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Unlearning Isn't Deletion: Investigating Reversibility of Machine Unlearning in LLMs

Xiaoyu Xu, Xiang Yue, Yang Liu, Qingqing Ye, Haibo Hu, Minxin Du

Year: 2025Venue: arXiv preprintArea: Model EditingType: EmpiricalEmbeddings: 130

Models: Qwen-2.5-7B, Yi-6B

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Summary

The paper investigates the reversibility of machine unlearning in Large Language Models (LLMs), demonstrating that standard task-level metrics like accuracy and perplexity are often misleading. The authors introduce a representation-level analysis framework using PCA-based similarity, CKA, and Fisher information to identify four distinct forgetting regimes (Reversible/Irreversible vs. Catastrophic/Non-Catastrophic). They find that most current unlearning methods are reversible, meaning 'forgotten' information can be easily restored, and propose a taxonomy to guide the development of more robust, irreversible erasure algorithms.

Entities (6)

LLM · technology · 100%Machine Unlearning · methodology · 100%Qwen-2.5-7B · model · 99%Yi-6B · model · 99%PCA-based similarity · diagnostic-tool · 95%Reversibility · concept · 95%

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Yi-6B → usedin → Machine Unlearning

confidence 99% · Applying this framework across six unlearning methods, three data domains, and two LLMs

Representation-level analysis framework → evaluates → Machine Unlearning

confidence 98% · To address this critical evaluation gap, we introduce a representation-level analysis framework.

Machine Unlearning → exhibits → Reversibility

confidence 95% · This phenomenon of reversibility suggests that information is merely suppressed, not genuinely erased.

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Abstract

Abstract:Unlearning in large language models (LLMs) aims to remove specified data, but its efficacy is typically assessed with task-level metrics like accuracy and perplexity. We demonstrate that these metrics are often misleading, as models can appear to forget while their original behavior is easily restored through minimal fine-tuning. This phenomenon of \emph{reversibility} suggests that information is merely suppressed, not genuinely erased. To address this critical evaluation gap, we introduce a \emph{representation-level analysis framework}. Our toolkit comprises PCA-based similarity and shift, centered kernel alignment (CKA), and Fisher information, complemented by a summary metric, the mean PCA distance, to measure representational drift. Applying this framework across six unlearning methods, three data domains, and two LLMs, we identify four distinct forgetting regimes based on their \emph{reversibility} and \emph{catastrophicity}. Our analysis reveals that achieving the ideal state--irreversible, non-catastrophic forgetting--is exceptionally challenging. By probing the limits of unlearning, we identify a case of seemingly irreversible, targeted forgetting, offering new insights for designing more robust erasure algorithms. Our findings expose a fundamental gap in current evaluation practices and establish a representation-level foundation for trustworthy unlearning.

