Relearning Resistance Under Adam: A Gradient-Entropy Certificate for Machine Unlearning

Relearning provides a direct test of whether machine unlearning has removed information or merely suppressed its immediate behavior. Relearning Convergence Delay ($\mathcal{R}\mathcal{C}\mathcal{D}$) quantifies this resistance by accumulating excess forget-set error along the relearning trajectory. Existing $\mathcal{R}\mathcal{C}\mathcal{D}$ theory focuses on gradient descent, despite the widespread use of adaptive optimizers such as Adam for training modern models. We develop a framework for analyzing $\mathcal{R}\mathcal{C}\mathcal{D}$ under Adam and derive an upper bound involving three quantities at the unlearning initialization: the forget-set loss gap, the local Hessian condition number, and Adam's second-moment condition number. Because large models are often fine-tuned by updating only a restricted subset of parameters, we further show that the expected certificate can be bounded in terms of the entropy and scale of the forget gradient. This analysis motivates Gradient-Entropy Maximization (GEM), a baseline-agnostic regularizer that promotes both gradient dispersion and magnitude. We establish an $O(1/ϵ)$ convergence rate to an $ϵ$-stationary point for the GEM objective and evaluate it empirically. Across classification, diffusion, and language-model unlearning, GEM can improve resistance to Adam relearning while largely preserving retain performance and forgetting quality.