Less Noise, Same Certificate: Retain Sensitivity for Unlearning

Abstract

Certified machine unlearning aims to provably remove the influence of a deletion set $U$ from a model trained on a dataset $S$, by producing an unlearned output that is statistically indistinguishable from retraining on the retain set $R := S\setminus U$. Many existing certified unlearning methods adapt techniques from Differential Privacy (DP) and add noise calibrated to global sensitivity, i.e., the worst-case output change over all adjacent datasets. We show that this DP-style calibration is often overly conservative: unlike DP, unlearning requires no indistinguishability across different retain sets, only between each deletion output and retraining on the same retain set $R$. Motivated by this distinction, we define retain sensitivity as the worst-case output change over deletions $U'$ from $R\cup U'$ while keeping $R$ fixed. While insufficient for DP, retain sensitivity is exactly sufficient for unlearning, allowing for the same certificates with less noise. We validate these reductions in noise theoretically and empirically across several problems, including the weight of minimum spanning trees, PCA, and ERM. Finally, we refine the analysis of two widely used certified unlearning algorithms through the lens of retain sensitivity, leveraging the regularity induced by $R$ to further reduce noise and improve utility.