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Under Review Unlearning Isn’t Deletion: INVESTIGATING REVERSIBIL- ITY OF MACHINE UNLEARNING IN LLMS Xiaoyu Xu 1 Xiang Yue 2 Yang Liu 3 Qingqing Ye 1 Huadi Zheng 4 Peizhao Hu 4 Minxin Du 1 Haibo Hu 1 1 The Hong Kong Polytechnic University 2 Carnegie Mellon University 3 University of California, Santa Cruz 4 Huawei Technologies ABSTRACT Unlearning in large language models (LLMs) aims to remove specified data, but its efficacy is typically assessed with task-level metrics like accuracy and per- plexity. We demonstrate that these metrics are often misleading, as models can appear to forget while their original behavior is easily restored through minimal fine-tuning. This phenomenon of reversibility suggests that information is merely suppressed, not genuinely erased. To address this critical evaluation gap, we intro- duce a representation-level analysis framework. Our toolkit comprises PCA-based similarity and shift, centered kernel alignment (CKA), and Fisher information, complemented by a summary metric, the mean PCA distance, to measure repre- sentational drift. Applying this framework across six unlearning methods, three data domains, and two LLMs, we identify four distinct forgetting regimes based on their reversibility and catastrophicity. Our analysis reveals that achieving the ideal state–irreversible, non-catastrophic forgetting–is exceptionally challenging. By probing the limits of unlearning, we identify a case of seemingly irreversible, targeted forgetting, offering new insights for designing more robust erasure algo- rithms. Our findings expose a fundamental gap in current evaluation practices and establish a representation-level foundation for trustworthy unlearning. 1 1INTRODUCTION Large language models (LLMs), trained on massive corpora, have achieved remarkable success across diverse tasks, yet their capacity to memorize training snippets poses acute ethical, legal, and security risks. Memorization can unintentionally disclose sensitive, harmful, or copyrighted text [28;14;35], conflicting with emerging regulations, such as the EU’s Right to be Forgotten [8]. Machine unlearning seeks to address this challenge by algorithmically erasing the influence of specified data, making a model behave as if it had never been trained on that data [2]. While numerous unlearning methods have been developed for LLMs [40;12;29;20;19;22;37], their efficacy is typically assessed using task-level metrics, such as accuracy on a held-out “forget set.” However, these evaluations overlook a pivotal question: Does LLM unlearning achieve genuine erasure, or does it merely suppress information, leaving it poised to resurface? Emerging evidence suggests that many unlearning methods may only be superficially effective. For instance, after unlearning, a model may exhibit poor performance on the forget set, yet this “forgotten” knowledge can often be rapidly restored through a brief fine-tuning step, even on unrelated data [24; 26] (see Figure 1). Similarly, low-bit quantization of model weights has been shown to recover forget-set performance without any retraining, offering a simple probe for this reversibility [44]. These findings reveal a critical gap between task-level metrics and the model’s internal state. If purportedly erased information can be easily recovered, the unlearning process may amount to little more than a shallow perturbation, failing to provide meaningful safety or compliance guarantees. 1 https://github.com/XiaoyuXU1/Representational_Analysis_Tools 1 arXiv:2505.16831v2 [cs.CL] 26 Sep 2025 Under Review Reversible(Catastrophic) Forgetting Unlearned Original Relearned Irreversible (Catastrophic) Forgetting UnlearnedOriginal Relearned CKA Representation Subspace (Activation Subspace) CKA Representation Subspace (Activation Subspace) TaskLevel Accuracy TaskLevel Accuracy (a) PCA Similarity Original Updated PCA Shift Centroid Updated Original CKA Original Activation Subspace Updated FIM Parameter Loss Original Updated Analysis Tools (b) Figure 1: (a) task-level accuracy and CKA subspaces of reversible (top) vs. irreversible (bottom) catastrophic forgetting due to continual unlearning then relearning, (b) Our four diagnostic tools While prior work has noted the phenomenon of reversibility [44], it has not been systematically inves- tigated, particularly in the context of continual unlearning, a more realistic scenario where deletion requests arrive sequentially over a model’s lifecycle. This paper presents the first systematic analysis of the reversibility of LLM unlearning. We argue that standard task-level metrics are insufficient, as performance can collapse (a.k.a. catastrophic forgetting) even when the underlying feature repre- sentations remain largely intact. To move beyond superficial evaluations, we introduce a diagnostic toolkit for representational analysis, which includes PCA subspace similarity and shift [45], centered kernel alignment (CKA) [16], and Fisher information (FIM) [3]. Building on this, we propose the mean PCA distance as a quantitative measure to summarize the degree of representational drift. With our toolkit, we build a taxonomy characterizing unlearning along two axes: reversibility and catastrophicity (collateral damage to retained knowledge). This allows us to distinguish four regimes: 1) Reversible, Catastrophic: Global performance collapse that is fully recoverable via relearning. 2) Reversible, Non-Catastrophic: Targeted performance modest degradation that is easily restored. 3) Irreversible, Catastrophic: Permanent and unrecoverable global performance collapse. 4) Irreversible, Non-Catastrophic: Ideally, permanent erasure of target data without collateral damage. Crucially, we find that task-level metrics (e.g., forget accuracy) alone cannot distinguish between reversible and irreversible (catastrophic) forgetting, underscoring the necessity of our representation- level analysis; the evaluation must transcend task-level performance. We further show that transitions between these regimes depend on the unlearning method, the volume of unlearning requests, and key hyperparameters. In safety-critical applications, the central criterion for success should be the demonstrable achievement of irreversible erasure, not merely superficial performance degradation. Contributions. We summarize our main contributions as follows: • We present the first systematic study of reversibility in both single and continual LLM unlearning. We introduce a representation-level diagnostic toolkit and a quantitative metric, the mean PCA distance, to analyze representational drift and distinguish four regimes of forgetting. •We conduct extensive experiments with six unlearning methods on two LLMs (Yi-6B and Qwen- 2.5-7B) across three distinct domains. Our results demonstrate that standard task-level metrics (e.g., accuracy, perplexity, MIA susceptibility) are insufficient for assessing the true extent of unlearning. •We theoretically analyze weight perturbations to explain how widespread vs. localized parameter changes relate to (ir)reversible forgetting. Small perturbations near the logits can distort task-level metrics despite intact features, hence leading to misleading assessments. •We identify a case of seemingly irreversible, non-catastrophic forgetting, offering insights for designing more robust unlearning algorithms. We also highlight the potential for unlearning to serve as a form of data augmentation, improving model representations upon relearning. 2 Under Review 2PRELIMINARIES AND OUR FORMULATION LLM unlearning aims to remove the influence of specific data from a trained model to enhance privacy, improve safety, or mitigate bias [40;12;29;20;19;22]. The standard paradigm involves a training corpusD, from whichD f ⊆Dis designated as the forget set. A modelMis first trained on Dvia an algorithmA. An unlearning procedureUthen transformsMinto unlearnedM f , which should ideally behave as if it were trained only on the retain setD r =D f . Formally, the goal is to statistically approximate the retrained modelM r : M f =U (M,D f )≈M r =A(D r ). Retraining LLMs is prohibitively costly, so most studies rely on empirical proxies rather than formal statistically-indistinguishable guarantees [27;20;6]. Evaluations track forget quality on the forget set, utility, and downstream task accuracy on the retain set, aiming to preserve both dimensions. While current methods can achieve reasonable balances between forgetting and utility insingle-shot scenarios [1;7], they often falter in the practical continual setting, where removal requests arrive sequentially over time [1]. For a sequence of forget setsD (1) f ,D (2) f ,...,D (t) f (the union isD f ), the retain set isD (t) r aftertrounds. The model is then updated recursively:M (t) f =U (M (t−1) f ,D (t) f ) , which should be similar toM r = A M,D (t) r ,∀t . However, empirically, it often leads to catas- trophic forgetting–a severe decline in performance on both forgotten and retained knowledge [1;32]. Single-shot unlearning is “fragile:” fine-tuning, even on benign, unrelated data, can rapidly restore the supposedly “forgotten” knowledge [1;26;24]. Such fragility persists in continual unlearning as well. Prior work has noted this phenomenon but has not deeply investigated its underlying mechanics. 