Adversarial Forget Set in Machine Learning

• Adversarial forget sets are deliberately constructed data subsets that suppress nuisance features to challenge machine unlearning and ensure representation invariance.
• They enable robust evaluation by exposing vulnerabilities in continual learning and revealing potential backdoor and poisoning attack pathways.
• Empirical studies show that forget-gate architectures maintain target task accuracy while reducing unwanted signal detection to chance levels.

An adversarial forget set is a deliberately defined or constructed subset of data—or a neighborhood around it—intended to challenge, subvert, or measure the robustness of machine unlearning, continual learning, and representation invariance mechanisms. In the literature, the adversarial forget set appears both as a mechanism for inducing invariance to unwanted factors in representation learning (notably in (Jaiswal et al., 2019)) and as a focal point for understanding and benchmarking the vulnerabilities and unintended side effects of forgetting, unlearning, and targeted data removal. It is central in adversarial training, backdoor/poisoning attacks, robust model evaluation, and unlearning algorithm design.

1. Mechanisms for Adversarial Forgetting and Invariance

A central paradigm, as proposed in (Jaiswal et al., 2019), centers on inducing invariance to nuisance or bias factors by actively “forgetting” the corresponding information via a dedicated adversarial mechanism. This is realized via an adversarial forget-gate architecture. An input $x$ is encoded into a latent representation $z$. A parallel forget-gate network generates a continuous mask $m$ (with $0 \leq m_k \leq 1$), and the invariant representation is then computed as $\, \hat{z} = z \odot m \,$ (elementwise multiplication). This mask is adversarially optimized:

• A discriminator $D$ attempts to recover the undesired signal $s$ from $\, \hat{z} \,$, but its gradients are allowed only to modify the mask $m$.
• The forget-gate thus learns to minimize mutual information $I(\hat{z}:s)$ while preserving $I(\hat{z}:y)$ for target variable $y$.

Mathematically, the mechanism forms an adaptive information bottleneck:

$$I(\hat{z}_i : z_i) \leq \frac{1}{2} \log \left( m_i^2 \operatorname{Var}(z_i) + \operatorname{Var}(\epsilon_i)\right) - \frac{1}{2} \log \operatorname{Var}(\epsilon_i)$$

where $\epsilon$ is small i.i.d. Gaussian noise.

This architecture and objective combine to define an adversarial forget set: the effective set of features or components actively suppressed or masked by the adversarial forget-gate in response to adversarial feedback.

2. Adversarial Forget Sets in Attacks, Learning Dynamics, and False Memories

Beyond representation invariance, the adversarial forget set is leveraged as a practical attack target and as a diagnostic tool to measure model vulnerabilities in continual, incremental, and unlearning settings:

• In continual learning, adversarial poisoning can implant false memories through sparse inserted backdoor samples (e.g., adding a small visual pattern and relabeling). Regularization-based models (e.g., EWC) and generative replay models are especially susceptible. When such triggers are presented at test time, the model is coerced into forgetting legitimate prior knowledge and misclassifying according to the adversary’s plan (Umer et al., 2020, Umer et al., 2021, Umer et al., 2022).
• Worst-case and adversarial benchmarking: Studies like (Fan et al., 12 Mar 2024) frame adversarial forget sets as subsets that, once selected for unlearning, maximally challenge or degrade unlearning algorithms. Through bi-level optimization, these sets expose methods’ limitations that are not apparent under random data deletion, providing a rigorous adversarial benchmark.
• Backdoor activation via unlearning: In (Arazzi et al., 14 Jun 2025), unlearning is manipulated to “activate” an otherwise dormant backdoor. An attacker injects a weak, distributed backdoor trigger during training. A subsequent clean unlearning request, targeting a carefully chosen forget set, realigns gradients so as to amplify the backdoor’s effect—the forget set itself becomes an unintentional adversarial vector.

3. Empirical Results and Evaluation Metrics

Empirical studies demonstrate that adversarial forget sets can achieve:

• Targeted forgetting or invariance: The forget-gate framework of (Jaiswal et al., 2019) achieves state-of-the-art invariance on datasets such as MNIST-ROT, Chairs, Extended Yale-B, Adult, and German—removing bias or nuisance factors so that predictive accuracy for $y$ remains high ($A_y$) while accuracy for $s$ is reduced to chance levels ($A_s$).
• Catastrophic forgetting in continual learning: Only 1% poisoning can reduce target task accuracy from nearly 98% to below 10% in the presence of triggers (Umer et al., 2021, Umer et al., 2022). Membership inference attack efficacy also drops to chance when forgetting is properly enforced via adversarial unlearning (Ebrahimpour-Boroojeny et al., 2 Mar 2025).
• Tamper-resistance failure: Even sophisticated unlearning may be vulnerable to “relearning” attacks where fine-tuning solely on the retained set revives forget-set accuracy from 50% (post-unlearning) to nearly 100%—pointing to residual memory of the adversarial forget set in the weights, unless specific regularization is applied (Siddiqui et al., 28 May 2025, Ha et al., 2 Jun 2025).

