🤖 machine learning

Shape of Memory: a Geometric Analysis of Machine Unlearning in Second-Order Optimizers

The paper argues that current machine unlearning definitions are insufficient for second-order optimizers because, while they appear to achieve performance goals, they retain detectable residual information within their optimizer states that can only be fully erased through controlled geometric perturbations.

Original authors: Kennon Stewart

Published 2026-04-28

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Kennon Stewart

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are training a professional chef to cook a signature dish.

Most people think "learning" is just about the chef memorizing the final recipe (the parameters). If you want the chef to "unlearn" a specific ingredient—say, they accidentally learned to use salt when they should have used sugar—most current methods just tell the chef, "Hey, change the final recipe; don't use salt."

But this paper argues that learning is much deeper than just the recipe. It’s about the chef’s intuition (the second-order optimizer state).

The Core Problem: The "Muscle Memory" of Math

When a chef learns, they don't just learn the ingredients; they learn the rhythm of the kitchen. They learn how heavy the knife is, how much pressure to apply to the garlic, and how the heat of the stove reacts to different oils. This "rhythm" is the second-order state. It’s a complex, invisible map of how every action affects the next.

Current "unlearning" techniques only fix the recipe. They change the final output so the dish tastes right. But they forget to fix the chef's intuition.

The paper’s big discovery is this: Even if the chef produces a perfect, salt-free dish (the model performs well), their rhythm is still "salty." Their muscle memory is still shaped by that mistake. If you watch them work, you can see the "ghost" of the deleted information in the way they move their hands.

The "Ghost in the Machine" (Volatility)

The researchers looked at two types of learners:

The Simple Learner (First-Order): This is like a chef who only follows a checklist. If you tell them to stop using salt, they stop. They don't have much "intuition" to mess up. They recover quickly.
The Intuitive Learner (Second-Order): This is the high-level chef. They use "curvature" (the shape of the kitchen) to work faster. When you tell them to "unlearn" an ingredient, they can fix the recipe easily, but their internal geometry—their sense of rhythm—gets jittery and weird.

The researchers found that even when the "Intuitive Learner" starts making correct predictions again, their internal state shows "volatility." It’s like a person who has been told to forget an ex-partner; they might act perfectly normal in public, but their heart rate spikes or their hands shake whenever a certain song plays. The information is "gone" from the output, but it’s still vibrating in the system.

The Solution: "Geometric Erasure"

The paper suggests that if we want to truly unlearn something, we can't just change the recipe. We have to perform a "geometric perturbation."

In our kitchen analogy, this isn't just telling the chef "don't use salt." It’s like slightly rearranging the kitchen layout or changing the weight of the knives. By shaking up the chef's physical environment (the optimizer's state), you force them to break their old, "salty" muscle memory and build a brand-new rhythm from scratch.

Summary in a Nutshell

The Old Way: "Change the answer so the mistake isn't visible."
The Paper's Way: "The mistake is still hiding in the way the machine thinks. To truly forget, you have to shake up the machine's internal sense of direction, not just its final answer."

The takeaway: If we want true privacy (making sure data is truly deleted), we can't just look at what the AI says; we have to look at how the AI thinks.

Technical Summary: Shape of Memory

Title: Shape of Memory: a Geometric Analysis of Machine Unlearning in Second-Order Optimizers
Author: Kennon Stewart
Core Theme: Investigating how second-order optimization states (curvature information) retain "memory" of deleted data, even when standard performance metrics suggest successful unlearning.

1. The Problem: Underspecified Unlearning Definitions

Current machine unlearning research primarily focuses on first-order methods (like Gradient Descent), where "forgetting" is defined by the similarity of the model's parameters ( $w$ ) to a counterfactual model trained without the deleted data.

However, the author argues that this definition is insufficient for second-order optimizers (like Online Newton Step). These algorithms maintain an auxiliary internal state ( $A_t$ )—a matrix representing accumulated curvature and historical geometry. Even if the parameters ( $w$ ) appear to align with the ideal counterfactual, the internal geometry ( $A_t$ ) may still encode information about the deleted observations. This creates a "hidden" privacy risk where the model's learning dynamics (how it responds to future data) still reflect the deleted information.

2. Methodology

The study employs a controlled experimental framework comparing two learners in an Online Convex Optimization (OCO) setting under adversarial deletions:

Learners:
- Online Gradient Descent (OGD): A first-order baseline with no auxiliary state.
- Online Newton Step (ONS): A second-order learner that maintains a positive definite matrix $A_t$ to define a time-varying metric for updates.
Experimental Setup: A 2D linear convex learning problem over $T=400$ rounds. A deletion event occurs at $t=200$ , removing the 10 most recent observations. The study tests both stationary and drifting (non-stationary) data streams.
Interventions (Second-Order State Manipulation): To attempt "true" unlearning, the author applies spectral treatments to the ONS state matrix $A_t$ $A_{t}$ :
- Partial Reset: Subtracting a constant $\alpha$ from the eigenvalues.
- Eigenvalue Decay: Rescaling the spectrum by a factor $\beta$ .
Metrics:
- Regret ( $R_T$ ): Cumulative loss relative to the best fixed parameter.
- Tracking Error ( $E_t$ ): Euclidean distance between the observed and counterfactual parameters.
- Parameter Shock ( $\Delta_\tau$ ): Instantaneous deviation at the moment of deletion.
- Spectral Diagnostics: Trace, condition number, and cosine alignment of the optimizer state.

3. Key Results

The "Performance Paradox": ONS shows nearly immediate recovery in terms of regret (the loss scale returns to normal almost instantly). However, this is deceptive. While the model performs well, its tracking error (the distance from the ideal counterfactual path) remains significant and does not converge to zero.
Geometric Hysteresis: Deletion in second-order methods manifests as "geometric hysteresis." The internal state $A_t$ undergoes a discontinuity in its condition number and orientation (cosine similarity) that persists long after the parameter values have stabilized.
Ineffectiveness of Mild Interventions: Mild spectral resets do not align the state with the counterfactual. In fact, the models with the lightest interventions showed the highest "cosine volatility," meaning they retained the most residual information in their geometry.
The Role of Aggressive Erasure: Stability and information loss are only regained under controlled state perturbation (aggressive eigendecay). Essentially, to truly forget, the optimizer must undergo a significant geometric "reset" that erases the stored curvature.

4. Key Contributions

Identification of a New Failure Mode: The paper identifies that "parameter-centric" unlearning is an incomplete metric for stateful optimizers.
Characterization of Second-Order Memory: It demonstrates that information persists as volatility in the optimizer state rather than error in the parameter vector.
Geometric Framework: It shifts the discourse of unlearning from simple parameter matching to state alignment in higher-order information spaces.

5. Significance and Future Directions

The paper concludes that for stateful, second-order learners, unlearning is a state alignment problem, not a performance problem. This has profound implications for privacy: a model might pass a standard "performance-based" unlearning audit while still leaking sensitive data through its internal curvature geometry.

Future Research Directions:

Developing certified unlearning for online settings that provides guarantees for both parameter similarity and state alignment.
Extending this analysis to more complex quasi-Newton methods like L-BFGS and Hessian Vector Products.
Framing unlearning within the context of Markov Decision Processes (MDPs) and regret control policies.