2.1A TAXONOMY OF FORGETTING REGIMES We hypothesize that this performance collapse does not necessarily equate to true information erasure; the knowledge might merely become latent or suppressed. To formalize this hypothesis, we introduce a taxonomy of forgetting based on two axes: catastrophicity (the extent of collateral damage to retained knowledge) and reversibility (whether forgotten knowledge can be recovered). Letθ 0 be the initial model parameters,θ u be the parameters after unlearning, andθ r be the parameters after a subsequent relearning phase (defined below). We useE(θ,T )to denote a performance metric (e.g., accuracy) evaluated on a task setT, which can be partitioned into a forget-related taskT f and a retain-related taskT r . We define four distinct regimes of forgetting, summarized in Table 1. Definition 1 (Four Regimes of Forgetting). Let∆ u (T ) = E(θ 0 ,T )−E(θ u ,T )be the performance drop after unlearning, and∆ r (T ) = E(θ 0 ,T )− E(θ r ,T )be the residual drop after relearning. The nature of forgetting is determined by these drops on the forget set (T f ) and retain set (T r ). Catastrophic vs. Non-Catastrophic: Forgetting is catastrophic if utility on the retain set degrades significantly (both ∆ u (T r ) and ∆ u (T u )≫ 0) and non-catastrophic otherwise (∆ u (T r )≈ 0). Reversible vs. Irreversible: Forgetting is reversible if relearning almost recovers initial performance on forget set (∆ r (T f )≈ 0) and irreversible if a significant performance drop persists (∆ r (T f )≫ 0). The combination of these two properties yields four regimes, among which the irreversible, non- catastrophic forgetting is deemed ideal, but remains challenging to achieve in practice. Relearning Restriction. To test reversibility, we must define a constrained relearning process that is distinct from a full retrain. After unlearning yieldsθ u , we obtainθ r by briefly fine-tuningθ u on a small relearning set, without access to raw pre-training data. This set is size-matched toD f (i.e., total budget is|D f |) and drawn from one of three sources: (i) the forget setD f itself, representing a worst- case recovery scenario, (i) a retain subsetD (t) r from domains similar toD f , or (i) general-domain, out-of-distribution data. This ensures a fair and direct assessment of knowledge recoverability. 3 Under Review Table 1: Four regimes of forgetting characterized by the reversibility and catastrophicity: denotes regimes commonly observed in practice, and G# denotes the ideal but elusive regime. RegimeObservedDescription Reversible, Catastrophic Performance on both forget and retain sets collapses, but is recoverable via relearning. Reversible, Non-Catastrophic Targeted performance drops on the forget set, which can be easily restored. Irreversible, Catastrophic Global, unrecoverable performance collapse on both forget and retain sets. Irreversible, Non-Catastrophic G#Targeted, permanent erasure of forget-set knowledge with no collateral damage. Table 2: Yi-6B: MIA / F.Acc / R.Acc (%) simple task using three LRs under single unlearning PhaseMethod LR=3×10 −6 LR=4×10 −6 LR=5×10 −6 MIAF.AccR.AccMIAF.AccR.AccMIAF.AccR.Acc Original–70.978.965.570.978.965.570.978.965.5 Unlearn GA45.565.454.043.862.452.341.260.350.9 GA+GD 65.475.164.658.273.865.855.368.563.5 GA+KL48.971.058.547.670.658.144.868.455.4 NPO67.276.264.765.275.862.862.275.262.7 NPO+KL66.576.364.867.276.463.264.575.661.2 RLabel69.677.764.769.276.564.568.775.463.3 Relearn GA67.276.665.268.677.662.867.676.965.5 GA+GD68.677.065.368.876.965.368.877.265.3 GA+KL 67.977.665.368.375.565.267.777.265.2 NPO68.277.165.368.277.265.268.377.065.1 NPO+KL68.977.165.367.976.363.068.676.965.2 RLabel 68.378.865.668.976.465.368.878.965.2 3CLASSIC (TASK-LEVEL) EVALUATION CAN BE DECEPTIVE 3.1EXPERIMENT SETUP Models and Datasets. We adopt two open-source LLMs: Yi-6B [41] and Qwen-2.5-7B [39]. To assess the generality of our findings, we employ two distinct dataset types for unlearning: (i) simple tasks, comprising arXiv abstracts and GitHub code from [40], and (i) a complex task, NuminaMath- 1.5, a recent benchmark for mathematical reasoning [18]. All experiments are performed on NVIDIA H100 GPUs. (Additional results on TOFU [27] and the Traditional-Chinese corpus 2 are in Appendix.) Unlearning algorithms. We evaluate six canonical unlearning methods, organized into three families. 1) Gradient-Ascent (GA) family. The unified goal isL = L forget D f + λL retain D r ,whereL forget maximizes the loss on the forget set via GA,L retain (optional) preserves utility on the retain set, and λ > 0balances the two. Choices forL retain give three variants: i) GA (L retain = 0), i) GA+GD (standard cross-entropy onDr), and i) GA+KL (KL divergence to the reference model onDr) [40]. 2) Negative Preference Optimization (NPO) family. GA is replaced by an NPO loss that penalizes agreement with the forget set [43]:L = L NPO D f + λL retain D r , Variants mirror those above: NPO (L retain = 0) and NPO+KL (retain-set KL regularization). 3) Random Label (RLabel). To mimic a model that never sawD f , true labels are replaced with random ones: L = L RLabel D f , inducing near-uniform predictions without GA/negative rewards [40]. Unlearning Scenarios. We consider two standard settings: i) Single unlearning: A trained modelM receives exactly one request to removeD f ⊂D, and i) Continual unlearning: The model processes a stream of requestsD (1) f ,...,D (t) f , yielding a sequence of models whereM (t) =U (M (t−1) ,D (t) f ). For simple tasks, we benchmark all six algorithms. For the complex math reasoning task, where a well-defined retain set is not available, we evaluate the core GA, NPO, and RLabel methods. Evaluation Metrics. In single unlearning (simple tasks), we measure forget-set accuracy (F.Acc), retain-set accuracy (R.Acc), and privacy leakage via min-k%-prob MIA AUC [31]. In continual unlearning (both task types), we provide a more comprehensive evaluation. For simple tasks, we report: F.Acc / R.Acc, forget/retain perplexity (F.Ppl / R.Ppl), downstream accuracy on 2 https://huggingface.co/datasets/taide/taide-bench 4 Under Review Table 3: Yi-6B: MIA / F.Acc / R.Acc (%) for simple task under four unlearning settings. Bold numbers indicate improvements over the Original baseline in F.Acc or R.Acc. PhaseMethodLR=3×10 −5 , N=100LR=5×10 −6 , N=100LR=3×10 −6 , N=100LR=3×10 −5 , N=6 MIAF.AccR.AccMIAF.AccR.AccMIAF.AccR.AccMIAF.AccR.Acc Original—70.878.965.570.878.965.570.878.965.570.878.965.5 Unlearn GA26.10.00.023.29.16.225.216.814.429.636.336.1 GA+GD16.89.72.328.73.63.169.478.865.566.977.064.0 GA+KL17.89.06.227.39.16.218.93.83.229.552.941.5 NPO60.137.837.950.651.052.368.478.364.168.771.659.4 NPO+KL59.064.355.965.477.664.366.778.865.567.967.656.1 RLabel65.10.00.063.60.10.461.40.40.762.772.761.1 Relearn GA74.52.11.868.080.065.068.680.865.268.270.558.7 GA+GD68.12.22.669.881.265.170.081.865.567.061.654.4 GA+KL70.71.71.668.381.164.870.781.063.265.066.656.2 NPO70.057.045.668.082.765.569.981.265.568.471.259.4 NPO+KL67.760.754.269.583.865.669.983.865.569.067.656.1 RLabel69.54.32.870.480.865.370.080.565.365.272.761.1 CommonsenseQA (CSQA )and GSM8K 0-shot [33;4], and min-k%-prob MIA AUC. For the complex task, we employ MATH 0-shot [9] and GSM8K 0-shot as the primary math utility benchmarks. Relearning Setting. To assess the reversibility and of unlearning, each run is followed by a controlled relearning phase. The unlearned model is briefly fine-tuned on specific data without access to the full pre-training corpus. For single unlearning, we fine-tune once on the entire forget setD f . For continual unlearning, we evaluate three conditions: (i) the cumulative forget set S t D (t) f , representing a worst-case adversarial scenario, (i) the corresponding retain subsetD (t) r , as a proxy for the data distribution, and (i) unrelated out-of-distribution data (general-domain samples explicitly different fromD f ). Each relearning dataset is size-matched to its corresponding unlearning request. Hyperparameter Configuration. To comprehensively evaluate the effects of unlearning, we design multiple hyperparameter configurations that vary both the learning rate and the number of unlearning requests. For single unlearning we sweep the learning rate overLR∈3, 4, 5×10 −6 while fixing the request count toN = 1. For continual unlearning we vary both knobs: on the simple task (Yi-6B) we testLR∈3, 5×10 −6 ∪3× 10 −5 withN ∈6→ 100; on the complex task (Qwen-2.5-7B) we useLR ∈ 3, 5× 10 −6 and3× 10 −5 together withN ∈ 6 → 100. All runs adopt the optimizer settings of [34]: AdamW [25] (β 1 = 0.9,β 2 = 0.95,ε = 10 −8 ), a cosine schedule with 10% warm-up followed by decay to 10% of peak, weight decay 0.1, and gradient clipping at 1.0. 