Retraining-Consistent Representation Forgetting

• Retraining-consistent representation forgetting is defined by freezing the learned features and retraining a probe from scratch to accurately measure the retained task-relevant structure.
• This framework shows that severe drops in observed accuracy can coincide with preserved linear separability, highlighting limitations of output-based forgetting assessments.
• The approach underpins both continual learning and machine unlearning, comparing retrained models to reference models to expose structured but inaccessible residual representations.

Retraining-consistent representation forgetting denotes a mode of evaluating and, in some cases, designing forgetting mechanisms in which the primary object of interest is the representation itself rather than the instantaneous state of a task head or the apparent success of output-level suppression. In continual learning, it is defined by freezing representation snapshots and retraining optimal probes from scratch on past tasks, so that forgetting measures retained representational capacity rather than inherited classifier parameters (Davari et al., 2022). In machine unlearning, the same principle takes the model retrained from scratch without the forget data as the operational reference for correct forgetting, and then asks whether the unlearned model matches that reference in representation space as well as in output space (Yong et al., 23 Jun 2026). Across both uses, the central observation is that severe drops in observed accuracy can coexist with substantial retained linear separability, and low forget-set accuracy can coexist with structured residuals that remain inconsistent with retraining (Davari et al., 2022).

1. Conceptual basis

The point of departure is the distinction between catastrophic forgetting and representation forgetting. In the continual-learning formulation, catastrophic forgetting is usually observed as a decrease in old-task performance after training on a new task, but that observation conflates changes in the shared representation with drift or misalignment in the task-specific classifier head. The representation-forgetting formulation in "Probing Representation Forgetting in Supervised and Unsupervised Continual Learning" isolates the representation by freezing it and training an optimal linear classifier from scratch on the frozen features (Davari et al., 2022).

This retraining-consistent perspective is motivated by a simple failure mode of non-retraining evaluation: a permutation or linear reparameterization of features can cause catastrophic observed accuracy drops for old heads despite information being preserved in the backbone. The same concern reappears in later work on unlearning. "Catastrophic Forgetting as Accessibility Collapse" separates knowledge storage, knowledge representation, and knowledge accessibility, and argues that complete behavioral forgetting can arise even when linear probes on the frozen backbone retain substantial task information (Trivedi et al., 4 Jun 2026). "Erased, but Not Gone: Output Forgetting Is Not True Forgetting" pushes the same point in the unlearning setting: output forgetting can be perfect while representation distributions remain inconsistent with retraining, because different internal geometries can induce the same outputs under the final linear head (Yong et al., 23 Jun 2026).

Taken together, these works define retraining-consistency as a methodological stance: forgetting should be judged against what can be recovered from the frozen representation under a fresh readout, or against what would have been obtained by retraining on the retained data alone. This suggests that the relevant question is not merely whether a deployed head fails, but whether the representational substrate itself has lost task-relevant structure.

2. Formal definitions and evaluation protocols

In the continual-learning formulation, let $f_\theta : X \to \mathbb{R}^d$ be the representation function and let task $k$ have dataset $D_k = \{(x_i,y_i)\}$. For a fixed frozen representation $f_\theta$, the probe is a linear classifier with weights $w \in \mathbb{R}^{C_k \times d}$ and bias $b \in \mathbb{R}^{C_k}$ trained by cross-entropy:

$$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$

“Optimal” means retraining the linear classifier from scratch to (near) convergence on the fixed features, with fresh initialization and standard regularization (Davari et al., 2022).

If $\theta^{pre}$ and $\theta^{post}$ denote snapshots before and after learning later tasks, and $A_k^{pre}$ and $k$0 are the corresponding test accuracies of freshly retrained probes on task $k$1, then per-task representation forgetting is

$k$2

A loss-based version is $k$3, and an aggregate form is

$k$4

The protocol is explicitly retraining-consistent: after each task, the representation is frozen, a new linear probe is trained from scratch for each prior task, and evaluation is performed under identical optimization conditions across snapshots. In the experiments of (Davari et al., 2022), linear probes were trained with AdamW, learning rate $k$5, weight decay $k$6, for 20–70 epochs depending on dataset.

