Forgetting Probe: Methods & Applications

• Forgetting Probe is a methodological tool that quantifies loss of information by measuring performance decrease or representational drift in machine learning models.
• It employs metrics such as KL divergence and average forgetting scores to diagnose catastrophic forgetting and guide continual learning.
• Applications span from optimizing privacy-driven unlearning in federated systems to enhancing model reliability in neuro-inspired and logic-based frameworks.

A forgetting probe is a methodological, algorithmic, or theoretical instrument used to measure, induce, and analyze the removal of specific information or the loss of task-relevant representations in machine learning models, neural networks, knowledge bases, or human memory. Probes can be used to evaluate catastrophic forgetting in sequential learning, inspect the efficacy of machine unlearning mechanisms, quantify representational drift, diagnose multi-level loss of predictive power, and guide both theoretical analysis and practical algorithm design. The forgetting probe encompasses both performance-driven (continual learning, domain adaptation) and privacy-driven (compliance, selective erasure) settings, across supervised, unsupervised, federated, logical, and neuro-inspired domains.

1. Formal Definitions and Theoretical Foundations

Forgetting probes are formally characterized through two axes: (1) the operational definition of forgetting (performance decrement, representational drift, inferential weakening, predictive self-inconsistency); and (2) the measurement protocol (metric, loss, or theoretical divergence).

• In machine learning, for a model $M_\theta$ trained on $D$, given a forget set $D_f \subset D$, a forgetting probe $P$ evaluates the effect of an unlearning operation resulting in a new model $M_{\theta'}$. Key desiderata are that for $x \in D_f$, $M_{\theta'}(x) \approx M_{\theta_0}(x)$ (where $M_{\theta_0}$ is unexposed to $D_f$), and for $y \notin D_f$, $M_{\theta'}(y) \approx M_\theta(y)$ (Sha et al., 31 May 2024). Typical metrics include KL divergence or $L_2$ distance between prediction distributions over $D_f$ and $D \setminus D_f$.
• In continual and federated learning, the average forgetting metric is:

$$F = \frac{1}{T-1} \sum_{i=1}^{T-1} [\max_{t=1\dots T} a_i(t) - a_i(T)]$$

where $a_i(t)$ is task $i$ accuracy after training on task $t$ (Davari et al., 2022, Aljahdali et al., 8 Feb 2024).

• From an information-theoretic perspective, forgetting is operationalized as predictive self-inconsistency: given a learner's predictive distribution $q_{\mathrm{pre}}(H^{t+1:\infty})$ before and $q_{\mathrm{post}}(H^{t+1:\infty})$ after an update, forgetting is quantified by divergence $D(q_0 \| q_k^*)$ over future trajectories (Sanati et al., 6 Nov 2025).
• In logic and knowledge representation, model-counting-based and probability-weighted losses are assessed after applying a forgetting operator $\mathcal{F}$:

$$loss_m^{NC,\bar{p}}(\Phi) = \frac{|\mathrm{Mod}(\mathcal{F}(\Phi;\bar{p}))| - |\mathrm{Mod}(\Phi)|}{|W|}$$

where $|\mathrm{Mod}(\cdot)|$ counts classical models, and $W$ denotes the set of worlds (Doherty et al., 3 Apr 2024).

2. Methodologies and Algorithms

Forgetting probes manifest as concrete workflows in multiple settings:

• Linear-Probe Forgetting Metrics: In continual learning, representational forgetting is measured by retraining a linear classifier $W^*_a$ on the frozen feature extractor $f_{\theta_b}$ for old task data $(X_a, Y_a)$:

$$\text{RepForgetting}(a \rightarrow b) = T(W^*_a f_{\theta_a}(X_a), Y_a) - T(W^*_b f_{\theta_b}(X_a), Y_a)$$

where $T(\cdot)$ denotes classification accuracy (Davari et al., 2022).

• Layerwise Probing in Transformers: Decoding heads (e.g., MLM classifiers) are trained atop each layer of the base model (e.g., BERT), freezing the encoder. Changes in probe accuracy across layers and tasks directly quantify not only if, but where, knowledge is forgotten (Wallat et al., 2021, Wallat et al., 2020, Tao et al., 2023).
• Retain-Free Unlearning via Decision-Boundary Probes: Probing via gradient ascent near the decision boundary for a class to be forgotten, followed by “push–pull” collaborative fine-tuning using only to-be-forgotten data (PTE framework), achieves sharp erasure of targeted classes while preserving all other knowledge (Chen et al., 12 Nov 2025).
• Propositional Logic Forgetting: Forgetting is algorithmically realized via strong unfoldings—systematic clause resolution and elimination in CNF to excise all mention of a set of atoms, preserving logical entailments outside the forgotten signature (Wang, 2015).
• Federated Learning Forgetting Metrics: Round-wise class-wise drops in accuracy are accumulated:

$$\mathcal{F}_t = -\frac{1}{C} \sum_{c=1}^C \min(0, A_t^c - A_{t-1}^c)$$

enabling fine-grained, non-compensatory tracking of knowledge loss per communication round (Aljahdali et al., 8 Feb 2024).

• XAI-Based Dissection Probes: The Catastrophic Forgetting Dissector (CFD) leverages feature-visualization and IoU metrics to localize “where” (e.g., which convolutional block) forgetting is most severe in neural architectures, informing layer freezing strategies (Nguyen, 2022).