3.2EVALUATION RESULTS We report quantitative results for single and continual unlearning on Yi-6B and Qwen-2.5-7B under various configurations. Complete results are provided in Appendix Tables 6 and 7. Single Unlearning. On Yi-6B, all six methods successfully reduce MIA and F.Acc, indicating a certain degree of forgetting (Table 2). The impact on the retain set is modest, with R.Acc dropping by only 2–5%. However, relearning often restores original performance; for instance, GA+KL and RLabel recover R.Acc to approximately 65% and F.Acc above 77%. These findings suggest that single unlearning achieves superficial forgetting, as the underlying representations remain largely intact (Section 4.2.1). This outcome characterizes the reversible, non-catastrophic forgetting regime. Continual Unlearning. Post-relearning analysis (Tables 3, 6, and 7) reveals two forms of reversible forgetting. In reversible, catastrophic forgetting, both utility (e.g., F.Acc, R.Acc) and privacy metrics drop sharply during unlearning but are fully restored after relearning. This is observed in GA and RLabel with moderate hyperparameters. Besides, reversible, non-catastrophic forgetting entails only a mild, easily recoverable performance degradation, as seen with NPO at LR = 3× 10 −5 , N = 6. Conversely, irreversible, catastrophic forgetting occurs when relearning fails to restore utility, leaving F.Acc and R.Acc low despite partial MIA recovery. This pattern is common for GA and RLabel under aggressive hyperparameters (e.g.,LR = 3× 10 −5 ,N = 100), where cumulative updates lead to irreversible representational collapse. The MIA AUC metric behaves erratically in this regime: it may fall below 50% during unlearning but misleadingly rebounds to high values after relearning, 5 Under Review even after the model’s capabilities have been permanently lost. These empirical results on single and continual unlearning are consistent with the theoretical framework in Section 5, which shows that small perturbations to the model weights can trigger disproportionately large drops in accuracy. 4REPRESENTATION-LEVEL EVALUATION 4.1REPRESENTATIONAL ANALYSIS TOOLS To analyze representational drift, we employ four hidden state diagnostics, as summarized in Fig- ure 1(b). Their precise definitions and implementation details are deferred to Appendix A.3. PCA Similarity, Shift, and Mean Distance. For each layeri, we collect activation matricesH orig i , H unl i , andH rel i on a probe setXfor the original, unlearned, and relearned models, respectively. Let c (∗) i,1 andp (∗) i,12 be the first principal direction and its mean projection for state(∗)∈orig, unl, rel. PCA Similarity is the cosine betweenc orig i,1 andc (∗) i,1 , while PCA Shift is the signed differencep (∗) i,12 . Small values for these metrics indicate stable features, whereas large, unrecovered shifts signify irreversible changes [45]. We also introduce the mean PCA distance, the average Euclidean distance on p (∗) i,12 across layers, to provide a single scalar measure of representation drift. Centered Kernel Alignment (CKA). Given centered activation matricesX orig i andY (∗) i , we compute CKA(X orig i ,Y (∗) i )∈ [0, 1]. Values≈ 1 mean nearly identical subspaces, those≈ 0 are orthogonal. Fisher information (FIM). We estimate the diagonal of the empirical FIM by averaging squared gradients over the probe setX. ComparingFIM orig ,FIM unl , andFIM rel reveals how unlearning alters the loss landscape and whether relearning restores parameter importance [15; 10]. All diagnostics are computed not only on the forget set but also on the retain set and unrelated data to distinguish targeted unlearning from general representational degradation. 4.2REPRESENTATIONAL RESULTS 4.2.1SINGLE UNLEARNING 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (a) Simple (Single Unlearning) 50050 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (b) Simple (Single Unlearning) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (c) Simple (Single Unlearning) 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (d) Simple (Single Unlearning) Figure 2: Single unlearning analysis on Yi-6B with GA un- der a simple task. In reversible non-catastrophic forgetting, PCA Similarity, PCA Shift, CKA, and FIM across layers show only minor changes with slight accuracy drops. Input queries are drawn from the forget set. Figure 2 demonstrates feature-level changes under single unlearning. (a) PCA Similarity remains near 1.0 across all layers, with minor, reversible dips, indicating that dominant activation di- rections are preserved. Slight dips in shallow and final layers are rapidly re- stored after relearning, suggesting min- imal and reversible drift. (b) PCA shifts are minimal, and relearned representa- tions closely realign with the original. (c) CKA values are nearly 1.0 for all model states, confirming that subspace structures remain intact. (d) FIM spec- tra show only mild, temporary shifts that are fully restored after relearning. These results, combined with the task-level evaluation in Section 3.2, demonstrate that single unlearning induces reversible, non-catastrophic forgetting. This high- lights the limitation of classic (task- level) metrics, which fail to capture the superficial nature of the forgetting. 6 Under Review 0102030 Layer 0 2 Cosine Similarity Reversible PCA Similarity Unlearned Relearned (a) (LR = 3× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity Reversible PCA Similarity Unlearned Relearned (b) (LR = 5× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity Irreversible PCA Similarity Unlearned Relearned (c) (LR = 3× 10 −5 , N = 100) 50050100 (PC1 ) 0 50 (PC2) Reversible PCA Shift Original Unlearned Relearned (d) (LR = 3× 10 −6 , N = 100) 0100200 (PC1 ) 0 50 (PC2) Reversible PCA Shift Original Unlearned Relearned (e) (LR = 5× 10 −6 , N = 100) 025005000750010000 (PC1 ) 0 2000 4000 (PC2) Irreversible PCA Shift Original Unlearned Relearned (f) (LR = 3× 10 −5 , N = 100) Figure 3: Layer-wise PCA Similarity and Shift for GA on Yi-6B (simple task). Vary LR3× 10 −6 , 5× 10 −6 , 3× 10 −5 atN = 100. Sustained low similarity or large shifts signal severe, irreversible catastrophic forgetting, whereas partial similarity or small shifts indicate mild, reversible catastrophic forgetting. Input queries are drawn from the forget set. 4.2.2CONTINUAL UNLEARNING As shown in Figures 3 and 5, PCA Similarity and Shift offer complementary views of representational change: similarity reflects global alignment, and shift is more sensitive to local variations. Relying on PCA similarity alone can obscure subtle effects; employing both avoids overlooking fine-grained distinctions, enabling a more comprehensive assessment. Higher learning rates or more requests cause sharp drops in similarity and large, unrecovered shifts, which is characteristic of irreversible catastrophic forgetting. In contrast, milder hyperparameters lead to high similarity and bounded shifts that are restored after relearning, consistent with reversible, catastrophic forgetting. This pattern is consistent across probe sets from forget set, retain set, and unrelated data (Figures 10 and 14). Figure 4 integrates CKA (top) and FIM (bottom) analyses. CKA reveals that mild unlearning main- tains stable alignment that recovers post-relearning, while aggressive unlearning causes irreversible degradation. The FIM spectra complement this by showing that continual unlearning flattens the loss landscape. Extreme hyperparameters induce a permanent leftward shift in sensitivity distributions, whereas moderate settings permit recovery. Together, these diagnostics suggest that observed perfor- mance loss is often due to temporary suppression rather than permanent erasure of knowledge. For conciseness, we present results on forget-set queries in the main text; retain-set queries, which yield similar conclusions under catastrophic forgetting, are in Appendix A.5 (e.g., Figures 12 and 10). Mean PCA Distance Analysis. To quantify representation-level drift with a single metric, we use the mean PCA distance. To assess the metric’s sensitivity, we compute its mean and standard deviation across four random seeds, as well as shuffling the order of unlearning requests. As shown in Table 4, higher learning rates consistently increase the mean PCA distance for both unlearned and relearned models At a low learning rate (e.g.,3× 10 −6 ), the mean and variance of the distance remain low after relearning, indicating stable and reproducible recovery. In contrast, a high learning rate (e.g., 3× 10 −5 ) sharply increases both values, reflecting greater variability and incomplete recovery. Combining these empirical findings with the task-level results in Section 3, we further connect to the theoretical analysis in Section 5, whose conclusions show that small weight perturbations induce only limited changes in the feature space, thereby enabling recovery, whereas larger perturbations accumulate into substantial representational drift that underlies irreversibility. 