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Model Sparsity Can Simplify Machine Unlearning

Jinghan Jia, Jiancheng Liu, Parikshit Ram, Yuguang Yao, Gaowen Liu, Yang Liu, Pranay Sharma, Sijia Liu

arXiv:2304.04934v13cs.LG

TL;DR

Machine unlearning must remove targeted data influence without incurring the cost of retraining, yet approximate methods can fall short across multiple criteria. This paper uses weight-pruning-induced model sparsity to narrow that gap, introducing prune-first and sparsity-aware unlearning; experiments report consistent benefits, including a 77% efficacy gain for fine-tuning, while limitations remain around provable sparse-MU guarantees and NLP validation.

  • Problem

    Approximate unlearning is computationally efficient but can lack strong effectiveness guarantees, require sensitive parameter tuning, and show trade-offs across evaluation criteria.

  • Method

    The paper connects machine unlearning with weight-pruning-induced model sparsity and proposes “prune first, then unlearn” plus sparsity-aware unlearning with soft sparsity regularization.

  • Results

    Across diverse unlearning setups, model sparsity reduces the gap between approximate and exact unlearning; sparsity-aware fine-tuning achieves a 77% unlearning efficacy gain.

  • Takeaways & Limitations

    Model sparsity can serve as a model prior that simplifies approximate machine unlearning while retaining computational efficiency.

  • Takeaways & Limitations

    The study lacks provable guarantees for ℓ1-sparse unlearning and mainly evaluates computer vision tasks, leaving NLP effectiveness unverified.

Abstract

from arXiv · show

In response to recent data regulation requirements, machine unlearning (MU) has emerged as a critical process to remove the influence of specific examples from a given model. Although exact unlearning can be achieved through complete model retraining using the remaining dataset, the associated computational costs have driven the development of efficient, approximate unlearning techniques. Moving beyond data-centric MU approaches, our study introduces a novel model-based perspective: model sparsification via weight pruning, which is capable of reducing the gap between exact unlearning and approximate unlearning. We show in both theory and practice that model sparsity can boost the multi-criteria unlearning performance of an approximate unlearner, closing the approximation gap, while continuing to be efficient. This leads to a new MU paradigm, termed prune first, then unlearn, which infuses a sparse model prior into the unlearning process. Building on this insight, we also develop a sparsity-aware unlearning method that utilizes sparsity regularization to enhance the training process of approximate unlearning. Extensive experiments show that our proposals consistently benefit MU in various unlearning scenarios. A notable highlight is the 77% unlearning efficacy gain of fine-tuning (one of the simplest unlearning methods) when using sparsity-aware unlearning. Furthermore, we demonstrate the practical impact of our proposed MU methods in addressing other machine learning challenges, such as defending against backdoor attacks and enhancing transfer learning. Codes are available at https://github.com/OPTML-Group/Unlearn-Sparse.

1 Introduction

Machine unlearning removes targeted training-data influence, but approximate methods trade retraining’s effectiveness for speed and can perform unevenly across criteria. This paper proposes model sparsification as a theoretically grounded way to narrow that gap and introduces sparsity-based unlearning methods.

  • Motivation: Machine unlearning removes targeted data influence to address privacy requirements, while exact retraining is effective but computationally intensive.Approximate methods are faster alternatives motivated by the cost of retraining from scratch on the remaining dataset.
  • Motivation: Approximate unlearning can depend heavily on parameter tuning and exhibit poorly understood trade-offs across efficacy and fidelity criteria.The introduction highlights Fisher forgetting’s tuning sensitivity and the fact that privacy efficacy does not determine remaining-data fidelity.
  • Research question: The paper asks whether a theoretically grounded, broadly applicable method can improve approximate unlearning across different evaluation criteria.This question follows from the performance gap and cross-criterion variability identified for approximate methods.
  • Core insight: Model sparsity achieved through weight pruning is shown in theory and practice to reduce the gap between approximate and exact unlearning.The proposed viewpoint treats sparsification as a model-based factor influencing unlearning rather than relying only on data-centric procedures.
  • Contributions: The paper introduces “prune first, then unlearn” and a sparsity-aware framework using soft sparsity regularization to enhance approximate unlearning.It also studies how pruning methods influence unlearning performance and evaluates the proposals across diverse datasets, models, and scenarios.

2 Revisiting Machine Unlearning and Evaluation

The paper formalizes machine unlearning as generating a model that removes a forgetting set’s influence while preserving remaining-data information and efficiency. It reviews exact and approximate methods, then defines a multi-metric evaluation spanning efficacy, fidelity, generalization, and runtime.