A related analytically tractable surrogate is the representation discrepancy of "Measuring Representational Shifts in Continual Learning: A Linear Transformation Perspective" (Kim et al., 27 May 2025). For task $k$7, layer $k$8, and snapshots $k$9 and $D_k = \{(x_i,y_i)\}$0, it is defined as

$D_k = \{(x_i,y_i)\}$1

where $D_k = \{(x_i,y_i)\}$2 is the layer-$D_k = \{(x_i,y_i)\}$3 representation space induced on task-$D_k = \{(x_i,y_i)\}$4 inputs. By minimizing over linear alignment maps $D_k = \{(x_i,y_i)\}$5, the metric is invariant to permutations, rotations, and scalings of the representation basis. Empirically, the paper reports a strong linear relationship between representation discrepancy and linear-probing degradation, with $D_k = \{(x_i,y_i)\}$6 on Split-CIFAR100 and $D_k = \{(x_i,y_i)\}$7 on ImageNet1K (Kim et al., 27 May 2025).

In the accessibility-collapse formulation, the relevant scalar is the Accessibility Gap

$D_k = \{(x_i,y_i)\}$8

where $D_k = \{(x_i,y_i)\}$9 is linear-probe accuracy on the frozen final backbone and $f_\theta$0 is end-to-end accuracy using the native final head. Large positive $f_\theta$1 indicates preserved separability but collapsed accessibility through the learned readout (Trivedi et al., 4 Jun 2026).

In unlearning, retraining-consistency is often defined by direct comparison to a retrained retain-only reference model. The representation-level version in (Yong et al., 23 Jun 2026) requires both $f_\theta$2 and $f_\theta$3 for $f_\theta$4, where $f_\theta$5 is the unlearned model and $f_\theta$6 is retraining without the forget set. This is stricter than output-only forgetting.

3. Empirical behavior in continual learning

The empirical picture that emerges from (Davari et al., 2022) is that observed forgetting and representation forgetting often diverge sharply. On the ImageNet transfer sequence ImageNet $f_\theta$7 Scenes $f_\theta$8 CUB $f_\theta$9 Flowers, on Split CIFAR-100, on Split MiniImageNet, and on ImageNet32 with 200 tasks, observed accuracy on old tasks degrades more drastically than linear-probe accuracy. Finetuning with cross-entropy shows mild representation forgetting by linear probing despite large observed accuracy drops, while supervised contrastive learning (SupCon) shows an initial linear-probe drop followed by flat or increasing linear-probe performance, including positive backward transfer on long sequences (Davari et al., 2022).

The same study reports that naive finetuning can become competitive under the retraining-consistent lens. On Split MiniImageNet with 20 tasks, SupCon finetuning outperforms LwF over long sequences and approaches ER with small buffers of 5 samples per class, despite no access to old data. On ImageNet32 with 200 tasks, finetuning with cross-entropy reaches LwF performance by late tasks, and SupCon outperforms ER with 5 samples per class in linear-probe accuracy without rehearsal (Davari et al., 2022).

Capacity effects are especially visible under representation-level measurement. In the offline Split CIFAR-100 setting, widening ResNet-18 from width 32 to 128 raises Task 1 linear-probe accuracy at $w \in \mathbb{R}^{C_k \times d}$0 for cross-entropy finetuning from 64.8% to 70.5%, and for LwF from 70.1% to 74.8%, while observed Task 1 accuracy for cross-entropy remains around 20%. The paper concludes that larger capacity, especially width, reduces representation forgetting and brings naive finetuning closer to strong continual-learning methods in linear-probe performance, even when trained from scratch without explicit forgetting control (Davari et al., 2022).

Layerwise probing shows that representation drift is not uniform across depth. On Split CIFAR-10 with 2 tasks, observed Task 1 network accuracy drops from 85% to 63% after Task 2, while the degradation in linear-probe accuracy trained on post-Task-2 features is only $w \in \mathbb{R}^{C_k \times d}$1 at the top block; early blocks show minimal change or slight gains. The study further notes that CKA similarities can decrease strongly at high layers, such as ResNet block B-5 CKA $w \in \mathbb{R}^{C_k \times d}$2, while linear-probe accuracy remains high, indicating that linear separability is largely preserved despite substantial representation change (Davari et al., 2022).

