On Optimization Complexity of Second-Order Certified Unlearning

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 $(\varepsilon, δ)$-unlearning; our bounds are tight up to a condition-number factor. For mean estimation over the unit ball, our upper and lower bounds match. In fact, our algorithm achieves $\varepsilon$-unlearning, which implies a notable separation between differential privacy and unlearning: $(\varepsilon, δ)$-unlearning has no statistical advantage over pure $\varepsilon$-unlearning. The optimal rate is the usual sampling error plus an unlearning penalty that interpolates between the retraining from scratch rate and an exponentially smaller term as $\varepsilon/d$ grows, where $d$ is the dimension of the model. The retraining penalty dominates the sampling error for large unlearning requests. In particular, retraining from scratch is information theoretically optimal up to $\varepsilon \lesssim d$. On the other hand, for $\varepsilon \gg d$ and large unlearning requests, our $\varepsilon$-unlearning algorithm offers an exponential accuracy improvement over retraining the model from scratch and differentially private baselines.

We argue that current definitions of machine unlearning are underspecified for second-order optimizers. We compare first-order and second-order learners for their ability to handle the data deletion task with varying degrees of eigendecomposition to mimic the loss model memory. While both first and second-order methods realign with the ideal counterfactul in terms of performance and gradient, the second-order optimizer shows significant volatility in the optimizer state. This indicates residual information, supposedly deleted, that isn't detectable by first-order analysis. Various eigendecay treatments show that stability and information loss is regained only under controlled state pertubation where geometric information (or memory) is erased.

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.