  • Problem setup: Machine unlearning seeks to generate θu from θo after removing the influence of forgetting dataset Df, with Dr denoting the remaining data.The setup includes class-wise forgetting and random data forgetting scenarios.
  • Unlearning methods: Exact unlearning retrains from scratch on Dr, whereas approximate methods reduce computation by acting as faster proxies with lower unlearning efficacy.The reviewed approximate methods include fine-tuning, gradient ascent, Fisher forgetting, and influence unlearning.
  • Approximate methods: Fine-tuning updates θo on Dr for a few epochs, gradient ascent increases loss on Df, and Fisher forgetting perturbs θo with Fisher-informed Gaussian noise.These methods use different mechanisms to approximate the effect of removing the forgetting data.
  • Evaluation: Table 1 summarizes which metrics prior approximate methods use and reports RTE as run-time cost reduction relative to Retrain on CIFAR-10 with ResNet-18.The table notes that GA appears better in RTE but is less effective in unlearning.
  • Influence unlearning: Influence unlearning estimates θu − θo using influence functions, while the paper revisits its formula through a weighted ERM formulation and normalized influence weights.The derivation uses an inverse-Hessian gradient product, approximated in practice with first-order WoodFisher.
  • Evaluation: Evaluation uses UA and MIA-Efficacy for unlearning efficacy, RA for fidelity, TA for generalization, and RTE for computation efficiency.Disparity Average measures the mean performance gap between an approximate method and Retrain across the considered metrics.

3 Model Sparsity: A Missing Factor Influencing Machine Unlearning

Model sparsity, introduced through weight pruning, reduces the gap between approximate and exact machine unlearning. Theory and experiments show that increasing sparsity improves unlearning efficacy across methods, while fidelity and generalization can remain close to retraining except at ultra-high sparsity.

  • OMP-based pruning produces sparse models whose testing accuracy is compared with the dense ResNet-18 model on CIFAR-10.OMP is used as the computationally lightest pruning method.
  • Model sparsity reduces the unlearning error between GA-based approximate unlearning and gold-standard retraining.The bound relates error to the sparse model pattern and the weight distance accumulated during training.
  • The theoretical analysis connects sparse unlearning error to the Hessian spectrum over unmasked parameter dimensions.Increasing sparsity reduces the number of active singular values considered by the bound.
  • As sparsity increases, approximate unlearning efficacy improves and its gap from Retrain decreases in class-wise forgetting.The pattern is reported for UA and MIA-Efficacy across FT, GA, FF, and IU.
  • FT and FF achieve fidelity and generalization closest to Retrain, while 99% sparsity introduces some trade-off with RA and TA.The comparison uses RA for fidelity and TA for generalization.

4 Sparsity-Aided Machine Unlearning

The paper extends sparsity-based unlearning through pruning-method selection and simultaneous sparsity regularization. SynFlow and OMP improve efficacy relative to dense and IMP-based alternatives, while a decaying regularization schedule improves efficacy, fidelity, and generalization.

  • Prune first, then unlearn: The prune-first-then-unlearn paradigm motivates selecting pruning methods by forgetting-data dependence, generalization preservation, and pruning efficiency.These criteria favor sparse priors that avoid incorporating the forgetting dataset while maintaining testing accuracy and runtime efficiency.
  • Prune first, then unlearn: OMP is selected as the default pruning method because it is computationally light and generally provides better generalization than SynFlow.OMP has weaker forgetting-dataset dependence than IMP-based methods, though it may still depend on the forgetting dataset.
  • Prune first, then unlearn: SynFlow and OMP improve FT-based UA and MIA-Efficacy over dense and IMP-generated sparse models.IMP can yield the best TA but its unlearning metrics worsen with increasing sparsity because it relies heavily on the training set, including forgotten data.
  • Sparsity-aware unlearning: ℓ1-sparse MU adds an ℓ1 penalty to the unlearning objective to reduce the magnitudes of unimportant weights.The regularization parameter γ controls the penalty level.
  • Sparsity-aware unlearning: A linearly decreasing γ schedule minimizes the efficacy gap to Retrain while improving preservation of RA and TA after unlearning.The schedule prioritizes sparsity early, then shifts toward fine-tuning accuracy on the remaining dataset.

5 Experiments

Experiments evaluate sparsity-based unlearning across datasets, models, scenarios, metrics, and applications. Model sparsity improves approximate unlearning, while sparsity-aware unlearning closes the gap with retraining and supports backdoor defense and transfer learning.