3. Empirical Benchmarks and Applications

Empirical evaluation of forgetting probes spans several domains:

• Continual Learning: Standard benchmarks include SplitCIFAR-100, SplitCIFAR-10, SplitMiniImageNet, and task-incremental ImageNet sequences. Representational forgetting is typically modest for supervised contrastive (SupCon) methods, often outstripping regularization and replay schemes when measured with linear probes (Davari et al., 2022).
• Transformers and NLP: Sequential task learning on BERT in text classification and QA reveals that catastrophic forgetting predominantly affects final-task decoders; frozen encoders retain intra-task geometry, as recoverable by linear probes (Tao et al., 2023). Factual knowledge probing (e.g., LAMA suite) shows that intermediate layers often maximize probe accuracy, with task-specific objectives (QA vs ranking) dictating the fraction of relations forgotten (Wallat et al., 2021, Wallat et al., 2020).
• Federated Learning: Flashback algorithm uses round-wise forgetting probes to mitigate both client-local and server-aggregation forgetting, reducing knowledge loss and accelerating convergence on federated tasks (CIFAR-10, CINIC10, FEMNIST) (Aljahdali et al., 8 Feb 2024).
• Knowledge Representation and Logic: Forgotting probes parameterized by model-counting or probability loss are used to assess the inferential degradation due to symbol elimination, guiding abstraction and privacy policies in reasoning systems (Doherty et al., 3 Apr 2024, Wang, 2015).
• Industrial/IIoT Unlearning: The PTE retain-free probe-edit paradigm enables class erasure with zero membership leakage and minimal loss on retained tasks, verified via membership inference attack rates and accuracy recovery (Chen et al., 12 Nov 2025).
• Neurocognitive Modeling: Mathematical probes fitted to Ebbinghaus's forgetting data parameterize dual-phase (fast/slow) retention and capacity-limiting interference, providing a normative model for probe schedule design in memory studies (Yu et al., 2018).

4. Categories and Taxonomy of Forgetting Probes

Forgetting probes can be grouped by operational goal and underlying methodology (Sha et al., 31 May 2024):

Category	Examples	Typical Use Case
Active (performance)	EWC, rehearsal buffer, SupCon, CFD	Continual learning
Passive (privacy)	Unlearning (SISA, PTE, influence-removal)	Data erasure, GDPR
Probing/diagnostic	Linear, layerwise, feature-visualization	Model analysis
Logical/inferential	Model-counting loss, CNF forgetting	Knowledge bases

Active probes often regularize or constrain learning to protect prior knowledge; passive probes seek to erase data traces with formal guarantees. Probing and diagnostic algorithms serve both as measurement tools and design guides.

5. Challenges, Limitations, and Theoretical Implications

Open challenges and subtleties surround the use and validation of forgetting probes:

• Verification: There is no fully general “forgetting certificate” that guarantees all traces of specific data or knowledge have been excised; quantitative metrics can be degraded by “collateral amnesia” or insufficient erasure (Sha et al., 31 May 2024).
• Task- and Objective-Sensitivity: The amount and quality of retained or lost knowledge is highly sensitive to the objective function (e.g., QA vs ranking in transformers), as opposed to mere dataset size or training protocol (Wallat et al., 2021, Wallat et al., 2020).
• Efficiency-Forgetting Tradeoff: Moderate, nonzero forgetting (as measured by predictive-divergence probes) sometimes accelerates learning and reduces final loss compared to both under- and over-forgetting, revealing a “Goldilocks zone” (Sanati et al., 6 Nov 2025).
• Interpretability: Feature-level probes (e.g., CFD) clarify which network blocks account for forgetting, but their insights are architecture- and input-dependent (Nguyen, 2022).
• Scalability and Computational Overhead: Probes based on retraining or layerwise evaluation may not scale to extremely large networks or logic bases, motivating the use of approximate or hybrid diagnostics (Doherty et al., 3 Apr 2024).

6. Future Directions

Research continues to advance forgetting probes along multiple dimensions:

• Unified Theoretical Frameworks: Algorithm- and task-agnostic formalisms based on predictive self-consistency and information loss provide a common probe for continual, supervised, generative, and RL paradigms (Sanati et al., 6 Nov 2025).
• Hybrid and Adaptive Mechanisms: Simultaneous deployment of active and passive probes, guided by forgetting monitoring, is an emerging paradigm for balancing plasticity, retention, and privacy (Sha et al., 31 May 2024).
• Neuroscientific and Cognitive Inspiration: Insights from interference-based memory models are being incorporated into probe design, aligning empirical human forgetting with artificial memory systems (Yu et al., 2018).
• Trust, Governance, and Ethics: Robust verification, bias auditing, and “forgetting-by-design” architectures are required to satisfy evolving social and legal requirements for regulatory compliance in machine unlearning (Sha et al., 31 May 2024).

7. Key Insights and Practical Guidelines

• Layerwise probing reveals substantial knowledge hidden in intermediate representations; single-point or final-layer probes systematically under-report capacity and retention (Wallat et al., 2020, Wallat et al., 2021).
• Linear-probe metrics can distinguish representational drift versus mere classifier-head misalignment; observed catastrophic forgetting may be mitigated by simple re-probing or re-aligning (Davari et al., 2022, Tao et al., 2023).
• Probing for task-agnostic prediction divergence enables real-time, architecture-independent forgetting diagnosis and guides hyperparameter tuning (Sanati et al., 6 Nov 2025).
• Privacy-centric unlearning probes benefit from adversarial boundary exploration and self-consistent editing to balance erasure and utility, especially in retain-free settings (Chen et al., 12 Nov 2025).
• Verification and quantification of forgetting through model-based, information-theoretic, or logic-based probes are essential for trustworthy, interpretable, and compliant machine learning systems (Sha et al., 31 May 2024, Doherty et al., 3 Apr 2024).