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Machine Unlearning for Gibbs Supervised Learning Algorithms
Yaiza Bermudez, Samir M. Perlaza, Iñaki Esnaola
TL;DR
Machine unlearning aims to remove the influence of training data without the computational cost of retraining, while exact guarantees can be difficult to obtain. The paper formulates exact unlearning for Gibbs algorithms as a relative-entropy-regularized optimization, yielding an algorithm that coincides with retraining on retained data and a broader data-reweighting perspective.
Problem
Exact unlearning is difficult to guarantee, while practical approaches often trade computational efficiency for approximate removal.
Source: §I · p. 1Method
The method maximizes empirical risk on the data to be unlearned while regularizing relative to the original Gibbs algorithm, with the retained data represented through the reference measure.
Source: §I · p. 1, §IV · p. 4Results
Under the corresponding parameter scaling, the new Gibbs algorithm coincides with the algorithm obtained by retraining from scratch on the retained dataset.
Source: §IV · pp. 3–4, §V · p. 5Takeaways & Limitations
Exact unlearning is the zero-weight case of a broader ERM-RER reweighting framework that can also up-weight or down-weight data points for purposes such as controlling Gibbs-algorithm generalization error.
Source: §I · p. 1
Abstract
from arXiv · showhide
In this paper, a method for achieving exact unlearning for Gibbs supervised learning algorithms is proposed using a variational formulation inspired by empirical risk minimization subject to relative entropy regularization (ERM-RER). Such a method consists of maximizing the expected empirical risk over the dataset to be unlearned subject to a regularization by relative entropy with respect to the original algorithm. The optimization variable is a probability measure on the models; and the solution is another Gibbs probability measure that represents a new Gibbs supervised learning algorithm. The method guarantees exact unlearning in the sense that the new Gibbs algorithm coincides in distribution with the algorithm that would have been obtained by retraining from scratch on the dataset to be retained. As a byproduct, a framework for reweighting data points in ERM-RER by strategically choosing both the reference measure and the regularization factor is obtained. In this framework, exact unlearning is the special case in which zero-weight is assigned to the contribution of the data points to be unlearned. More generally, depending on the choice of certain parameters, data points can be up-weighted or down-weighted in ERM-RER problems for particular purposes, e.g., controlling the generalization error of Gibbs algorithms. This paves the way for new constructive or adversarial views on classical reweighting data points in ERM-RER.
I. INTRODUCTION
Machine unlearning seeks to remove the influence of training data that may become unavailable or impermissible to use, while avoiding retraining from scratch. The paper develops exact unlearning for Gibbs algorithms and frames it as a special case of data reweighting in ERM-RER.
I. INTRODUCTION
- The introduction notes that exact-unlearning guarantees are difficult to obtain and practical approaches often trade efficiency for approximate removal.Source: §I · p. 1
- The paper proposes exact unlearning for Gibbs algorithms through a relative-entropy-regularized variational formulation.The formulation draws on empirical risk minimization with f-divergence regularization.Source: §I · p. 1
- The method maximizes empirical risk on data to be unlearned using the original Gibbs algorithm as the reference measure.With the corresponding choice of regularization factor, the result coincides with the Gibbs algorithm trained on retained data.Source: §I · p. 1
- Exact unlearning is a zero-weight case of a broader ERM-RER reweighting framework that can also up-weight or down-weight points for generalization-oriented or adversarial purposes.Source: §I · p. 1
II. SUPERVISED MACHINE LEARNING
The supervised-learning setup represents training data as partitions of labeled examples, defines model losses and empirical risks, and represents a learning algorithm as a conditional probability measure.
II. SUPERVISED MACHINE LEARNING
- The available training data are partitioned into K datasets, each containing labeled data points.Source: §II · p. 2
- The aggregate training dataset is formed by collecting the partition datasets together.Source: §II · p. 2
- A model incurs a Borel-measurable loss on a labeled example, and empirical risk is defined with respect to a dataset.Source: §II · p. 2
- A supervised machine-learning algorithm is represented by a conditional probability measure over models given a dataset.The instance trained on a particular dataset is denoted by conditioning on that dataset.Source: §II · p. 2
III. GIBBS ALGORITHMS
Gibbs algorithms are probability measures parameterized by empirical risk, a reference measure, and a regularization parameter. Their optimization characterization depends on the parameter's sign, and for positive values the measure is also identified as a long-run stochastic-gradient-descent distribution.