  • Experiment setups: Experiments primarily use CIFAR-10 with ResNet-18, while additional datasets and VGG-16 results show consistent benefits from model sparsification.The main unlearning scenarios are class-wise forgetting and random forgetting of 10% of the training data.
  • Experiment setups: Evaluation covers unlearning efficacy, fidelity, generalization, computation efficiency, and disparity from the retrained model.Disparity Average is lower when an approximate method is closer to Retrain across the considered metrics.
  • Model sparsity improves approximate unlearning: Model sparsity consistently improves UA and MIA-Efficacy without much RA loss, reducing approximate methods’ performance gaps with Retrain.Table 3 compares dense and 95%-sparse models across class-wise and random data forgetting.
  • Model sparsity improves approximate unlearning: 51% UA improvement and 8% MIA-Efficacy improvement are obtained for FT on class-wise forgetting at 95% sparsity.FT and IU also preserve TA better than other methods in the 95%-sparse setting, while approximate methods show different trade-offs.
  • Effectiveness of sparsity-aware unlearning: Sparsity-aware ℓ1-sparse MU outperforms FT in efficacy and closes the gap with Retrain without losing approximate unlearning’s computation advantage.The comparison covers class-wise and random data forgetting on CIFAR-10 with ResNet-18.
  • Applications: FT-based unlearning reduces backdoor ASR with little SA loss, especially at 90% sparsity, while ℓ1-sparse MU removes the backdoor effect and largely preserves generalization.The original Trojan model maintains 100% ASR across sparsity levels.
  • Applications: ℓ1-sparse MU achieves comparable or slightly better transfer accuracy than retraining and provides a 2× speedup across ImageNet transfer-learning cases.Evaluation uses SUN397 and OxfordPets with linear probing after removing 100, 200, or 300 ImageNet classes.

6 Related Work

Related work spans machine unlearning, data-influence analysis, and model pruning. This paper differs by systematically studying foundational connections between pruning and unlearning rather than tailoring pruning to one application.

  • Machine unlearning: Machine unlearning includes exact and approximate methods, probabilistic approaches using differential privacy, and applications in federated, graph, and adversarial learning.Differential-privacy approaches can provide provable error guarantees but typically rely on strong assumptions and may lack practical effectiveness.
  • Understanding data influence: Unlearning is closely related to understanding data influence, with influence functions serving as an algorithmic backbone for many unlearning methods.Data influence perspectives also motivate unlearning for defense against poisoning backdoor attacks.
  • Model pruning: Model pruning addresses computation, energy, and memory constraints while seeking a balance between model sparsity and generalization.The lottery ticket hypothesis demonstrates that sparse subnetworks can match or exceed the test accuracy of dense models.

7 Conclusion

The paper presents model sparsification as a foundation for improving machine unlearning. Its findings support pruning-first and sparsity-aware paradigms across unlearning setups and suggest future study of unlearning in weight or architecture space.

  • Conclusion: Weight-pruning-based model sparsification is shown theoretically and practically to close the gap between exact and approximate unlearning.The paper frames model sparsity as a foundational factor in approximate unlearning efficacy.
  • Conclusion: The proposed ‘prune first, then unlearn’ and ‘sparsity-aware unlearn’ paradigms significantly improve approximate unlearning efficacy.Their effectiveness is demonstrated through extensive experiments across different unlearning setups.
  • Conclusion: The study indicates that model modularity traits such as weight sparsity could simplify machine unlearning and motivate research in weight or architecture space.This is presented as a prospect for future research rather than a completed evaluation direction.

A Proof of Proposition 1

The proof analyzes implicit unlearning under average- and sum-ERM formulations. It shows that simplex-constrained average ERM yields a more accurate first-order Taylor approximation as the number of forgotten points grows.

  • Implicit gradient: Implicit differentiation derives the gradient dθ(w)/dw through the optimization problem defining θ(w), assuming θ(w) is the unique minimizer.The closed form is evaluated at the average-ERM linearization point w = 1/N.
  • Weight configurations: Under average ERM, unlearning weights remain simplex-constrained, assigning zero weight to forgotten samples and renormalized weight 1/(N−k) to retained samples.Under sum ERM, the retained samples remain at weight 1 while forgotten samples receive weight zero.
  • Setup: The model update is defined as Δ(w) = θ(w) − θo, where θo is the original ERM-trained model and θ(w) is the weighted-ERM solution.The update captures the parameter adjustment induced by the unlearning weight configuration.
  • Weight configurations: The simplex constraint causes average- and sum-ERM to produce different unlearning weight configurations and corresponding model updates.The proof explicitly attributes the difference to the simplex constraint on w.
  • Approximation accuracy: As k increases, sum-ERM can make the first-order Taylor expansion in influence unlearning less accurate than the average-ERM expansion.The stated error orders are k^2 for sum ERM and k/[N(N−k)] for average ERM.

B Proof of Proposition 2

The proof analyzes approximate unlearning error for a sparse model encoded by a fixed binary pruning mask. It derives an error bound whose dependence on retained dimensions suggests pruning can reduce the error relative to no pruning.

