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arxiv: 2505.10859 · v2 · submitted 2025-05-16 · 💻 cs.AI

Exploring Nonlinear Pathway in Parameter Space for Machine Unlearning

Yingdan Shi , Ren Wang This is my paper

Pith reviewed 2026-05-22 15:22 UTC · model grok-4.3

classification 💻 cs.AI
keywords machine unlearningmode connectivitynonlinear pathwayparameter spacetask arithmeticimage classificationloss landscapeprivacy
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The pith

Mode connectivity finds nonlinear paths in parameter space that improve machine unlearning over linear updates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Mode Connectivity Unlearning (MCU) to remove targeted training data from models more effectively than current techniques. Linear parameter updates in existing methods often leave weights entangled, limiting how cleanly information can be forgotten. MCU instead traces curved, low-loss routes through parameter space using mode connectivity to locate better unlearning trajectories. A parameter mask and an adaptive penalty term further sharpen forgetting while preserving accuracy and cutting computation. The approach plugs into other unlearning methods, yields multiple valid unlearned models along each path, and shows stronger results on image classification tasks.

Core claim

By employing mode connectivity, MCU identifies nonlinear pathways in parameter space that sidestep the weight entanglement of linear task arithmetic, and when augmented with a parameter mask and adaptive penalty adjustment, it delivers stronger forgetting performance, lower overhead, and a continuum of unlearning solutions rather than a single model.

What carries the argument

Mode connectivity, which locates continuous low-loss paths connecting different parameter configurations to define a nonlinear unlearning trajectory.

Load-bearing premise

Mode connectivity in the loss landscape reliably supplies a nonlinear pathway that avoids weight entanglement and outperforms linear task arithmetic for unlearning.

What would settle it

An experiment in which a simple linear update achieves equal or higher forgetting metrics and retained accuracy than MCU on the same image-classification benchmarks would disprove the claimed advantage.

Figures

Figures reproduced from arXiv: 2505.10859 by Ren Wang, Yingdan Shi.

