Exact Unlearning from Proxies Induces Closeness Guarantees on Approximate 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.

Unlearning in diffusion models aims to remove undesirable data or concepts while preserving the utility of pretrained models -- two fundamentally conflicting objectives. We propose a principled constrained optimization framework that formulates unlearning as minimizing the deviation from a pretrained model, subject to explicit separation constraints from the unlearning distributions. Specifically, we formulate three constrained optimization problems based on reverse and forward KL divergences, and likelihood constraints. The first two generalize existing approaches for concept and data unlearning, while the third offers a novel and natural formulation for unlearning. Despite the nonconvexity of the KL constraints, we establish strong duality for all three problems, enabling us to explicitly characterize their optimal solutions as unlearning targets and develop primal-dual algorithms for each formulation. Experimental results demonstrate that our KL-constrained approach achieves superior retention-unlearning tradeoffs compared to weight-based baselines for concept and data unlearning, and that our likelihood-based approach matches unlearning effectiveness while better preserving retained concepts compared to baselines.

Machine unlearning aims to remove the influence of selected training examples without full retraining. Standard evaluations often summarize unlearning quality with aggregate metrics, such as accuracy- and forgetting-based scores, which can hide localized failures. We study this failure mode at the example level by comparing the predictions of an unlearned model to those of the model retrained after deletion. We show that this pointwise discrepancy can be highly non-uniform: for gradient-ascent and random-labeling methods, with and without retain-set fine-tuning, it grows with geometric proximity to the forget set. We call this phenomenon localized collateral forgetting. Our analysis identifies a mechanism behind the effect: surrogate targets used during unlearning can be inconsistent with the local prediction structure induced by retraining, and this inconsistency propagates through shared representations to nearby examples. Motivated by this mechanism, we propose Local Teacher Distillation, a simple mitigation strategy that replaces random targets with soft labels from a small teacher trained only on retained neighbors of the forget set. On CIFAR-100 partial-class deletion, this local teacher brings the unlearned model substantially closer to retraining, especially near the forget set, while maintaining competitive aggregate unlearning metrics.