7 Under Review 051015202530 Layer index 0 2 Linear CKA Reversible CKA Unlearned Relearned (a) (LR = 3× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA Reversible CKA Unlearned Relearned (b) (LR = 5× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA Irreversible CKA Unlearned Relearned (c) (LR = 3× 10 −5 , N = 100) 10 9 10 5 10 1 10 3 10 7 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Reversible FIM @ Layer 31 Original Unlearned Relearned (d) LR=3× 10 −6 , Layer 31 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Reversible FIM @ Layer 31 Original Unlearned Relearned (e) LR=5× 10 −6 , Layer 31 10 21 10 14 10 7 10 0 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Irreversible FIM @ Layer 31 Original Unlearned Relearned (f) LR=3× 10 −5 , Layer 31 Figure 4: CKA for GA on Yi-6B, simple task. Vary LR3× 10 −6 , 5× 10 −6 , 3× 10 −5 with N = 100. High CKA ( 1) and concentrated FIM spectra indicates reversible catastrophic forgetting, while persistently low CKA and large-shifted, flattened spectra denote severe representational drift and irreversible catastrophic forgetting. Input queries are drawn from the forget set. Table 4: Yi-6B (GA): Mean±standard deviation of PCA distance on forget and retain sets across varying learning rates and random seeds; the number of unlearning requests is fixed. ModelLearning RatePhaseSeedMean PCA distance (forget set)Mean PCA distance (retain set) Yi-6B (GA) 3× 10 −6 Unlearn12, 22, 32, 429.62± 5.666.85± 4.10 3× 10 −6 Relearn12, 22, 32, 422.11± 1.421.64± 1.12 5× 10 −6 Unlearn12, 22, 32, 4211.52± 6.198.79± 5.20 5× 10 −6 Relearn12, 22, 32, 421.37± 0.741.05± 0.58 3× 10 −5 Unlearn12, 22, 32, 42133.20± 45.81121.45± 38.60 3× 10 −5 Relearn12, 22, 32, 42104.58± 39.7095.34± 32.40 5THEORETICAL ANALYSIS 5.1A PERTURBATION MODEL OF UNLEARNING To explain the empirical distinction between reversible and irreversible (catastrophic) forgetting, we introduce a perturbation model that links unlearning updates to representational changes across layers. Consider anL-layer feedforward networkf (x) = σ(W L σ(·σ(W 1 x)· ))with activation σand weightsW i L i=1 . We model unlearning as a layer-wise perturbation f W i = W i + E i , where the magnitude of the error term,|E i | = O(LR,N ), scales with the learning rate (LR) and the number of unlearning requestsN. A Neumann-series expansion of the network’s output shows that the total change e f (x)− f (x) is defined as P ∅̸=S⊆1,...,L (W L ◦·◦ E i k ◦·◦ W 1 )(x). When perturbations are small and localized to a few layers, first-order terms dominate, leading to reversible (catastrophic) forgetting. In contrast, when comparable perturbations are distributed across many layers, higher-order interaction terms accumulate, causing structural degradation that results in irreversible (catastrophic) forgetting. We can formalize the impact on our diagnostic tools: PCA Similarity. LetX i andY i = X i + E ′ i be the centered activations at layeribefore and after un- learning. By the Davis–Kahan theorem [5],cos∠(c orig i ,c upd i )≈ 1−O(∥E ′ i ∥/(λ 1,i −λ 2,i )), with top two eigenvalues λ 1,i ,λ 2,i . The layer-averaged PCA similarity is ̄ S PCA ≈ 1− O((1/L) P i ∥E ′ i ∥). PCA Shift. Along the first principal component, the activation-centroid shift is expressed as p i,12 = O(∥E ′ i ∥). Large perturbations∥E ′ i ∥propagating across multiple layers lead to irreversible representational drift, whereas smaller perturbations remain localized and thus reversible. 8 Under Review Table 5: Yi-6B (GA+GD+WAGLE) performance under different relearning settings. F.Acc/R.Acc are forget/retain accuracy, with mean PCA distances on the forget and retain sets. PhaseF.AccR.AccMean PCA distance (forget set)Mean PCA distance (retain set) Original model78.965.500 LR=2× 10 −5 , N = 50, relearned by retain set (N = 25) Unlearn37.855.911.846.28 Relearn46.958.39.005.91 LR=4× 10 −5 , N = 50, relearned by unrelated data (N = 50) Unlearn27.851.426.028.37 Relearn31.553.524.568.12 CKA. Let e K Y i = e K X i + ∆K i denote the perturbed Gram matrix at layeri. The corresponding CKA score is computed asCKA i = 1− O ∥∆K i ∥ ∗ /∥ e K X i ∥ ∗ . Averaging across layers yields ̄ C ≈ 1− O 1 L P i ∥∆K i ∥ ∗ , where ̄ C denotes the layer-averaged CKA. Fisher Information. Given updateδw i = O(∥E i ∥), the Fisher diagonal behaves asF i (w + δw) = F i (w) +O(∥δw i ∥), so the average Fisher becomes ̄ F = (1/P ) P i F i = F 0 −O((1/P ) P i ∥E i ∥) . 5.2BRIDGING REPRESENTATIONAL DRIFT AND TASK-LEVEL METRICS Classic (task-level) metrics can be misleading. They are highly sensitive to small weights’ changes, particularly in the final layers, which can cause large shifts in output probabilities without altering the model’s deeper representations. For a softmax output, a small perturbationδθto the model parameters yields a large change in log-probability:logp(y|x;θ + δθ)≈ logp(y|x;θ) +∇ θ logp(y|x;θ) ⊤ δθ + O(∥δθ∥ 2 ). A minor update to the logits can dominate this first-order term, causing a sharp drop in accuracy that suggests catastrophic forgetting, even if the underlying geometry is preserved. This aligns our theoretical model with the empirical findings in Sections 3 and 4.2. WhenLRorN is small, changes are confined to first-order effects, feature spaces remain intact, and forgetting is reversible. WhenLRorNis large, higher-order perturbations accumulate across layers, making recovery impossible and leading to irreversible forgetting. Figure 3 illustrates such a transition. Interestingly, relearning can sometimes lead to performance that exceeds the original model’s accuracy on the forget set (Table 3). This suggests that unlearning can act as a form of contrastive regularization, reinforcing salient features related to the forgotten data, which a brief relearning can then exploit. 5.3PROBING THE LIMITS OF IRREVERSIBILITY In our primary experiments, we did not observe irreversible non-catastrophic forgetting; even a small fraction (e.g.,10%) of the forget set was sufficient to restore performance. To explore this regime, we conducted extra experiments with more constrained relearning conditions. We used the GA+GD+WAGLE method [13], which selectively updates influential parameters, and limited the relearning data to either (i) 50% of the retain set or (i) an equal-sized, unrelated dataset (Table 5). Under these conditions, the method exhibited seemingly irreversible, non-catastrophic forgetting. The forget set showed large, unrecoverable PCA distances, while the retain set experienced only modest, partially recoverable degradation. This demonstrates that achieving the ideal of targeted, permanent unlearning without collateral damage remains an open challenge. Defining precise thresholds to distinguish the forgetting regimes (reversible vs. irreversible and catastrophic vs. non-catastrophic) is non-trivial, as they depend on the unlearning method, task complexity, and other factors. Our representational diagnostics reveal structural drift on each layer, pointing toward avenues for more robust and effective unlearning. A promising future direction is the design of new loss functions, such as a forget component penalization term to dismantle target representations and a retain component regularization term to preserve the geometry of desired representations. 9 Under Review 6CONCLUSION This work demonstrates that class (task-level) evaluations of LLM unlearning are insufficient, as performance collapse often masks the reversibility of forgetting. Models may appear to have erased data while their internal representations remain intact and easily recoverable. Our representation-level toolkit reveals that genuine forgetting requires substantial, coordinated weight perturbations. Minor updates often create only a superficial, reversible effect. We find that achieving the ideal goal of irreversible, non-catastrophic forgetting remains an open challenge, exposing a fundamental limitation in current methods. 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[45]Junhao Zheng, Xidi Cai, Shengjie Qiu, and Qianli Ma. Spurious forgetting in continual learning of language models. In ICLR, 2025. [46]Andy Zou, Zifan Wang, J. Zico Kolter, and Matt Fredrikson. Universal and transferable adversarial attacks on aligned language models. arXiv:2307.15043, 2023. 13 Under Review AAPPENDIX A.1LIMITATIONS Our experiments target two LLMs and a handful of tasks and unlearning methods; although our diagnostic framework is model-agnostic and designed to scale, empirical validation on much larger models and production-scale pipelines remains to be done. The constrained relearning protocol and selected metrics provide clear insights into representational drift but are not exhaustive and do not offer formal privacy guarantees. In this work, we primarily relied on four diagnostic tools—PCA similarity, PCA shift, CKA, and Fisher information—to capture different aspects of representational drift. Other feature-level methods, such as correlation-based approaches (e.g., SVCCA [30]), offer similar perspectives on subspace similarity. Incorporating a broader suite of analytic tools is an important direction for future work. A.2RELATED WORK Machine Unlearning. Machine unlearning has emerged as a critical direction for addressing privacy, safety, and bias in large language models (LLMs) [40;12;29;20;23;7;32;37;44;42; 38;36;2]. It is typically defined as either exact or approximate [2]. Exact unlearning requires the resulting model to be indistinguishable from one retrained from scratch on the retain set, fully eliminating any statistical trace of the forget set. Approximate unlearning relaxes this requirement to distributional or behavioral similarity, demanding only comparable outputs (e.g., perplexity or forget-set accuracy) between unlearned and retrained models [27; 32]. For modern