arXiv:2509.05865v1 Announce Type: cross 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 nullity 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.
Original: https://arxiv.org/abs/2509.05865