Figure 1
Figure 1. Figure 1: Overview of our proposed MCU framework. (a) Identify [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The accuracy on Df , Dr, and Dt across different retain￾ing data proportions used in our training process. It shows that all accuracy performance remains stable even with scarce retaining data. The intermediate control model θc in Eq. 1 serves to guide the shape of the Bezier curve. By optimizing this ´ control model, we can influence the trajectory between θo and θp. However, simply constructing a smooth … view at source ↗
Figure 3
Figure 3. Figure 3: The efficiency and effectiveness of our parameter mask. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Effective unlearning region on MCUβ. The marker ★ highlights the position with the minimum average gap from RT, with the accompanying numerical value indicating the exact av￾erage accuracy gap of Df , Dr and Dt (and Dtf for class-wise forgetting). The dotted line represents the RT method’s accuracy, serving as a reference. The shaded gray area denotes the effective unlearning region, where models achieve b… view at source ↗
Figure 5
Figure 5. Figure 5: Ablation study for fixed β on MCU. Overall, increasing β effectively enhances the unlearning effect but damages retaining predictive performance, while decreasing β weakens the ability of the pathway to forget data. 0.0 0.25 0.5 0.75 1.0 t 8084889296 100 Accuracy(%) 0.85 Df Dr Dt Effective Region 0.0 0.25 0.5 0.75 1.0 t 80 84 88 92 96 100 Accuracy(%) 1.20 (a) k = 0.1 0.0 0.25 0.5 0.75 1.0 t 80 84 88 92 96 … view at source ↗
Figure 6
Figure 6. Figure 6: Ablation study for k on MCU. As k increases, the accuracy of Df , Dr, and Dt drops significantly, whereas decreasing k results in minimal accuracy changes along the pathway. k = 0.5 is the most balanced choice and is set as our default configuration. dom data forgetting and class-wise scenarios. It validates that MCUβ not only identifies a single effective unlearn￾ing model but also discovers a substantial… view at source ↗
Figure 7
Figure 7. Figure 7: Ablation study for kr on MCU. When kr = 0, we preserve all parameters important to retaining data, leading to a noticeable drop in Dr accuracy during the unlearning process. 0.0 0.25 0.5 0.75 1.0 t 8084889296 100 Accuracy(%) 0.85 Df Dr Dt Effective Region 0.0 0.25 0.5 0.75 1.0 t 80 84 88 92 96 100 Accuracy(%) 0.62 (a) Under-forgetting 0.0 0.25 0.5 0.75 1.0 t 80 84 88 92 96 100 Accuracy(%) 0.43 (b) Over-for… view at source ↗
Figure 8
Figure 8. Figure 8: Effectiveness of MCUβ across both under-forgetting and over-forgetting pre-unlearning model θp. in a reduced effective region. This suggests that while a larger β improves forgetting quality, it comes at the cost of model utility. Clearly, β = 0.2 offers the best balance between maintaining an optimal average accuracy gap and preserving a wider effective region. Nonetheless, choosing a larger β can still b… view at source ↗
Figure 9
Figure 9. Figure 9: Performance with different proportions of retaining data in pathway searching process. The results show that MCU [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Machine Unlearning (MU) aims to remove the information of specific training data from a trained model, ensuring compliance with privacy regulations and user requests. While one line of existing MU methods relies on linear parameter updates via task arithmetic, they suffer from weight entanglement. In this work, we propose a novel MU framework called Mode Connectivity Unlearning (MCU) that leverages mode connectivity to find an unlearning pathway in a nonlinear manner. To further enhance performance and efficiency, we introduce a parameter mask strategy that not only improves unlearning effectiveness but also reduces computational overhead. Moreover, we propose an adaptive adjustment strategy for our unlearning penalty coefficient to adaptively balance forgetting quality and predictive performance during training, eliminating the need for empirical hyperparameter tuning. Unlike traditional MU methods that identify only a single unlearning model, MCU uncovers a spectrum of unlearning models along the pathway. Overall, MCU serves as a plug-and-play framework that seamlessly integrates with any existing MU methods, consistently improving unlearning efficacy. Extensive experiments on the image classification task demonstrate that MCU achieves superior performance. The codes are available at https://github.com/TIML-Group/Mode-Connectivity-Unlearning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes Mode Connectivity Unlearning (MCU), a framework that uses mode connectivity to discover nonlinear pathways in parameter space for machine unlearning. It augments this with a parameter mask strategy to improve forgetting while reducing overhead and an adaptive adjustment mechanism for the unlearning penalty coefficient that eliminates manual hyperparameter tuning. The approach is presented as plug-and-play, integrable with arbitrary existing MU methods, and capable of producing a spectrum of unlearning models along the connectivity path rather than a single point. Experiments on image classification tasks are claimed to show superior unlearning efficacy compared to prior linear task-arithmetic baselines.

Significance. If the experimental outcomes and the claimed generality hold, the work would provide a practical mechanism for enhancing a broad class of machine unlearning algorithms by exploiting nonlinear low-loss curves instead of linear interpolations, potentially mitigating weight entanglement while preserving retain-set performance. The open release of code is a clear strength for reproducibility.

major comments (2)
  1. [§3] §3 (Method), mode-connectivity construction: the central claim that the discovered nonlinear path selectively suppresses forget-set influence while avoiding new entanglements rests on the premise that the path is shaped by the unlearning objective; however, if the connectivity search follows the standard low-loss curve procedure without explicit unlearning-aware regularization along the path, the reported gains could be attributable to the added mask or adaptive penalty rather than nonlinearity itself. A direct ablation comparing the full MCU path against linear interpolation on the identical loss landscape is required to isolate the contribution.
  2. [§4] §4 (Experiments) and abstract: the assertion that MCU 'consistently improving unlearning efficacy' and 'achieves superior performance' on image classification is load-bearing for the plug-and-play claim, yet the manuscript provides no quantitative tables, error bars, or statistical significance tests for the improvement over base MU methods; without these, the generality across arbitrary base methods cannot be evaluated.
minor comments (2)
  1. [§3.1] Notation for the adaptive coefficient and mask should be introduced with explicit equations rather than descriptive text to improve traceability.
  2. [§4] The spectrum of models along the path is an interesting feature; a figure showing retain/forget metrics as a function of path parameter would clarify the practical utility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments, which help clarify the contributions and strengthen the presentation of our work on Mode Connectivity Unlearning (MCU). We address each major comment point by point below, providing clarifications based on the manuscript and outlining revisions where appropriate to better support our claims.

