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On the Plasticity Collapse in Continual Machine Unlearning
Yingdan Shi, Xiang Xu, Kaize Ding, Alfred O. Hero, Ren Wang
TL;DR
Continual unlearning exposes plasticity collapse, a fundamental limitation in which models progressively lose their ability to forget. Through theoretical analysis and experiments, the paper shows that sequential unlearning creates restrictive parameter subspaces, causing deteriorating forgetting and spontaneous re-memorization.
Problem
Continual unlearning can progressively impair models’ ability to forget, threatening the long-term reliability of sequential unlearning systems.
Method
The paper combines theoretical analysis of sequential unlearning dynamics with experiments across architectures, datasets, and image-classification unlearning methods.
Results
The analysis and experiments identify plasticity collapse as a pervasive phenomenon, with deteriorating forgetting quality and spontaneous re-memorization during sequential unlearning.
Takeaways & Limitations
Plasticity-preserving approaches are needed, while the Energy Ratio and Projection Coefficient can monitor impending collapse and provide regularization signals.
Takeaways & Limitations
The empirical analysis focuses on image classification, while broader contexts such as generative modeling and large language models remain more complex extensions.
Abstract
from arXiv · showhide
Machine unlearning enables deep neural networks to selectively remove the influence of specific data in response to privacy and regulatory requirements. While prior work largely studies single-shot unlearning, real-world systems must accommodate continual unlearning, where multiple unlearning requests occur sequentially over time. In this work, we identify a fundamental limitation of this setting: plasticity collapse, a progressive breakdown in a model's ability to effectively forget. Through theoretical analysis of continual unlearning dynamics, we show that continual unlearning operations accumulate geometric constraints in parameter space, leading to saturated subspaces that restrict future updates. This structural effect induces two distinct failure modes: (1) Forward failure -- diminishing forgetting quality for subsequent tasks, and (2) Backward failure -- spontaneous re-memorization of previously forgotten information. Extensive experiments across multiple architectures, datasets, and methods in image classification confirm that plasticity collapse is not an artifact of specific implementations, but a pervasive phenomenon inherent to continual unlearning. Our findings reveal a critical barrier to the long-term reliability of machine unlearning systems and motivate the development of plasticity-preserving unlearning algorithms. Our code is available at https://github.com/TIML-Group/Continual-Machine-Unlearning-Plasticity-Collapse
1 Introduction
The paper examines why sequential unlearning must remain reliable over time and identifies plasticity collapse as a structural limitation. Geometric constraints accumulate in parameter space, producing forward forgetting degradation and backward re-memorization.
- Motivation: Machine unlearning removes information encoded in model parameters to address privacy, regulatory, harmful-content, and bias concerns.This extends beyond deleting records from training datasets.
- Motivation: Sequential unlearning is needed because deployed models must honor multiple forgetting requests over time, not just one request.The paper frames this requirement in settings such as evolving content-moderation policies and recommendation systems.
- Core finding: Plasticity collapse occurs when successive unlearning operations accumulate geometric constraints, creating saturated subspaces that restrict future parameter updates.The paper presents this as a fundamental limitation of continual unlearning rather than a single implementation artifact.
- Core finding: Forward failure progressively reduces forgetting quality for subsequent tasks, whereas backward failure causes previously forgotten tasks to spontaneously re-emerge.These are the two failure modes attributed to plasticity collapse.
- Evidence: Experiments across diverse architectures, datasets, and image-classification unlearning methods indicate that plasticity collapse is pervasive in continual unlearning.The authors report that the phenomenon is not an artifact of specific implementations.
- Scope: The analysis focuses on image classification while relating its optimization-based mechanisms to broader sequential-unlearning contexts, including generative models and LLMs.The broader applicability is presented as a theoretical and diagnostic foundation, not as direct experimental validation.
2 Related Work
Prior work provides exact or approximate unlearning methods and studies related continual learning-forgetting settings, but mostly does not address purely sequential unlearning. This paper positions plasticity collapse as a distinct, theoretically grounded phenomenon in that setting.
- Unlearning methods: Exact unlearning methods provide certified guarantees through data partitioning and selective submodel retraining, but require substantial storage and computation.SISA is presented as an example of this approach.
- Unlearning methods: Approximate methods improve deployment efficiency through fine-tuning, parameter perturbation, or selective gradient manipulation, but have been studied mostly in single-shot settings.These methods trade formal guarantees for efficiency.