LLMs, however, exact unlearning is computationally infeasible, as full retraining or partition-based schemes scale poorly [2]. Consequently, approximate methods dominate practice in LLMs. Single-Shot Unlearning. Most existing approaches are designed for single deletion events. Gradient-based strategies (e.g., GA) enforce forgetting directly but often incur significant util- ity loss [40]. Recent advances such as WAGLE augment these methods with weight attribution (e.g., GA+GD+WAGLE), selectively updating the most influential parameters to enhance forgetting efficacy while mitigating utility degradation [13]. Prompt-based steering avoids parameter updates, reducing cost, but typically achieves only superficial forgetting with vulnerability to reactivation [23]. Model-editing methods, such as AlphaEdit [21], are lightweight and potentially robust, yet their behavior under sequential or heterogeneous requests remains underexplored. Continual Unlearning. When unlearning requests arrive sequentially, naive extensions of single- shot methods tend to compound damage, leading to catastrophic forgetting and unstable dynamics [1; 32]. Each request operates on an already modified model, magnifying utility loss. Recent work has attempted to mitigate this through orthogonal updates (e.g., LoRA-based unlearning [11]) and OOD detectors. ALKN [36] advances this line by providing a principled framework for continual unlearning, introducing parameter-level interventions and adaptive modules to counteract accumulative decline. Evaluations. Evaluating unlearning efficacy remains an open challenge. Existing studies rely on three main classes of metrics. First, classic (task-level) metrics such as accuracy, perplexity [40;20] are widely used but can be misleading, since performance degradation does not guarantee removal of knowledge. Second, memorization probes [17] assess verbatim recall, offering finer granularity but failing to capture semantic or paraphrased knowledge. Third, robustness-based evaluations examine vulnerabilities to jailbreaking [46], relearning attacks [24], and even quantization attacks [44]. For quantization attacks, low-bit compression restores forget-set behavior without direct access to the forget. While existing approaches expose weaknesses in reversibility, they often conflate forgetting with class task-level degradation and lack structural insight. Our representation-level toolkit closes this gap by jointly diagnosing reversibility and catastrophicity, yielding a more faithful understanding of what is truly forgotten. We apply this toolkit to both single unlearning and continual unlearning, the latter of which has not been systematically investigated despite being a more realistic scenario where deletion requests arrive sequentially over a model’s lifecycle. 14 Under Review A.3DETAILED ANALYSIS TOOLS PCA Similarity and PCA Shift. For each Transformer layer, we perform PCA on the hidden activations of the original and updated models. Letc orig i,1 andc upd i,1 denote the first principal component (PC1) directions of layer i. The PCA Similarity is defined as PCA-Sim(i) = cos c orig i,1 ,c upd i,1 = (c orig i,1 ) ⊤ c upd i,1 ∥c orig i,1 ∥c upd i,1 ∥ ∈ [−1, 1], where values near1indicate stable directional alignment, and values near−1suggest a near- orthogonal shift in dominant directions. To capture translational drift, we also compute the mean projection of activations along PC1 and PC2: PC1 ∆(i) = c i,1 upd − c i,1 orig , PC2(i) = c i,2 upd , p i,12 = (PC1 ∆(i),PC2(i)) wherep i,12 quantifies displacement along PC1 and captures orthogonal deviation along PC2. These metrics reflect how the representation center drifts within the top subspace. Centered Kernel Alignment (CKA).To assess subspace alignment, we use linear Centered Kernel Alignment (CKA) [16], which compares activation matricesX,Y ∈ R N×D from before and after unlearning. First, we compute the centered Gram matrices: e K X = HXX ⊤ H, e K Y = HY Y ⊤ H, H = I N − 1 N 11 ⊤ . The CKA score is then given by: CKA(X,Y ) = Tr( e K X e K Y ) q Tr( e K 2 X ) q Tr( e K 2 Y ) ∈ [0, 1], where values near1indicate highly overlapping subspaces, and values near0signal near- orthogonality. Fisher Information. To measure parameter-level importance, we compute the diagonal of the empirical Fisher Information Matrix (FIM). For each parameterw i and input distributionD dis , the diagonal entry is approximated as: FIM i ≈ E (x,y)∼D dis h (∂ w i logp(y | x;w)) 2 i . Larger values indicate thatw i has a stronger influence on the model’s predictions. A substantial leftward shift in the Fisher spectrum after unlearning implies a flattened loss landscape and diminished parameter sensitivity. Together, these tools form a feature-space diagnostic suite: FIM captures global sensitivity, CKA measures subspace preservation, and PCA-based metrics expose fine-grained geometric drift across layers—enabling a robust assessment of representational degradation during unlearning. A.4MEAN PCA DISTANCE UNDER DIFFERENT DATASET To examine the role of distributional alignment, we evaluate unlearning and relearning under two dataset settings. First, we use the TOFU benchmark [27], where both unlearning and relearning occur within the same distribution. Second, treating different languages as out-of-distribution (OOD), we include a Traditional-Chinese corpus for relearning. This setup enables us to probe whether cross-lingual signals can drive effective recovery, and how their efficacy compares with in-distribution. Table 8 confirms that cross-lingual relearning improves the model but achieves less complete restoration than English data: mean PCA distance and related summary metrics move closer to baseline values, yet remain substantially higher. Greater linguistic or domain dissimilarity therefore reduces the efficacy of recovery, though partial restoration is still attainable. For the TOFU dataset, the overall pattern holds: learning rate and the number of unlearning requests (N) effectively regulate feature drift and reversibility. However, the representational shifts induced by TOFU are milder than those observed in our simple and complex tasks. We attribute this to the smaller and less diverse nature of TOFU’s corpus; many entries are short and contain only author metadata, making its impact on the model’s feature space comparatively limited. 15 Under Review Table 6: Yi-6B simple-task metrics under four(LR,N )settings. For each block: forget/retain per- plexity (F.Ppl / R.Ppl), forget/retain accuracy (F.Acc / R.Acc), CommonsenseQA (CSQA), GSM8K, and membership-inference AUC (MIA). PhaseMethodF.PplR.PplF.AccR.AccCSQAGSM8KMIA LR=3× 10 −5 , N = 100 Original—3.87.878.965.573.139.670.9 Unlearn GA∞ ∞0.00.019.30.026.1 GA+GD∞ ∞9.72.319.70.016.8 GA+KL∞ ∞9.06.219.60.017.8 NPO31296.5597.937.837.962.21.060.1 NPO+KL348080.24482.064.355.964.91.459.0 Rlable63791.765903.40.00.020.90.065.1 Relearn GA137094.5758443.52.11.819.70.074.5 GA+GD5274.59568.62.22.619.60.068.1 GA+KL5037.115019.91.71.620.60.070.7 NPO16.641.757.045.651.80.670.0 NPO+KL21.816.260.754.348.00.967.7 Rlable4056.115048.64.32.819.70.069.5 LR=5× 10 −6 , N = 100 Unlearn GA∞ ∞9.16.219.60.023.2 GA+GD∞ ∞3.63.124.50.028.7 GA+KL∞ ∞9.16.219.60.027.3 NPO3017.71110.650.152.372.937.550.6 NPO+KL38.5232.477.664.373.137.665.4 Rlable57035.453377.10.10.419.10.063.6 Relearn GA3.77.880.064.970.239.968.0 GA+GD3.67.681.265.172.139.069.8 GA+KL3.68.481.164.871.640.768.3 NPO3.57.682.765.574.039.768.0 NPO+KL3.57.883.865.674.139.769.5 Rlable3.67.780.865.371.839.270.3 LR=3× 10 −6 , N = 100 Unlearn GA∞ ∞16.814.469.512.325.2 GA+GD3.37.678.865.577.037.569.4 GA+KL∞ ∞35.440.663.218.318.9 NPO3.77.978.365.073.338.768.4 NPO+KL3.88.178.465.173.638.666.7 Rlable36794.732562.03.83.219.32.261.4 Relearn GA3.77.680.865.273.439.968.6 GA+GD3.67.481.865.572.139.070.0 GA+KL3.610.381.063.367.240.770.7 NPO3.57.581.265.472.939.769.9 NPO+KL3.57.583.865.573.039.769.9 Rlable3.67.680.565.372.239.270.0 LR=3× 10 −5 , N = 6 Unlearn GAinfinf36.336.169.15.829.6 GA+GD209.320.677.064.070.037.866.9 GA+KLinfinf53.041.568.32.029.5 NPO12.310.771.659.471.724.768.7 NPO+KL8.910.774.762.172.832.267.9 Rlable51589.240622.90.40.719.80.062.6 Relearn GA6.811.470.558.764.518.468.2 GA+GD12.311.561.654.461.37.367.1 GA+KL17.111.666.656.260.63.065.0 NPO6.011.671.259.459.42.068.4 NPO+KL7.311.667.656.142.91.669.0 Rlable6.411.472.761.167.528.965.2 A.5DETAILED ANALYSIS RESULTS A.5.1PRINCIPAL COMPONENT ANALYSIS: SIMILARITY AND SHIFT Across the same hyper-parameter grid, Figure 7 (PCA–Similarity) and Figure 11 (PCA–Shift) provide complementary views of representational drift. For GA, higher learning rates drive unlearned states (orange) far from the original (blue), while relearning (green) fails to return, producing long rays of irreversible drift. GA+GD narrows the spread but still collapses at 3×10 −5 . 