Statistical Unlearning of Distributions: A Hypothesis Testing Approach
Authors: Aaradhya Pandey, Sanjeev Kulkarni
Organizations: Princeton University
Abstract
Machine learning systems increasingly face requirements to forget not only individual data points, but entire domains of information, such as toxic language, copyrighted corpora, or demographic biases. This raises a fundamental dilemma of statistical-computational tradeoffs: removing all samples from an unwanted domain may be computationally prohibitive, while randomly removing a subset may not provide distribution-level statistical guarantees. We propose a statistical framework for distributional unlearning, in which domains are modeled as probability distributions, and the goal is to remove a carefully chosen subset of samples that reduces the effect of an unwanted distribution while preserving performance on a desired one. We formalize this using a hypothesis test of the edited data with the desired and unwanted domains, leading to an interpretable and robust criterion for selecting samples to remove. Within this statistical framework, we characterize the fundamental region of the allowable edited data distributions and the removal-preservation Pareto frontier for a broad class of distribution families. This includes parametric families such as shifted Gaussians of arbitrary dimension, a one-dimensional location family with log-concave noise, and the one-dimensional Poisson family. It also includes nonparametric families such as the Gaussian white noise model, a canonical model for nonparametric regression. We prove composition rules that describe how distributional unlearning behaves across multimodal unwanted domains, and introduce a central-limit behavior for the removal-preservation baselines when composing a large number of such families. Finally, we provide finite sample guarantees by providing Pareto frontiers for some selection algorithms, and observe an information-computation gap.
This paper proposes a paradigm shift linking machine unlearning directly to the structure of the data distributions rather than a mere update of the neural network parameters. We show that inferring these distributions with precision enables distilling the exact unlearning signal induced by the modeling. Theoretical bounds on the Kullback-Leibler divergence from the ideal retrained model to our unlearned model, under verifiable admissibility criterion, reveal the soundness of our framework. This method is experimentally validated over three forgetting scenarios as reaching the closest classifier to the ideal retrained model when compared to competitors.
Machine unlearning is motivated by legal and user-facing requirements to remove the influence of individuals' data from trained models, such as the right to be forgotten. Prior work has developed algorithms and error bounds for unlearning in smooth strongly convex stochastic optimization, but the fundamental statistical cost of unlearning has remained unclear. We nearly resolve this problem by proving upper and lower bounds on the excess population risk of approximate ε-unlearning; our bounds are tight up to a condition-number factor. For mean estimation over the unit ball, our upper and lower bounds match. The optimal rate is the usual statistical error plus an unlearning penalty that interpolates between the retraining-from-scratch rate and an exponentially smaller term as ε/d grows, where d is the dimension of the model. In particular, when ε≫d, our ε-unlearning algorithm offers an exponential accuracy improvement over retraining the model from scratch and differentially private baselines. On the other hand, when ε≤d, retraining from scratch is optimal.
How can we effectively remove or ``unlearn'' undesirable information, such as specific features or the influence of individual data points, from a learning outcome while minimizing utility loss and ensuring rigorous guarantees? We introduce a unified mathematical framework based on information-theoretic regularization to address both data-point unlearning and feature unlearning. For data-point unlearning, we introduce the \emph{Marginal Unlearning Principle}, an auditable and provable framework. Moreover, we provide an information-theoretic unlearning definition based on the proposed principle and provable guarantees on sufficiency and necessity of marginal unlearning. We then show that the proposed framework provides a natural solution to the marginal unlearning problem and yields auditable high-probability marginal-unlearning guarantees. For feature unlearning, the framework applies to deep learning with flexible training objectives. By combining flexibility in learning objectives with simplicity in regularization design, our approach is highly adaptable and practical for a wide range of machine learning and AI applications. From a mathematical perspective, we provide a unified analytic solution to the optimal feature unlearning problem with a variety of information-theoretic training objectives. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. Numerical simulations support our theoretical findings.