- Continual settings: Continual learning-forgetting methods interleave new training data with selective forgetting, which helps preserve model utility during the process.This mixed setting differs from purely sequential unlearning without new learning.
- Continual settings: Incremental unlearning studies sequential forgetting on a fixed model while retaining non-targeted knowledge, and this paper extends that setting by characterizing plasticity collapse.The paper focuses on successive forgetting operations without interleaved learning.
- Plasticity: Prior plasticity research concerns declining capacity to learn new information, whereas this work argues that unlearning plasticity has distinct optimization objectives and parameter-space constraints.The paper identifies unlearning plasticity as previously unexplored and theoretically distinct.
3 Plasticity Collapse in Continual Unlearning
Continual unlearning is modeled as sequential parameter updates whose shared geometric directions can become increasingly restrictive. This produces forward failure through deteriorating forgetting quality and backward failure through spontaneous re-memorization.
- Setup: Each request updates model parameters from θ_t−1 to θ_t using forget and optional retain losses, while methods differ in loss design.The general framework does not presuppose a particular unlearning method.
- Failure Modes: Plasticity collapse occurs when forgetting quality progressively deteriorates or earlier forgotten information re-emerges without explicit re-exposure.These are defined as forward and backward failure, respectively.
- Mechanism: Sequential unlearning accumulates geometric constraints because update directions share subspaces in parameter space.Composed update operators can repeatedly act along shared directions and amplify them geometrically.
- Mechanism: If shared-subspace operators expand exponentially, bounded parameter trajectories require suppressing motion or introducing cancellation within that subspace.Theorems state growth at least as (1+ρ)^|T| or (1+ρ)^(t−s+1), forcing stabilization mechanisms.
- Failure Modes: Suppressing shared-subspace motion reduces later forgetting ability, whereas opposing updates can partially undo earlier forgetting-induced parameter changes.The first mechanism yields forward failure; cancellation yields backward failure through re-memorization.
- Diagnostics: Energy Ratio measures update energy projected onto the shared subspace, and its decline toward 0 signals diminishing forgetting quality for subsequent tasks.The shared subspace is estimated from top-r right singular vectors of stacked update increments.
4 Experiments
Experiments on image-classification benchmarks show that continual unlearning progressively degrades forgetting quality or re-memorizes earlier forgotten tasks. Diagnostic analyses link these failures to shrinking parameter updates and geometric saturation of shared update subspaces.
- Experimental setup: The evaluation uses Tiny-ImageNet and CIFAR-100 with VGG-16-BN and PreResNet-110, testing five representative unlearning methods.The reported results average three independent trials; CIFAR-100 results appear in the appendix.
- Forward failure: Random data forgetting causes all five methods to exhibit increasing forgetting accuracy as tasks accumulate, with the final task averaging 33.2% above the single-task baseline.Higher forgetting accuracy indicates worse forgetting quality; NegGrad+ shows the most severe degradation, while MUNBa is also affected.
- Mechanistic diagnostics: The L2 norm of consecutive parameter updates decreases as tasks accumulate, indicating that the model becomes increasingly frozen in parameter space.For RL and SalUn, re-memorizing tasks have lower update norms than other tasks.
- Two failure modes: Under class-wise forgetting, FT, NegGrad+, and MUNBa show forward failure, whereas RL and SalUn lack pronounced forward failure but exhibit backward failure.Class-wise forgetting is near 0% in the single-task setting for most methods, so continual performance no longer preserves that strong baseline.
- Two failure modes: Historical-task heatmaps expose forward failure on current-task diagonals and backward failure in off-diagonal entries when previously forgotten tasks re-emerge.The heatmaps record forgetting accuracy for task F_i on later model θ_j, where j ≥ i.
- Mechanistic diagnostics: Energy-ratio decay toward zero tracks degraded forgetting accuracy, while near-zero and oscillating coefficient values diagnose saturated forgetting directions and antagonistic interference.These diagnostics provide theory-grounded evidence for forward and backward failure through geometric saturation of the shared subspace W.
5 Conclusion
The paper identifies loss of unlearning plasticity as a fundamental challenge in continual machine unlearning. It combines theoretical analysis with empirical validation and motivates plasticity-preserving approaches for deployment.
- Sequential unlearning causes deteriorating forgetting quality and spontaneous re-memorization.These findings expose severe limitations of methods designed for single-shot unlearning in continual settings.
- The findings motivate fundamentally new plasticity-preserving approaches for practical deployment.
