SAUL: Sharpness-Aware Augmented-Lagrangian Unlearning

• URL: http://arxiv.org/abs/2608.16249v1
• Date: Mon, 17 Aug 2026 08:23:05 GMT
• Title: SAUL: Sharpness-Aware Augmented-Lagrangian Unlearning
• Abstract summary: We propose SAUL (Sharpness-Aware Augmented-Lagrangian Unlearning), which formulates unlearning as a constrained minimization problem.<n>At its core, SAUL formulates forgetting as an explicit constraint with a prescribed satisfaction criterion.<n>We show that applying the augmented-Lagrangian controller as a drop-in modifier to representative baselines improves their post-forgetting utility.
• Score: 4.268049805078337
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Machine unlearning in Large Language Models (LLMs) faces a critical trade-off between erasing target knowledge and preserving general utility. We propose SAUL (Sharpness-Aware Augmented-Lagrangian Unlearning), which formulates unlearning as a constrained minimization problem following the principle of "forget enough, but no more than necessary." At its core, SAUL formulates forgetting as an explicit constraint with a prescribed satisfaction criterion, whereas prior unlearning methods typically specify the desired level of forgetting implicitly through optimization objectives. An augmented Lagrangian controller adaptively adjusts forget-side pressure according to constraint violation and can eventually deactivate the forget-side update as the prescribed criterion remains satisfied. Sharpness-aware updates on both retain and forget objectives, together with a dual-optimizer design that maintains role-separated states, further stabilize the resulting unlearning dynamics. We evaluate SAUL on the TOFU, WMDP, and MUSE benchmarks, demonstrating favorable forgetting-utility trade-offs over representative sharpness- and perturbation-based baselines under benchmark-specific forgetting criteria. Beyond the complete SAUL framework, we further show on TOFU that applying the augmented-Lagrangian controller as a drop-in modifier to representative baselines improves their post-forgetting utility, demonstrating the practical value of explicit forgetting control.

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