16 Under Review Table 7: Qwen-2.5-7B: MIA / MATH / GSM8K Accuracy (%) for complex task under four settings. Bold numbers indicate improvements over the Original baseline in MATH or GSM8K. PhaseMethodLR=3× 10 −5 , N =6LR=3× 10 −6 , N =6LR=5× 10 −6 , N =6LR=5× 10 −6 , N =100 MIAMATHGSM8KMIAMATHGSM8KMIAMATHGSM8KMIAMATHGSM8K Original—–99.39.080.199.39.080.199.39.080.199.39.080.1 Unlearn GA5.90.00.00.90.00.03.80.00.05.50.00.0 NPO95.90.00.297.421.574.167.424.171.894.70.00.4 RLabel35.50.00.069.60.01.511.20.00.02.90.00.0 Relearn GA97.60.01.199.35.183.299.49.377.899.20.00.0 NPO95.80.00.099.44.782.699.416.575.799.20.00.0 RLabel99.50.00.099.35.383.399.310.077.299.60.00.0 Table 8: Yi-6B (GA): Mean PCA distance under different learning rates. The left block uses China Taiwan for relearning only, while the right block uses TOFU for both unlearning and relearning. Relearning with China TaiwanUnlearning + Relearning with TOFU Learning RatePhaseMean PCA distance (forget set)PhaseMean PCA distance (forget set) 3× 10 −6 Unlearn17.12Unlearn0.51 3× 10 −6 Relearn4.98Relearn0.27 5× 10 −6 Unlearn20.27Unlearn2.41 5× 10 −6 Relearn10.77Relearn1.08 3× 10 −5 Unlearn193.13Unlearn11.96 3× 10 −5 Relearn167.32Relearn11.02 On Qwen-2.5-7B, GA shifts span thousands of PC1 units and drive PC2 to extreme negatives (Figure 13c,f,i), consistent with the multi-layer perturbations predicted in Section 4. In complex tasks such as mathematical reasoning, even small perturbations in hidden states can lead to substantial performance differences. This is reflected in our PCA–Similarity analysis, where seemingly minor changes in hidden state geometry correspond to meaningful behavioral variations. Besides, PCA- Similarity captures global alignment, whereas PCA–Shift highlights fine-grained translational drift. This distinction also explains why Figure 9h,i show only moderate misalignment under similarity but reveal pronounced displacements under shift (cf. Figure 13). Using both metrics thus provides a more complete characterization of reversibility. Overall, these results confirm that GA, with or without GD or KL, induces large and often irreversible displacements, whereas NPO variants, and to a lesser extent RLabel, constrain less shifts, consistent with our utility findings. A.5.2CENTERED KERNEL ALIGNMENT ANALYSIS Figures 15–17 report layer-wise linear CKA between the original model and its unlearned or relearned counterparts. Across both Yi-6B and Qwen-2.5-7B, GA stands out: as the learning rate orNincreases, its CKA curve drops close to zero in most layers and fails to recover, revealing a deep subspace fracture consistent with the irreversible PCA trends. GA+GD and GA+KL mitigate this decline to some extent but do not restore full alignment after relearning. Task complexity does not alter the ordering but amplifies the differences. On the math-heavy Qwen benchmark, GA’s tail layers fall almost to zero at high learning rates, whereas NPO maintains significantly higher alignment. Taken together with the PCA-Shift results, these findings show that GA-style objectives consistently break subspace alignment, NPO variants preserve much greater stability, and RLabel induces moderate but partly recoverable distortions. A.5.3FISHER INFORMATION ANALYSIS Figures 19–33 plot the empirical Fisher spectra layer by layer. Across both Yi-6B (simple) and Qwen-2.5-7B (complex), GA and its variants exhibit a pronounced leftward shift of the diagonal histogram as LR orNincrease. The peaks move several orders of magnitude in middle and deep layers, reflecting a flattened loss surface and diminished parameter salience. Crucially, these shifts persist after relearning, marking the onset of irreversible forgetting. 17 Under Review 0102030 Layer 0 2 Cosine Similarity Reversible PCA Similarity Unlearned Relearned (a) (LR = 3× 10 −5 , N = 6) 0102030 Layer 0 2 Cosine Similarity Reversible PCA Similarity Unlearned Relearned (b) (LR = 3× 10 −5 , N = 50) 0102030 Layer 0 2 Cosine Similarity Irreversible PCA Similarity Unlearned Relearned (c) (LR = 3× 10 −5 , N = 100) 0246 (PC1 ) 0 50 (PC2) Reversible PCA Shift Original Unlearned Relearned (d) (LR = 3× 10 −5 , N = 6) 50050100 (PC1 ) 0 50 (PC2) Reversible PCA Shift Original Unlearned Relearned (e) (LR = 3× 10 −5 , N = 50) 025005000750010000 (PC1 ) 0 2000 4000 (PC2) Irreversible PCA Shift Original Unlearned Relearned (f) (LR = 3× 10 −5 , N = 100) Figure 5: Layer-wise PCA Similarity and Shift for GA on Yi-6B (simple task). varyN ∈6, 50, 100 at LR= 3× 10 −5 . Sustained low similarity or large shifts signal severe, irreversible catastrophic forgetting, whereas partial similarity or small shifts indicate mild, reversible catastrophic forgetting. Input queries are drawn from the forget set. NPO, NPO+KL, and RL produce smaller leftward displacements under moderate LR orN, and their Fisher spectra recenter after relearning, indicating primarily reversible drift. Under extreme settings (e.g.,LR = 3× 10 −5 orN = 100), these methods also show persistent displacement in some layers, suggesting milder but still irreversible forgetting. Figures 14, 10, 18, and 34 examine relearning dynamics when the fine-tuning data and input query are drawn from the forget set, the retain set, or an unrelated data: i) across all sources, the overall trends are similar: alignment can be partially restored, but recovery is consistently weaker with unrelated data, underscoring that effective relearning depends on both the size and the relevance of the training set; i) the observed behavior also varies with the choice of input queries. In the case of reversible catastrophic forgetting, all forget set, retain set, and unrelated data undergo the similar feature drifts. 18 Under Review 051015202530 Layer index 0 2 Linear CKA Reversible CKA Unlearned Relearned (a) (LR = 3× 10 −5 , N = 6) 051015202530 Layer index 0 2 Linear CKA Reversible CKA Unlearned Relearned (b) (LR = 3× 10 −5 , N = 50) 051015202530 Layer index 0 2 Linear CKA Irreversible CKA Unlearned Relearned (c) (LR = 3× 10 −5 , N = 100) 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Reversible FIM @ Layer 31 Original Unlearned Relearned (d) N = 6, Layer 31 10 8 10 3 10 2 10 7 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Reversible FIM @ Layer 31 Original Unlearned Relearned (e) N = 50, Layer 31 10 21 10 14 10 7 10 0 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Irreversible FIM @ Layer 31 Original Unlearned Relearned (f) N = 100, Layer 31 Figure 6: CKA and FIM for GA on Yi-6B, simple task. Vary LR= 3× 10 −5 withN ∈6, 50, 100. High CKA ( 1) and concentrated FIM spectra indicates reversible catastrophic forgetting, while persistently low CKA and large-shifted, flattened spectra denote severe representational drift and irreversible catastrophic forgetting. Input queries are drawn from the forget set. 19 Under Review 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (a) Simple (LR = 3× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (b) Simple (LR = 5× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (c) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (d) Simple (LR = 3× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (e) Simple (LR = 5× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (f) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (g) Simple (LR = 3× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (h) Simple (LR = 5× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (i) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (j) Simple (LR = 3× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (k) Simple (LR = 5× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (l) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (m) Simple (LR = 3× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (n) Simple (LR = 5× 10 −6 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (o) Simple (LR = 3× 10 −5 , N = 100) Figure 7: PCA Similarity Across Layers. Each row shows results under different unlearning methods: GA+GD (a–c), GA+KL (d–f), NPO (g–i), NPO+KL (j–l), and Rlable (m–o). All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixedN = 100. 20 Under Review 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (a) Simple (LR = 3× 10 −5 , N = 6) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (b) Simple (LR = 3× 10 −5 , N = 50) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (c) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (d) Simple (LR = 3× 10 −5 , N = 6) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (e) Simple (LR = 3× 10 −5 , N = 50) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (f) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (g) Simple (LR = 3× 10 −5 , N = 6) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (h) Simple (LR = 3× 10 −5 , N = 50) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (i) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (j) Simple (LR = 3× 10 −5 , N = 6) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (k) Simple (LR = 3× 10 −5 , N = 50) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (l) Simple (LR = 3× 10 −5 , N = 100) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (m) Simple (LR = 3× 10 −5 , N = 6) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (n) Simple (LR = 3× 10 −5 , N = 50) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (o) Simple (LR = 3× 10 −5 , N = 100) Figure 8: PCA Similarity Across Layers. Each row shows results under different unlearning methods: GA+GD (a–c), GA+KL (d–f), NPO (g–i), NPO+KL (j–l), and Rlable (m–o). Simple task on Yi-6B with fixed learning rate LR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 21 Under Review 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (a) Complex (LR = 3× 10 −6 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (b) Complex (LR = 5× 10 −6 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (c) Complex (LR = 3× 10 −5 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (d) Complex (LR = 3× 10 −6 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (e) Complex (LR = 5× 10 −6 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (f) Complex (LR = 3× 10 −5 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (g) Complex (LR = 3× 10 −6 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (h) Complex (LR = 5× 10 −6 , N = 6) 01020 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (i) Complex (LR = 3× 10 −5 , N = 6) Figure 9: PCA Similarity Across Layers. Each row shows results under different unlearning methods: GA (a-c) NPO (d–f), Rlable (g–j). All plots are for the complex task on Qwen2.5-7B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 6. 