III. GIBBS ALGORITHMS
- Gibbs conditional probability measures are parameterized by empirical risk, a σ-finite reference measure, and the dataset.Source: §III · p. 2
- A Gibbs probability measure is characterized through an optimization problem whose unique solution depends on the sign of the regularization parameter.For positive λ_k, it solves the stated minimization problem; for negative λ_k, it solves the corresponding maximization problem.Source: §III · pp. 2–3
- For positive λ_k, the Gibbs measure minimizes training empirical risk over a neighborhood of the reference measure.Source: §III · p. 3
- For positive λ_k, the Gibbs probability measure is identified as the long-run distribution of a stochastic gradient descent algorithm.The passage attributes this observation to.Source: §III · p. 3
IV. MAIN RESULT
The main theorem gives exact unlearning for a Gibbs algorithm by optimizing over the dataset to be removed, using the original algorithm as reference; under a specified scaling, the resulting measure equals retained-only training.
- The unlearning optimization maximizes empirical risk on z1 using the original Gibbs algorithm as its reference, with the retained data entering through that reference measure.Source: §IV · pp. 3–4
- Under the theorem’s specified scaling, the new Gibbs measure coincides with training only on retained dataset z2, achieving exact unlearning.Source: §IV · pp. 3–4
- The common optimizer can be characterized either as a retained-data risk minimizer over G1 or as an unlearned-data risk maximizer over G2.Source: §IV · p. 4
- The parameter λ0 remains free and can be optimized, for example, to minimize the resulting Gibbs algorithm’s generalization error.Source: §IV · p. 4
A. Proof of Theorem 4
The proof of Theorem 4 relies on a preceding result and uses the exact-unlearning choice of λ1 to eliminate the contribution of the dataset to be unlearned.
A. Proof of Theorem 4
- The proof relies on a preceding result, and its full presentation is referred to.The section identifies the next result as the basis for Theorem 4's proof.Source: A. Proof of Theorem 4 · pp. 4–5
- Choosing λ1 to eliminate dataset z1's contribution yields the exact-unlearning identity in Theorem 4.Theorem 5 makes λ1's reweighting role explicit.Source: A. Proof of Theorem 4 · p. 5
- The proof substitutes (18b) and (31) into (30), then invokes [24, Theorem 5] and [24, Theorem 4] for equalities (33) and (34).The passage states that these steps complete the proof of Theorem 4.Source: A. Proof of Theorem 4 · p. 5
B. Reweighting Data Points in ERM-RER
Theorem 5 generalizes the unlearning construction by showing how λ0 and λ1 reweight the contribution of dataset z1 in ERM-RER.
B. Reweighting Data Points in ERM-RER
- Theorem 5's reweighting observation applies beyond the specific parameter choices used for exact unlearning.The passage frames the observation for λ0 > 0 and λ1 ∈ R \ {0}.Source: B. Reweighting Data Points in ERM-RER · p. 5
- Equation (35) highlights that both λ0 and λ1 reweight dataset z1 relative to the optimization problem whose solution is P(Q, λ0) in (22); in (39), its contribution is proportional to n1.Source: B. Reweighting Data Points in ERM-RER · p. 5
- When λ1 > 0, dataset z1's contribution is up-weighted; when λ1 < 0, it is down-weighted.The contribution of retained dataset z2 remains invariant in both cases.Source: B. Reweighting Data Points in ERM-RER · p. 5
V. CONCLUSIONS AND FINAL REMARKS
The paper presents exact unlearning for Gibbs algorithms as a special case of ERM-RER data reweighting, with broader robustness-oriented and adversarial implications.
V. CONCLUSIONS AND FINAL REMARKS
- The proposed method yields the same Gibbs algorithm as retraining from scratch on only the retained dataset.The optimization uses only the dataset to be unlearned, while the retained dataset affects it through the reference measure.Source: §V · p. 5
- In ERM-RER, exact unlearning is the special case that assigns zero weight to the data points being unlearned.Reweighting can also attenuate atypical or influential points to improve generalization, or up-weight outliers adversarially.Source: §V · p. 5
- Choosing parameters for reweighting beyond unlearning is outside the paper's scope.The theorem provides a theoretical foundation for both robustness-oriented reweighting and its adversarial counterpart.Source: §V · p. 5