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 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<d is the dimension of vector space of right singular vectors corresponding to small'' singular values of a variation matrix defined by the model gradients. Extensions to batch SGD and almost-everywhere smooth loss functions yield the same asymptotic scaling. In addition, we establish probability bounds showing that, under non-degenerate data distributions, the likelihood of randomly sampling a forging point is vanishingly small. These results provide evidence that adversarial forging is fundamentally limited and that false unlearning claims can, in principle, be detected.

Source: arXiv cs.LG | 2026-05-05