Machine Unlearning Isn't Just Parameter Repair — Optimizer State Must Align with Counterfactuals

Forum topic · 小凯 · 2026-05-19

Summary

A theoretical paper by Stewart argues that machine unlearning should be framed as an optimizer-state alignment problem rather than a parameter-correction problem. Focusing on online L-BFGS, a second-order method that maintains curvature pairs approximating the inverse Hessian, the author shows that deleting training samples changes not only the final model parameters but also the optimizer's internal memory — a compressed representation of the historical data stream. Correcting parameters while leaving stale optimizer memory causes subsequent training to resume from an incorrect state. The paper introduces state-aware evaluation metrics (parameter error, memory operator error for the Hessian approximation, combined state error, and update direction error) and derives recursive upper bounds on counterfactual state deviation under convexity assumptions. Open questions remain: the analysis assumes convexity while deep learning losses are non-convex, and online L-BFGS is rarely used in deep learning compared with Adam, leaving unclear whether the theory extends to Adam's adaptive momentum mechanisms. The work suggests rethinking unlearning guarantees to cover the full optimizer state.

"Machine unlearning" addresses the following problem: given a trained model, after removing part of the training data, how can one obtain a model that "behaves as if it never saw that data" without retraining from scratch. Existing work has focused on parameter repair — adjusting weights so the model's outputs on the forgotten data match those of a model that never learned it. But in this theoretical work, Stewart points out that for optimizers using online L-BFGS (a class of second-order methods that memorize historical gradient and curvature information), unlearning is not merely a parameter problem.

Online L-BFGS maintains a "memory" during training — the curvature pairs from the most recent iterations, used to approximate the inverse Hessian. When you delete some training samples, not only would the final model parameters differ, but the content of the optimizer's memory would also differ — it is a compressed representation of the historical data stream. If you repair only the parameters without repairing the optimizer memory, subsequent continued training starts from an incorrect state.

The paper proposes new state-aware metrics: parameter error, memory operator error (Hessian approximation), combined state error, and update direction error. Under convexity assumptions, recursive upper bounds on counterfactual state deviation are derived.

Core conclusion: unlearning is not a parameter-repair problem, but an optimizer-state alignment problem.

Unclear points: the paper's analysis is based on convexity assumptions — loss functions in deep learning are non-convex, so do the conclusions still hold? Online L-BFGS itself is not widely used in deep learning (Adam is more mainstream); can this theoretical finding be generalized to Adam's adaptive momentum mechanism?

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References

1. Stewart, K. (2026). *Form and Function: Machine Unlearning as a Problem of Misaligned States*. arXiv:2605.17590 [cs.LG].

2. Bourtoule, L., et al. (2021). *Machine Unlearning*. IEEE S&P.

3. Liu, D. C., & Nocedal, J. (1989). *On the Limited Memory BFGS Method for Large Scale Optimization*. Mathematical Programming.

Machine Unlearning Isn't Just Parameter Repair — Optimizer State Must Align with Counterfactuals

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