Fisher-LoRA: Per-Edit Rank-Adaptive Knowledge Editing via Causal Fisher Information

Knowledge editing of large language models patches individual factual associations through low-rank weight updates, and each such editor must commit in advance to how many weight-space directions the update may occupy. However, existing editors do not answer how many a particular edit fundamentally requires: each fixes a footprint decoupled from the edit, ROME imposing rank one, MELO rank eight, and ELDER rank thirty-two, so they over-pay on easy edits and silently fail on hard ones. We propose Fisher-LoRA, a per-edit rank-adaptive editor that sets LoRA rank from a measurement on the model rather than a method-level constant: the eigenrank of an attribution-patching estimate of the Causal Fisher Information Matrix on the edit's causal subspace, obtained from one backward pass on the unedited model and read as an estimate ${\hat{r}}^{\star }(e)$ of the edit's minimum rank. A Cram\'er-Rao-style heuristic motivates an energy estimate for the Fisher-weighted update along this subspace, and throughout the evaluation we observe an empirical rank floor at $r^{\star }(e)$ less a slack. The estimate is diagonal in the component basis, so its spectrum is read by sorting component scores and the LoRA factors start at its top eigenvectors. On COUNTERFACT and zsRE across three backbones, Fisher-LoRA reaches $96.4\%$ reliability and $91.3\%$ locality on LLaMA-2-7B, $12.1$ locality points above ELDER at an average rank of $11.4$ against its fixed $32$, cuts ELDER's hard-slice failure rate from $15.8\%$ to $7.6\%$, edits at $1.9\times$ lower cost, and predicts which edits defeat ROME, MEMIT and ELDER at Spearman $\rho =0.74$, and at $\rho =0.71$ when the joint failure also covers the memory editors GRACE and WISE. Edit difficulty is, to first order, a property of the (model, edit) pair, not of the editor, and readable before any edit is attempted.