Measuring Representational Shifts in Continual Learning: A Linear Transformation Perspective

We thank the reviewer cno5 for the detailed review and constructive suggestions. We appreciate your acknowledgements that the theoretical aspects of this paper are clear and adequately justified, and that the findings are interesting and novel. Below we show our reply on your comments and questions.

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[R4-1]In Section 6.1, why compute the representation forgetting \Delta P^k_t instead of directly computing your newly introduced metric D^k_t? I understand that the two are correlated as stated in 4.2 but all the theoretical analysis is made on D^k_t, which should be computable

As noted in [R3-3] and [R3-6], we evaluated $D_t^k$ directly and observed results consistent with Figures 5 and 6. These will be included in the revised manuscript.

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[R4-2]I am unsure how much should the analysis of \Delta P^k_t gives insight on the behavior of the maximum value of D^k_t. Could you provide more explanation on this choice and its consequences?

Thank you for your insightful comments. Our original intent in conducting experiments with $\Delta P^k_t$ was to demonstrate that the theoretical results derived from $D^k_t$ also holds in $\Delta P^k_t$. However, we agree that experiments based solely on $\Delta P^k_t$ are insufficient to support the theoretical claims concerning $D^k_t$. Accordingly, we have conducted additional experiments using $D^k_t$, as detailed in [R3-3] and [R3-6].

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[R4-3]How far is the theoretical upper bound U^k_t,\infty from an experimental upper bound? For example, what is the discrepancy between the model trained on the first task and a randomly initialized model? Such information could be interesting to understand how much the model can forget, as for now it is not clear if this saturation is the consequence of "not being able the forget more" or if it occurs before reaching such a stage, therefore maintaining some of the previous knowledge (and if so, how far are we from "maximum forgetting"?).

We thank the reviewer for the insightful question. To clarify, at each layer $k$, the below plot reports the representation discrepancy $D_t^k$ between the task-1 model $h_1$ and two models:

• a randomly initialized model $h_0$, i.e., $\Delta t = -1$
• the final model after training on all tasks $h_N$, i.e., $\Delta t = N-1$

Our results show that $D^k_1(h_1, N-1)$ is significantly smaller than $D^k_1(h_1, -1)$, indicating that the saturation does not correspond to complete forgetting. Thus, the model retains a nontrivial amount of task-1 information even after learning all subsequent tasks.

https://hackmd.io/_uploads/ByCfAQOTke.png

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[R4-4]The increase of forgetting in latest layers is intuitive and expected. This has similarly been discussed in previous studies such as [1]. While the theoretical findings are interesting, further discussion with previous studies would be included.

Thank you for the suggestion.

We will include a discussion comparing our results with [1], which empirically observed greater drift in deeper layers. Our work complements this by providing the first theoretical explanation of this phenomenon (see Corollary 1 and Fig. 5).

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[R4-5]I would like the code to be shared

You can find code here: link

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[R4-6]From a theoretical perspective, it seems to me that such analysis can be adequately applied beyond Class-Incremental Learning scenarios (which is the only one considered here). I would appreciate discussion in this regard. While I believe the current experiments to be sufficient, experiments in Domain Incremental Learning cases would further strengthen the theoretical findings.

We agree that our framework can extend to domain-incremental learning (DIL), provided an appropriate modeling of weight perturbation as done in Assumption 2 for class-incremental learning. While a formal theory for DIL is left for future work, we provide preliminary empirical evidence in [R2-2] using rotated Split-CIFAR100, where Corollary 1 continues to hold.

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[R4-7]What about when the representations are normalized? Such strategy is very common in representation learning, how would the current analysis be impacted?

While we do not yet have a formal analysis under normalization, we offer the following intuition: normalization compresses the representation space by reducing feature norms, which in turn may reduce the magnitude of $D_t^k$. If the shift in linear structure is preserved under normalization, our discrepancy measure would still reflect representational drift, but with smaller scale. We will mention this in the revised manuscript.

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[R4-8]Why not use centroid distances for definition 3, instead of the most distant features?

We appreciate the suggestion. We follow the max-discrepancy formulation as in Guha et al. (2024), who adopted a similar worst-case approach in their theoretical analysis of catastrophic forgetting.

We thank the reviewer 4iQx for the detailed review and constructive suggestions. We appreciate your acknowledgements that the problem addressed in this manuscript is potentially important and interesting, and that our results are clearly distinct from the previous works. Below we show our reply on your comments and questions.

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[R3-1]The introduction of a new measure for representation forgetting is motivated by the difficulty of deriving an optimal linear classifier for hidden layers. I’m confused by this argument.

We believe that there may have been a misunderstanding regarding our motivation for introducing the measure $D^k_t$.

The term difficulty in our statement refers to theoretical intractability rather than computational cost. Our intention was not to use $D^k_t$ as a more efficient alternative for computing representation forgetting in practical settings. Rather, we introduced $D^k_t$ to facilitate the theoretical analysis of $\Delta P^k_t$.