- Energy Ratio and Projection Coefficient can monitor impending plasticity collapse online.The paper also proposes incorporating them as regularization signals into gradient-based unlearning objectives.
A Unlearning Methods
The paper compares several unlearning baselines that use fine-tuning, gradient manipulation, random labels, salient-parameter optimization, or bargaining-based gradient resolution.
- Finetuning updates the pretrained model using the retain dataset.
- NegGrad+ combines retain-data fine-tuning with gradient ascent on the forgetting data.
- RandomLabeling fine-tunes on forgetting data with randomly assigned labels to enforce forgetting.
- SalUn optimizes only salient parameters identified from randomly labeled forgetting data.
- MUNBa formulates unlearning as bargaining between forgetting and preservation players to resolve gradient conflicts.It uses a Nash bargaining closed-form solution to reach an optimal Pareto stationary point.
B Proof for Theorem 1
The proof establishes a one-step norm-growth bound and iterates it over invariant updates, yielding the theorem’s cumulative lower bound.
- Step 1 (one-step bound): Positive semidefiniteness and assumption (B2) yield a one-step norm bound for (I + M_j)w.The proof derives ||(I + M_j)w|| ≥ (1+ρ)||w||.
- Step 2 (iteration): Assumption (B1) keeps intermediate iterates inside W during repeated updates.
- Step 2 (iteration): Applying the one-step bound across T updates gives ||w_t+1|| ≥ (1+ρ)^|T| ||v||.Because w_t+1 = A_t:s v, this establishes Eq. 10.
C Proof for Theorem 3
The proof uses positive-semidefinite linearized operators and invariant subspaces to establish repeated norm growth, while the accompanying figures report Tiny-ImageNet accuracy evaluations.
- Proof: Under NTK linearization, M_j^F is symmetric positive semidefinite by construction.
- Proof: Assumption (N2) gives A_j restricted to W a smallest singular value of at least 1 + ρ.This yields ||A_j w|| ≥ (1+ρ)||w|| for every w in W.
- Proof: Invariance under (N1) permits repeated application of the one-step bound and produces Eq. 12.Eq. 13 follows by unrolling the affine recursion.
- Experiments: Figures 11 and 12 report retain accuracy for random-data and class-wise forgetting on Tiny-ImageNet with VGG-16-BN.
- Experiments: Figures 13 and 14 report test accuracy for the same two forgetting scenarios and model.
D.1 Accuracy Results
Across CIFAR-100 evaluations, retain and test accuracy generally remain stable, preserving model utility. However, several unlearning methods exhibit forward or backward failure depending on the forgetting scenario, while some methods fall outside the analysis because their single-shot efficacy is insufficient.
- Retain and test accuracy remain stable across most evaluated cases, preserving overall model utility.This stability satisfies a fundamental assumption of the theoretical framework.
- In random data forgetting, FT and NegGrad+ show significant utility degradation and are excluded from plasticity-collapse analysis.Their degradation violates the framework’s assumption that utility remains relatively stable.
- In random data forgetting, SalUn and RL demonstrate forward failure, whereas MUNBa does not exhibit plasticity collapse.
- In class-wise forgetting, NegGrad+ exhibits forward failure, while RL and SalUn demonstrate pronounced backward failure.The backward-failure pattern is illustrated by the historical-task heatmaps.
- Theorems 1 and 2 predict diminishing update magnitudes as the shared subspace W approaches saturation and increasingly suppresses updates toward W.The update magnitude is defined as u_t := θ_t − θ_{t−1}; empirical forward-failure cases consistently exhibit this reduction.
D.3 Diagnostic Quantities for Plasticity Collapse (ER and CO)
The diagnostic analysis estimates the shared update subspace with PCA and tracks ER_t and CO_t across continual unlearning stages. In forward-failure scenarios, ER_t progressively decays toward zero, supporting the predicted suppression mechanism.
- The shared subspace W is estimated by applying PCA to the stacked update matrix U formed from sequential updates.For visualization, the analysis uses rank r = 1 and the leading right singular vector; patterns remain consistent for r > 1.
- The analysis tracks the L2 norm of parameter differences between consecutive updates to diagnose systematic reductions in update magnitude.These quantities are reported separately for random-data and class-wise forgetting scenarios.
- In forward-failure scenarios, ER_t progressively decays toward zero across all evaluated methods.This evolution is shown across sequential stages in Figs 25a–26a.
- The ER_t trend supports the predicted suppression mechanism as accumulated A_t induces exponential growth along W and routes updates away from W.