  • Sparse-model formulation: The proof represents the sparse model as θ′ := m ⊙ θ using a fixed binary pruning mask m.The initialization for sparse SGD training is θ′_0 = m ⊙ θ_0.
  • Approximate-unlearning error: GA-based approximate unlearning adds back first-order gradients of the sample to be unlearned, but retains an unlearning error from higher-order terms.The error is identified with the last term of (A12).
  • Error-bound derivation: The sparse model m ⊙ θ is written as diag(m)θ to derive an unlearning-error bound while retaining the η² terms in the recursive expansion.Terms with higher powers of η are ignored because they contribute at higher degree in the learning rate.
  • Spectral dependence: The bound uses the largest eigenvalue among dimensions left intact by the binary mask m.This quantity is taken over coordinates with nonzero mask entries.
  • Implication for pruning: The proof notes that the unlearning error might be large when m = 1, corresponding to no pruning.The resulting big-O expression follows from (A17).

C.1 Datasets and models

The appendix summarizes dataset and model setups, pruning configurations, and approximate-unlearning training and evaluation procedures. Fine-tuning and gradient-ascent settings are tuned per dataset and architecture, while MIA evaluates privacy-related unlearning performance.

  • Datasets and models: Dataset and model configurations are summarized in Table A1, with pruning-training details summarized separately in Table A2.The appendix references these tables for the experimental setup.
  • Pruning configuration: IMP, SynFlow, and OMP use SOTA implementation settings, step learning-rate decay at 50% and 75% of training, and an initial learning rate of 0.1.The scheduler decay rate is 0.1.
  • Unlearning configuration: Fine-tuning runs for 10 epochs and GA runs for 5 epochs, with learning rates tuned in [10^-5, 0.1] for each dataset and architecture.SGD is the default optimizer for FT and GA.
  • Unlearning configuration: For CIFAR-10 ResNet-18 class-wise forgetting, the selected learning rates are 0.01 for FT and 10^-4 for GA across sparsity levels.These settings are reported as a specific default configuration.
  • MIA evaluation: MIA trains a predictor on balanced samples from the remaining dataset Dr and a separate test dataset, then evaluates unlearning during testing.The evaluation uses a prediction-confidence-based attack.

C.4 Additional experiment results

Additional experiments show that sparsity improves privacy and narrows the gap between approximate and exact unlearning across datasets, architectures, and model sizes. Sparsity-aware unlearning likewise improves forgetting metrics with limited computational overhead while preserving retention performance.

  • Privacy: MIA-Privacy decreases as sparsity increases, and approximate unlearning can outperform Retrain in protecting the remaining dataset Dr.IU and GA achieve especially low MIA-Privacy because their unlearning procedures depend less on Dr.
  • Prune first, then unlearn: Across datasets and architectures, pruning reduces the discrepancy between approximate and exact unlearning while improving UA and MIA-Efficacy without substantial RA and TA degradation.The pattern is reported across multiple unlearning scenarios.
  • ImageNet: 83% model sparsity on ImageNet with ResNet-18 is used to preserve TA while sparsity reduces the gap between Retrain and FT or GA.The ImageNet experiments follow the class-wise-forgetting setting.
  • Sparsity-aware MU: ℓ1-sparse MU reduces the approximate-to-exact gap across CIFAR-100, SVHN, and ImageNet, notably outperforming other methods on UA and MIA-Efficacy.In class-wise and random forgetting, its UA and MIA-Efficacy are reported as on par with Retrain.
  • Efficiency and retention: ℓ1-sparse MU consistently improves forgetting metrics with an insignificant computational-cost increase compared with FT while preserving acceptable RA and TA.The method is presented as a practical choice across unlearning scenarios.
  • Model-size robustness: Sparsity consistently diminishes the unlearning gap with Retrain across ResNet-20s and ResNet-50, extending the pattern beyond ResNet-18.The reported performance gap is better when smaller.

D Broader Impacts and Limitations

The paper frames sparsity-inspired unlearning as a versatile approach for privacy, efficiency, robustness, and related model concerns. It also identifies missing formal guarantees for ℓ1-sparse MU and limited validation beyond computer-vision tasks, especially NLP.

  • Broader impacts: The proposed MU framework is presented as a way to forget arbitrary data points while addressing model privacy, efficiency, and robustness.The broader-impact discussion also connects it to regulatory compliance and bias mitigation.
  • Adaptation and sustainability: Unlearning outdated information is described as supporting model adaptation, relevance to evolving domains, and continued usefulness over time.The paper associates this capability with sustainability.
  • Limitations: ℓ1-sparse MU lacks provable guarantees, and the framework has mainly been evaluated on computer-vision tasks rather than NLP.Its effectiveness for NLP, including large language models, remains unverified.
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