The Measure of Deception: An Analysis of Data Forging in Machine Unlearning

arXiv:2509.05865v2 Announce Type: replace Abstract: Motivated by privacy regulations and the need to mitigate the effects of harmful data, machine unlearning seeks to modify trained models so that they effectively ``forget'' designated data. A key challenge in verifying unlearning is \emph{forging} -- adversarially crafting data that mimics the gradient of a target point, thereby creating the appearance of unlearning without actually removing information. To capture this phenomenon, we consider the collection of data points whose gradients approximate a target gradient within tolerance $\epsilon$ -- which we call an $\epsilon$-forging set -- and develop a framework for its analysis. For linear regression and one-layer neural networks, we show that the Lebesgue measure of this set is small. It scales on the order of $\epsilon$, and when $\epsilon$ is small enough, $\epsilon^d$. More generally, under mild regularity assumptions, we prove that the forging set measure decays as $\epsilon^{(d-r)/2}$, where $d$ is the data dimension and $r