Counterfactual Certification for Sequential Unlearning with Long-Run Performance Guarantees

Certified machine unlearning aims to provably remove the influence of a deletion set $U$ from a model trained on a dataset $S$, by producing an unlearned output that is statistically indistinguishable from retraining on the retain set $R := S\setminus U$. Many existing certified unlearning methods adapt techniques from Differential Privacy (DP) and add noise calibrated to global sensitivity, i.e., the worst-case output change over all adjacent datasets. We show that this DP-style calibration is often overly conservative: unlike DP, unlearning requires no indistinguishability across different retain sets, only between each deletion output and retraining on the same retain set $R$. Motivated by this distinction, we define retain sensitivity as the worst-case output change over deletions $U'$ from $R\cup U'$ while keeping $R$ fixed. While insufficient for DP, retain sensitivity is exactly sufficient for unlearning, allowing for the same certificates with less noise. We validate these reductions in noise theoretically and empirically across several problems, including the weight of minimum spanning trees, PCA, and ERM. Finally, we refine the analysis of two widely used certified unlearning algorithms through the lens of retain sensitivity, leveraging the regularity induced by $R$ to further reduce noise and improve utility.

Machine unlearning updates a trained model after records are deleted, aiming to match exact retraining without repeating the full training procedure. We develop Gaussian mechanisms for Newton updates under sequential deletion requests. Using Gaussian differential privacy (GDP) and its adaptive composition rule, we show that the full sequence of released models is statistically difficult to distinguish from matched exact retraining. To calibrate these mechanisms for empirical risk minimization, we derive upper bounds on the error of the Newton approximation relative to exact retraining and on how this error changes after each deletion batch. Independent Gaussian noise is calibrated using bounds on the full residual at each release, whereas Gaussian random walk noise uses smaller bounds on residual increments. These bounds yield allocations minimizing the worst-case maximum noise variance across releases under the resulting GDP certification constraints. With count-based bounds, the random walk asymptotically matches the worst-case variance of a single release at deletion cap $M$, while independent noise incurs an additional factor of order $M$. Set-based bounds can reduce the noise variances by using gradients and Hessians of the deleted records. For singleton deletion, we further show that count-based independent noise, count-based random walk noise, and set-based independent noise are minimax among fixed Gaussian covariances under their respective residual or increment bounds. With set-based bounds, allowing variances to adapt to deleted records can improve on every fixed covariance by a factor of order $(\log M)^2$ on some data sequences. The residual and noise bounds also yield parameter and predictive consistency relative to exact retraining, uniformly over deletion policies. Simulations and a credit default data analysis evaluate bounds, noise variances, and estimation errors.

Machine unlearning is needed when data must be removed because of deletion requests, outdated records, or data-quality concerns, while retraining from scratch can be costly. Certified machine unlearning methods provide mathematical guarantees, while deployed systems release concrete finite-precision artifacts produced by software. To bridge the gap between mathematical guarantees and practical deployment, we introduce Executable Release Certification (ExecCert), a release-time layer that certifies the candidate artifact considered for release. ExecCert either closes a method's native certificate for the executed candidate or applies Retraining-Reference Release Verification (RRV) to certify fidelity to current retain-set retraining. Sequential deletion makes the latter nontrivial because the exact retain-set reference and the stored numerical state evolve separately. For frozen representations with a mutable ridge head, we develop an incremental realization of RRV that maintains certified evidence across deletion requests rather than reconstructing it at each release. On four published unlearning implementations, ExecCert preserves valid certificates, changes release decisions, tightens conservative bounds, and identifies the retraining-reference fidelity supported by concrete outputs. In sequential-service experiments, RRV eliminates false releases caused by stored-equation verification while closely tracking realized error, and incremental certification remains cheaper than both fresh and maintained verified-factor alternatives once release checks become sufficiently frequent.