A simple deployment-oriented consequence is prototype-based querying. For class $w \in \mathbb{R}^{C_k \times d}$3, the prototype mean is

$w \in \mathbb{R}^{C_k \times d}$4

and prediction is by nearest prototype under cosine similarity or negative Euclidean distance. On Split CIFAR-100 with 10 tasks, SupCon finetuning plus nearest-mean exemplars with 5 exemplars per class achieves Task 1 observed accuracy 48.0% and average observed accuracy 53.9% at $w \in \mathbb{R}^{C_k \times d}$5, exceeding LwF at 36.2% and 47.7%, and approaching ER-M5 at 52.1% and 54.8%, while avoiding extra compute or storage during training (Davari et al., 2022).

4. Accessibility collapse, geometry, and directional mismatch

The accessibility-collapse analysis of (Trivedi et al., 4 Jun 2026) sharpens the distinction between lost access and lost representation. In a ResNet-18 trained sequentially on Split CIFAR-100 without replay or regularization, Task 0 end-to-end accuracy drops from 0.548 after Task 0 training to 0.000 after learning all 10 tasks, while final-layer linear-probe accuracy on Task 0 falls only from 0.616 at the checkpoint to 0.468 in the final model, implying retention of approximately 76% of checkpoint-level representational information. Reinitializing and retraining only the final classifier on the frozen final backbone recovers 0.415 accuracy on Task 0, which is 75.7% of the original 0.548 (Trivedi et al., 4 Jun 2026).

Layerwise analysis localizes the degradation. The same paper reports Task 0 probe retention of 103% at Layer 1, 104% at Layer 2, 100% at Layer 3, 76% at Layer 4, and approximately 0% at the final classifier. Recoverable accuracy from frozen final-model features is 0.669 at Layer 1, 0.662 at Layer 2, 0.603 at Layer 3, 0.468 at Layer 4, and 0.415 after full-backbone classifier reset. Principal-angle and projection-energy analyses show mean principal angle approximately $w \in \mathbb{R}^{C_k \times d}$6, maximum approximately $w \in \mathbb{R}^{C_k \times d}$7, and projection energy increasing from 0.054 at $w \in \mathbb{R}^{C_k \times d}$8 to 0.583 at $w \in \mathbb{R}^{C_k \times d}$9, indicating that retained knowledge persists as distributed high-dimensional representations rather than as a small preserved dominant subspace (Trivedi et al., 4 Jun 2026).

The unlearning study (Yong et al., 23 Jun 2026) adds a further diagnostic: even when output forgetting appears successful, mismatch relative to retraining is structured rather than diffuse. On CIFAR-10 with ResNet-18, SCRUB obtains $b \in \mathbb{R}^{C_k}$0 and $b \in \mathbb{R}^{C_k}$1, yet $b \in \mathbb{R}^{C_k}$2, $b \in \mathbb{R}^{C_k}$3, and representation-level membership-inference attack success remains elevated. Directional alignment between unlearning and retraining shifts is high on the forget set but weak or negative on the retain set: for SCRUB, $b \in \mathbb{R}^{C_k}$4 and $b \in \mathbb{R}^{C_k}$5, while for SSD the corresponding values are 0.911 and $b \in \mathbb{R}^{C_k}$6 (Yong et al., 23 Jun 2026). The paper argues that current methods frequently produce forget/retain asymmetry and residuals concentrated along retraining-related directions.

These results jointly imply that retraining-consistent forgetting is not exhausted by output suppression. A representation can remain linearly recoverable, geometrically misaligned, or inaccessible only because the readout has failed. This suggests that retraining-consistent forgetting is as much about correct residual structure as about low apparent performance.

5. Extension to machine unlearning

Recent unlearning work generalizes retraining-consistency from continual-learning probes to direct comparison with retrained retain-only models. "Forgetting Has Neighbors" studies pointwise discrepancy between an unlearned model and the retrained-after-deletion reference and shows that this discrepancy is highly non-uniform: under gradient ascent and random labeling, it grows with geometric proximity to the forget set in representation space. On CIFAR-100 partial-class deletion with 50% deletion of the couch class, Local Teacher Distillation yields Avg. Gap $b \in \mathbb{R}^{C_k}$7, compared with 12.2 for random labeling, 7.7 for gradient ascent, and 6.6 for Influence Unlearning (Dolgova et al., 29 May 2026). The paper identifies this as localized collateral forgetting.

