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

2025-09-06

AI Summary

Machine unlearning faces a critical challenge from adversarial data fabrication—attackers can construct gradient-similar synthetic samples that induce the *appearance* of forgetting target data, while preserving underlying information. Method: This paper introduces, for the first time, the concept of an ε-forgery set and establishes a measure-theoretic detection framework. Leveraging gradient analysis, regularity assumptions, and smooth loss function theory, our approach applies to linear regression, single-layer neural networks, and mini-batch SGD. Contribution/Results: We prove that the Lebesgue measure of the ε-forgery set decays at rate ε^(d−r)/2, revealing an intrinsic limitation on adversarial forgery under non-degenerate data distributions in high dimensions. Consequently, the probability that random sampling hits a forged point vanishes asymptotically—yielding the first theoretically grounded, computationally feasible criterion for verifying genuine unlearning.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $ε$ -- which we call an $ε$-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 $ε$, and when $ε$ is small enough, $ε^d$. More generally, under mild regularity assumptions, we prove that the forging set measure decays as $ε^{(d-r)/2}$, 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.

Problem

• Detecting adversarial data forging in machine unlearning verification
• Analyzing measure of ε-forging sets under different model architectures
• Establishing probability bounds for detecting false unlearning claims

Innovation

• Analyzes epsilon-forging sets for unlearning verification
• Measures forging set decay under regularity assumptions
• Establishes vanishing probability for random forging sampling