 conditions, and in these conditions every fact on which it was violated had R ≥ a, where a is the distance of the fact's margin to the boundary. The complete Freedman bound certifies only facts with R < a, and it held in every condition. On the violated facts, the spread of the margin across test runs was a median of 14.6 times the prediction of the response model, so the failures are breakdowns of the model, and in our data they occurred only where R ≥ a. We found this pattern post hoc and tested it in two preregistered confirmatory studies with 43 new conditions: the complete bound held in all of them, and the simplified bound failed there on only 3 facts, each with R ≥ a. A screen that decided for a whole condition whether to use the simplified bound failed on new starting states. First-order models of fine-tuning can thus bound forgetting on the facts whose response coefficients change by less than their distance to the boundary.