We will clarify this point in the revised manuscript.

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[R3-2]It remains unclear how D^k_t and P^k_t scale or whether one provides an upper bound for the other.

Below we added plots showing $D_t^k$ versus $\Delta P_t^k$ across different layers $k$. In both datasets, $\Delta P_t^k$ scales approximately linearly with $D_t^k$, supporting its role as a surrogate. We will include these results in the revised manuscript.

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[R3-3] The empirical results evaluate the properties of only P^k_t not D^k_t. Thus, the analytical results do not support their claims on forgetting convincingly.

We conducted additional experiments directly evaluating $D_t^k$. As shown below, we observed a strong linear relationship between $D_t^k$ and $R_t^k$, similar to our observation in Fig.5 of the submitted manuscript. We also confirm the linearity between $U_t^k$ and $R_t^k$ as predicted by Corollary 1.

These results will be included in the revised manuscript.

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[R3-4]The motivation behind the definition of the representation discrepancy is unclear. It is susceptible to outlier, raising concerns about its robustness.

Thank you for your comment. As noted in [R3-1], our primary motivation for introducing the metric $D^k_t$ is to facilitate the theoretical analysis of $\Delta P^k_t$. We adopted the maximum discrepancy formulation following the approach of Guha et al. (2024), who succesfully employed a similar method in their theoretical study of catastrophic forgetting in continual learning.

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[R3-5]The proof of Prop 1., claims d increases linearly with \Delta t, citing Thm 4.1 of Guha & Lakshman (2024), but the theorem only shows this dependence for the upper bound of d. Hence, the proof and the following arguments on the two-stage dynamics is not well supported.

Thank you for the sharp comment. The reviewer is correct that the linear dependency of $d$ with respect to $\Delta t$ is not theoretically proven in Guha & Lakshman (2024). As such, our proof in Proposition 1 requires an additional assumption (that the lower bound of $d$ increases without a limit as a function of $\Delta t$) for the result to hold.

Under this assumption, the unique peak described in Proposition 1 may not always be guaranteed. However, this assumption is sufficient for the two-phase dynamics to hold, as the upper bound $U^k_t$ will eventually reach a peak and then saturate.

We also note that if $d$ itself monotonically increases with $\Delta t$, then the uniqueness of the peak as stated in Proposition 1 follows. We will clarify these assumptions and their implications in the revised version.

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[R3-6]The convergence rate in Fig. 6 is defined using P^k_t, which is different from the original definition in Eq. 7.

As suggested, we report the convergence rate using $U_t^k$ instead of $\Delta P_t^k$. Similar to Figure 6, we find that $\Delta t_{\text{sat}}$ decreases with layer index $k$, so the convergence rate $r_t^k = 1 / \Delta t_{\text{sat}}$ increases with $k$, consistent with Theorem 2.

https://hackmd.io/_uploads/r1VnzXOp1l.png

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[R3-7]Naively thinking, Assumption 1 shouldn't hold when w_k < w_{k-1}. Why is that the case?

Assumption 1 compares across different tasks $t$ and $t'$ for the same layer $k$, not across different layers. Thus, the width $w_k$ remains fixed for each comparison, and the condition holds regardless of whether $w_k < w_{k-1}$. This is why Assumption 1 is satisfied in Figure 2.

We thank the reviewer 9B8C for the detailed review and constructive suggestions. We appreciate your acknowledgements that our theoretical and experimental results are generally well-presented and convincing, and that our proofs are methodologically sound. Below we show our reply on your comments and questions.

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[R2-1] Generalizability and limitations of Assumption 1

Thank you for raising this important point.

We agree that Assumption 1 (existence of a near-perfect linear transformation $T$ aligning $W_t^k$ and $W_{t'}^k$) may not universally hold. In the revised manuscript, we clarify its scope and limitations.

As shown in Figure 2, Assumption 1 holds empirically for MLPs and ResNets, the latter of which includes skip connections (asked by the reviewer). To test generality, we conducted experiments on a Vision Transformer (ViT) (9 layers, 1 head) trained on Split-CIFAR100, optimizing $T$ to align $W_1^9$ and $W_{50}^9$, which showed clear convergence:

https://hackmd.io/_uploads/SkzzjJ4p1x.png

We also compare alignment errors:

While an approximate version of the assumption (e.g., $\| W_t^k - T W_{t'}^k \| \leq \varepsilon$) is possible, understanding its effect on theory is left for future work. This limitation will be explicitly discussed.

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[R2-2] Ablations across architectures or continual learning scenarios

Thanks for the suggestion.

We added experiments under a domain-incremental setup, training ResNet on rotated Split-CIFAR100 (same classes, different input rotations). We checked $R_t^k$ vs. $U_t^k$ for $t=1$ and $k=1,\dots,9$, observing consistent linear trends as predicted by Corollary 1:

https://hackmd.io/_uploads/r14Vgb_Tyg.png

In addition, we ran additional experiments for Vision Transformer (ViT) architecture to check the validity of Assumption 1, as shown in our response in [R2-1].