Method/paper	Forget set handling	Retained utility
Adversarial forget-gate (Jaiswal et al., 2019)	Masked out in representation	Maintained/ state of art
Backdoor attacks (Umer et al., 2021, Umer et al., 2022)	False memory/poisoning	High for untargeted tasks
Adversarial unlearning (Ebrahimpour-Boroojeny et al., 2 Mar 2025)	Localized conf reduction	Test accuracy preserved
Bi-level worst-case (Fan et al., 12 Mar 2024)	Maximally challenging set selection	Benchmarks robustness of algorithms

4. Mathematical Formulations of Adversarial Forgetting

Formalizing adversarial forget sets often involves min–max or bi-level objectives:

• Adversarial invariance objective (Jaiswal et al., 2019):

$\min_{E,F,P,R}\max_D\; J(E,F,P,R,D)$

$$J(E,F,P,R,D) = L_y(y, P(\hat{z})) + \rho L_x(x, R(z)) + \delta L_s(s, D(\hat{z})) + \lambda m^T(1-m)$$

• Poisoning or worst-case unlearning (Fan et al., 12 Mar 2024):

$$\min_{w\in\mathcal{W}} f(\theta^*(w), w) \quad \text{s.t.}\quad \theta^*(w) = \arg\min_\theta L^{MU}(\theta; w)$$

where $w$ denotes the selection of the forget set (indicator vector), and $f$ computes the worst-case influence remaining after unlearning.

These formulations ensure adversarially selected features or instances are either maximally suppressed, made invariant, or exposed as worst-case tests for algorithmic robustness.

5. Security, Fairness, and Broader Implications

Adversarial forget sets are significant for:

• Fairness: By targeting the suppression of demographic or biasing signals, adversarial forgetting frameworks like (Jaiswal et al., 2019) allow equitable decision making (e.g., removing gender or age effects).
• Security and privacy: As models can be forced to “forget” specific data—sometimes at the adversary’s behest—this has clear applications in data privacy compliance and in building robust privacy guarantees (e.g., resistance to membership inference attacks (Ebrahimpour-Boroojeny et al., 2 Mar 2025)).
• Model assessment: Adversarial forget sets provide rigorous evaluation benchmarks (especially in worst-case construction) that expose vulnerabilities, over-unlearning effects, or the inability of standard unlearning methods to permanently erase information (Ha et al., 2 Jun 2025).

6. Limitations, Open Challenges, and Future Research

The literature reveals several critical limitations and points of ongoing research:

• Effectiveness and permanence: Many approximate methods may not fully remove forget-set information; latent knowledge can persist in the weight-space, making models susceptible to relearning attacks (Siddiqui et al., 28 May 2025).
• Collateral damage: Over-unlearning (i.e., deterioration of retained data near the forget set) is a practical risk that must be minimized via appropriately regularized objectives (Ha et al., 2 Jun 2025).
• Defense design: Defensive mechanisms—including information bottlenecks, adversarial decoys, regularization, and explicit weight-space displacement—remain an active area for improving tamper-resistance.
• Algorithmic evaluation: Adversarial selection of forget sets via bi-level optimization (Fan et al., 12 Mar 2024) is likely to become a standard for benchmarking the robustness of model unlearning.

7. Applications and Prospects

Adversarial forget sets have become integral to:

• The design of privacy-compliant adaptive models.
• Robustness evaluation for fairness and security.
• Defense constructions in continual, incremental, and federated learning settings.
• Practical machine unlearning frameworks for structured, vision, and LLMs.

Advances in min–max optimization, sparse and mask-based representation learning, and worst-case adversarial set identification are expected to further refine the boundaries between robust forgetting, resilient model behavior, and guaranteed privacy or fairness.

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In summary, the adversarial forget set is a foundational construct for both robust representation learning and secure model unlearning. It formalizes both the target of adversarial removal (for invariance or unlearning) and the mechanism of challenge and benchmarking (by probing weaknesses via worst-case, backdoor, or relearning scenarios), anchoring a spectrum of algorithmic design and analysis from information-theoretic foundations to practical defense engineering in modern machine learning.