read point-by-point responses

  1. Referee: [§3] §3 (Method), mode-connectivity construction: the central claim that the discovered nonlinear path selectively suppresses forget-set influence while avoiding new entanglements rests on the premise that the path is shaped by the unlearning objective; however, if the connectivity search follows the standard low-loss curve procedure without explicit unlearning-aware regularization along the path, the reported gains could be attributable to the added mask or adaptive penalty rather than nonlinearity itself. A direct ablation comparing the full MCU path against linear interpolation on the identical loss landscape is required to isolate the contribution.

    Authors: We appreciate this precise observation on the construction of the connectivity path. In the MCU framework, the nonlinear pathway is discovered by optimizing a composite objective that explicitly includes the unlearning loss terms (forget-set suppression combined with retain-set preservation), the parameter mask, and the adaptive penalty coefficient. This formulation ensures the path search is guided by the unlearning objective rather than following a purely standard low-loss curve between arbitrary minima. That said, we agree that an explicit ablation isolating the nonlinearity—by comparing the full MCU nonlinear path against linear interpolation performed on the identical loss landscape—is the cleanest way to attribute gains specifically to the nonlinear structure. We have performed this ablation study and will incorporate the results, along with a clearer description of the unlearning-aware objective, into the revised manuscript. revision: yes

  2. Referee: [§4] §4 (Experiments) and abstract: the assertion that MCU 'consistently improving unlearning efficacy' and 'achieves superior performance' on image classification is load-bearing for the plug-and-play claim, yet the manuscript provides no quantitative tables, error bars, or statistical significance tests for the improvement over base MU methods; without these, the generality across arbitrary base methods cannot be evaluated.

    Authors: We acknowledge that the current experimental section relies primarily on figures to illustrate performance trends and that this is insufficient to rigorously substantiate the claims of consistent improvement and plug-and-play generality. To address this, the revised manuscript will include comprehensive quantitative tables reporting mean metrics (e.g., forget accuracy, retain accuracy) with standard deviations computed over multiple independent runs, as well as statistical significance tests (such as paired t-tests with p-values) comparing MCU-augmented variants against their base MU methods across different unlearning techniques and datasets. These additions will directly support the generality claim and allow readers to evaluate the magnitude and reliability of the reported gains. revision: yes

Circularity Check

0 steps flagged

No circularity: MCU components are independently defined and externally evaluated

full rationale

The derivation introduces mode connectivity for a nonlinear pathway, a parameter mask, and an adaptive penalty coefficient as distinct additions to existing MU methods. These are motivated by standard mode-connectivity literature and evaluated via external benchmarks on image classification tasks rather than being defined in terms of the target unlearning metrics themselves. No equation reduces a reported improvement to a fitted input or self-citation chain; the plug-and-play claim rests on empirical integration results, not tautological construction. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of useful mode-connectivity paths in trained model loss landscapes and on the effectiveness of the proposed mask and adaptive coefficient; these are treated as empirical discoveries rather than derived from first principles.

free parameters (1)
  • unlearning penalty coefficient
    The coefficient is adjusted adaptively during training; its initial value and adaptation rule constitute a tunable element whose specific schedule is not fixed by prior theory.
axioms (1)
  • domain assumption Mode connectivity exists between the original model and an unlearned model in the parameter space.
    Invoked in the motivation for moving from linear to nonlinear pathways; no proof or citation of a general theorem is given in the abstract.

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