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[R2-3] Clarifying proofs and discussing limitations

Thank you. We revised the proofs to add explanations in non-trivial derivations:

For discussion on potential limitations of our assumption, please refer to our response in [R2-1].

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[R2-4] Referencing recent unsupervised and contrastive learning works

While our focus is on supervised continual learning, we agree that linking to recent related work adds context. We will include the following:

• Malviya et al., J. Comput. Sci., 2025
• Wen et al., arXiv:2405.18756, 2024
• Zhang et al., arXiv:2404.19132, 2024

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[R2-5] Practical implications of the theoretical findings

Thank you for the insightful suggestion.

While our metric is theoretical in nature, the insights derived from it suggest actionable strategies for improving continual learning (see [R1-5]). We will emphasize this connection more clearly in the revision.

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[R2-6] Hyperparameter sensitivity and dataset diversity

Thank you for the constructive suggestion.

We conducted experiments varying batch size ($B = 128, 256, 512$) and learning rate ($\gamma = 10^{-3}, 10^{-4}, 10^{-5}$) on Split-CIFAR100 using ResNet. The linear relationship between $R_t^k$ and $U_t^k$ remained consistent, demonstrating robustness:

As for dataset diversity, see [R2-2] for domain-incremental results on rotated Split-CIFAR100.

We thank the reviewer 1dag for the detailed review and constructive suggestions. We appreciate your acknowledgements that our proposed metric is a reasonable choice for facilitating theoretical analysis and that this paper is well-written. Below we show our reply on your comments and questions.

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[R1-1] In Definition 3, the self-distance is nonzero. Does this indicate that Definition 3 is not suitable for measuring distances between representation spaces?

We truly appreciate this comment and would like to thank all the reviewers for pointing this out. We found that there was a typo in Def.3 (as well as Eq.1 for recalling the definition in Guha et al. 2024) and revised Def.3 as follows. We assure you that our theoretical results and the corresponding proofs remain the same.

Previously, our definition of the distance between two representation spaces was based on computing the distance between the two most distant features in the respective representation spaces. Our newly revised definition only computes the maximum distance between the features of $\mathcal{R}^k_t(h_{t_1})$ and $\mathcal{R}^k_t(h_{t_2})$ with respect to the same input.

Thus, in our revised version, the self-distance is zero.

We will revise the manuscript accordingly.

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[R1-2] Without a discussion on the tightness of the upper bound, certain interpretations—such as the peak value in Proposition 1—become questionable.

We thank the reviewer for the insightful comments. In response, we conducted experiments using a ReLU network on Split-CiFAR100 dataset to assess the tightness of the upper bound $U^k_t$ with respect to the representation discrepancy $D^k_t$, when $t=1$ and $k=5$; see the below plot.

https://hackmd.io/_uploads/SyV_GCD6yg.png

Although there exists a gap between $U^k_t$ and $D^k_t$, we observe that U^k_t consistently tracks the trend of D^k_t, suggesting that its analysis still yields meaningful insights on D^k_t.

We will include this discussion in the revised manuscript.

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[R1-3] Other aspects of the paper may also raise concerns about precision and tightness, such as Definitions 3 and 4. The proposed distance/discrepancy measures are only approximations based on imprecise beliefs (as indicated by the use of “ \approx ” in Section 4.2).

We agree with the reviewer that our distance/discrepancy measures lack formal guarantees for exactly quantifying the representation forgetting. However, similar approximations have been employed in prior work, e.g., Guha et al. (2024), which used similar measures to analyze the catastrophic forgetting, and their empirical results were well-aligned with their theoretical insights. In our case as well, the empirical trends closely align with the theoretical predictions, supporting the practical relevance of our formulations.

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[R1-4] Figures 4 and 8 display multiple peaks, which contradicts the unique peak predicted by Proposition 1. Could this be evidence that the theoretical upper bound is not sufficiently tight?

Thank you for pointing this out.

The reviewer is true that the number of peaks is different from theory (in Proposition 1) and practice (in Figures 4 and 8), but we claim that the number of peaks is not an important factor in terms of figuring out the overall trend. More importantly, both curves (theory and practice) have similar trend of exibithing two phases -- forgetting phase and saturation phase -- where the amount of forgetting significantly increases at the first phase, and the amount of forgetting deviates in a smaller range in the second phase. We will mention this in the revised manuscript.

Regarding the tightness of our upper bound, please refer to our response in [R1-2].

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[R1-5] How useful are the insights in this paper for improving the performance of continual learning?

Our analysis reveals two key findings with direct implications for improving performance in continual learning: (1) representation forgetting tends to occur more rapidly and severely in deeper layers, and (2) increasing network width mitigates the degree and speed of forgetting. These insights suggest practical strategies such as allocating more regularization or memory resources to deeper layers, or designing architectures with wider representations in critical layers to slow down forgetting.