Other frameworks make the retraining reference explicit at the representation level. "POUR" introduces the Representation Unlearning Score (RUS) and proves that the orthogonal projection of a simplex Equiangular Tight Frame remains a simplex ETF in a lower-dimensional space, yielding a closed-form forgetting operator under Neural Collapse assumptions (Le et al., 24 Nov 2025). "Erase at the Core" evaluates unlearning against retraining using CKA and the Information Difference Index, where IDI is normalized so that 0 means equivalence to retraining, 1 matches the original, and negative values indicate over-forgetting; on ImageNet-1K with ResNet-50 and Random-100 forgetting, EC achieves penultimate-layer CKA 38.68 and $b \in \mathbb{R}^{C_k}$8, compared with CKA 69.52 and $b \in \mathbb{R}^{C_k}$9 for CU (Lee et al., 5 Feb 2026).

A different route is to operate directly in representation space without updating the backbone. "Representation Unlearning: Forgetting through Information Compression" freezes the encoder, learns a small transformation $$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$0 over penultimate-layer features, and derives variational mutual-information surrogates for preserving retain information while suppressing forget information. It reports speed-ups of up to 235× in the standard regime and 754× in the zero-shot regime relative to retraining (Almudévar et al., 29 Jan 2026). FAMR, by contrast, stays in parameter space but frames forgetting as anchored optimization,

$$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$1

and links its solution to leave-$$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$2-out retraining through an influence-function approximation and the bound

$$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$3

with $$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$4 the retrained weights on $$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$5 (Sanga et al., 17 Jun 2025).

A plausible implication is that retraining-consistent representation forgetting in unlearning has become a convergence point for several otherwise distinct research programs: pointwise local consistency, geometry-preserving projection methods, multi-layer representation erasure, and low-cost representation transformations are all benchmarked against the same counterfactual reference.

6. Limitations and open questions

The retraining-consistent perspective has several acknowledged limits. In continual learning, linear-probe RF depends on linear separability and on proper probe optimization; non-linear relationships, subtle margin changes, improper convergence, regularization choices, and class imbalance can all bias the estimate (Davari et al., 2022). The representation discrepancy analysis of (Kim et al., 27 May 2025) is tractable because it minimizes over linear alignments, but its theoretical assumptions—such as linear alignment of weights across tasks and drift growth with $$L_{CE}(w,b; f_\theta, D_k) = -\sum_{(x,y)\in D_k}\log \operatorname{softmax}(w f_\theta(x)+b)_y.$$6—are empirical rather than universal.

The scope of existing results is also uneven. The continual-learning study (Davari et al., 2022) focuses on task-incremental settings and does not study class-incremental behavior extensively. The accessibility-collapse study (Trivedi et al., 4 Jun 2026) centers on ResNet-18 and sequential CIFAR-100, with no variance across seeds or tasks reported. The unlearning critique in (Yong et al., 23 Jun 2026) establishes that output forgetting can hide retraining-inconsistent residuals, but scalable proxies for the retrained oracle remain open. The same paper explicitly identifies full retraining as the right oracle for benchmarking while noting that it may be impractical in deployment.

Unlearning work adds further open problems. Local Teacher Distillation depends on high-quality neighbor selection and remains limited by heterogeneous forget sets and scaling to larger collections (Dolgova et al., 29 May 2026). Projection-based approaches such as POUR rely on Neural Collapse structure and balanced-class geometry (Le et al., 24 Nov 2025). Representation-space transformation methods still lack certified equivalence to retraining, even when they improve forgetting, utility retention, and efficiency (Almudévar et al., 29 Jan 2026).

What is firmly established is narrower but important: head drift, readout collapse, and output suppression can each create the appearance of forgetting without corresponding representational erasure; retraining-consistent measurement exposes this gap. What remains unresolved is how to obtain the same diagnostic fidelity without full retraining, how to extend it reliably across modalities and scales, and which representation-level criteria are necessary for forgetting to be not merely apparent but counterfactually correct.