0102030 Layer 0 2 Cosine Similarity Reversible PCA Similarity Unlearned Relearned (a) Simple (Relearned by forget set) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (b) Simple (Relearned by Retain set) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (c) Simple (Relearned by unrelated data) 0102030 Layer 0 2 Cosine Similarity Reversible PCA Similarity Unlearned Relearned (d) Simple (input data = forget set) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (e) Simple (input data = retain set) 0102030 Layer 0 2 Cosine Similarity PCA Similarity Unlearned Relearned (f) Simple (input data = unrelated data) Figure 10: PCA Similarity Analysis for GA under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): PCA similarity is measured using the forget set, retain set, or unrelated data as evaluation input. 22 Under Review 02040 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (a) Simple (LR = 3× 10 −6 , N = 100) 1000100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (b) Simple (LR = 5× 10 −6 , N = 100) 020004000 (PC1 ) 250 0 250 (PC2) PCA Shift Original Unlearned Relearned (c) Simple (LR = 3× 10 −5 , N = 100) 50050100 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (d) Simple (LR = 3× 10 −6 , N = 100) 0100200300 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (e) Simple (LR = 5× 10 −6 , N = 100) 010002000 (PC1 ) 200 0 (PC2) PCA Shift Original Unlearned Relearned (f) Simple (LR = 3× 10 −5 , N = 100) 0102030 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (g) Simple (LR = 3× 10 −6 , N = 100) 20020 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (h) Simple (LR = 5× 10 −6 , N = 100) 0100200 (PC1 ) 0 50 100 (PC2) PCA Shift Original Unlearned Relearned (i) Simple (LR = 3× 10 −5 , N = 100) 02040 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (j) Simple (LR = 3× 10 −6 , N = 100) 50050100 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (k) Simple (LR = 5× 10 −6 , N = 100) 0100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (l) Simple (LR = 3× 10 −5 , N = 100) 1000100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (m) Simple (LR = 3× 10 −6 , N = 100) 1000100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (n) Simple (LR = 5× 10 −6 , N = 100) 0500100015002000 (PC1 ) 100 0 100 (PC2) PCA Shift Original Unlearned Relearned (o) Simple (LR = 3× 10 −5 , N = 100) Figure 11: PCA Shift Across Layers. Each row shows results under different unlearning methods: GA+GD (a–c), GA+KL (d–f), NPO (g–i), NPO+KL (j–l), and Rlable (m–o). All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixedN = 100. 23 Under Review 024 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (a) Simple (LR = 3× 10 −5 , N = 6) 0100200300 (PC1 ) 0 50 100 (PC2) PCA Shift Original Unlearned Relearned (b) Simple (LR = 3× 10 −5 , N = 50) 020004000 (PC1 ) 250 0 250 (PC2) PCA Shift Original Unlearned Relearned (c) Simple (LR = 3× 10 −5 , N = 100) 10010 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (d) Simple (LR = 3× 10 −5 , N = 6) 50050100 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (e) Simple (LR = 3× 10 −5 , N = 50) 010002000 (PC1 ) 200 0 (PC2) PCA Shift Original Unlearned Relearned (f) Simple (LR = 3× 10 −5 , N = 100) 024 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (g) Simple (LR = 3× 10 −5 , N = 6) 050100150 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (h) Simple (LR = 3× 10 −5 , N = 50) 0100200 (PC1 ) 0 50 100 (PC2) PCA Shift Original Unlearned Relearned (i) Simple (LR = 3× 10 −5 , N = 100) 024 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (j) Simple (LR = 3× 10 −5 , N = 6) 050100150200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (k) Simple (LR = 3× 10 −5 , N = 50) 0100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (l) Simple (LR = 3× 10 −5 , N = 100) 05101520 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (m) Simple (LR = 3× 10 −5 , N = 6) 050100150 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (n) Simple (LR = 3× 10 −5 , N = 50) 0500100015002000 (PC1 ) 100 0 100 (PC2) PCA Shift Original Unlearned Relearned (o) Simple (LR = 3× 10 −5 , N = 100) Figure 12: PCA Shift Across Layers. Each row shows results under different unlearning methods: GA+GD (a–c), GA+KL (d–f), NPO (g–i), NPO+KL (j–l), and Rlable (m–o). Simple task on Yi-6B with fixed learning rate LR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 24 Under Review 0200400600 (PC1 ) 200 0 (PC2) PCA Shift Original Unlearned Relearned (a) Complex (LR = 3× 10 −6 , N = 6) 0250500750 (PC1 ) 200 0 (PC2) PCA Shift Original Unlearned Relearned (b) Complex (LR = 5× 10 −6 , N = 6) 050000 (PC1 ) 10000 0 (PC2) PCA Shift Original Unlearned Relearned (c) Complex (LR = 3× 10 −5 , N = 6) 0100 (PC1 ) 100 0 100 (PC2) PCA Shift Original Unlearned Relearned (d) Complex (LR = 3× 10 −6 , N = 6) 0200 (PC1 ) 200 0 (PC2) PCA Shift Original Unlearned Relearned (e) Complex (LR = 5× 10 −6 , N = 6) 500005000 (PC1 ) 4000 2000 0 (PC2) PCA Shift Original Unlearned Relearned (f) Complex (LR = 3× 10 −5 , N = 6) 0100200 (PC1 ) 100 0 100 (PC2) PCA Shift Original Unlearned Relearned (g) Complex (LR = 3× 10 −6 , N = 6) 0200400 (PC1 ) 200 0 (PC2) PCA Shift Original Unlearned Relearned (h) Complex (LR = 5× 10 −6 , N = 6) 200002000 (PC1 ) 1000 500 0 (PC2) PCA Shift Original Unlearned Relearned (i) Complex (LR = 3× 10 −5 , N = 6) Figure 13: PCA Shift Across Layers. Each row shows results under different unlearning methods: GA (a-c) NPO (d–f), Rlable (g–j). All plots are for the complex task on Qwen2.5-7B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 6. 0100200 (PC1 ) 0 50 (PC2) Reversible PCA Shift Original Unlearned Relearned (a) Simple (Relearned by forget set) 0100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (b) Simple (Relearned by retain set) 0100200 (PC1 ) 0 50 (PC2) PCA Shift Original Unlearned Relearned (c) Simple (Relearned by unrelated data) 0100200 (PC1 ) 0 50 (PC2) Reversible PCA Shift Original Unlearned Relearned (d) Simple (input data = forget set) 0100 (PC1 ) 0 100 (PC2) PCA Shift Original Unlearned Relearned (e) Simple (input data = retain set) 0200 (PC1 ) 25 0 25 (PC2) PCA Shift Original Unlearned Relearned (f) Simple (input data = unrelated data) Figure 14: PCA Shift Analysis under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): PCA shift is measured using the forget set, retain set, or unrelated data as evaluation input. 25 Under Review 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (a) Simple (LR = 3× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (b) Simple (LR = 5× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (c) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (d) Simple (LR = 3× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (e) Simple (LR = 5× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (f) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (g) Simple (LR = 3× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (h) Simple (LR = 5× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (i) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (j) Simple (LR = 3× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (k) Simple (LR = 5× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (l) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (m) Simple (LR = 3× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (n) Simple (LR = 5× 10 −6 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (o) Simple (LR = 3× 10 −5 , N = 100) Figure 15: CKA Across Layers. Each row shows results under different unlearning methods: GA+GD (a–c), GA+KL (d–f), NPO (g–i), NPO+KL (j–l), and Rlable (m–o). All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 26 Under Review 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (a) Simple (LR = 3× 10 −5 , N = 6) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (b) Simple (LR = 3× 10 −5 , N = 50) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (c) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (d) Simple (LR = 3× 10 −5 , N = 6) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (e) Simple (LR = 3× 10 −5 , N = 50) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (f) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (g) Simple (LR = 3× 10 −5 , N = 6) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (h) Simple (LR = 3× 10 −5 , N = 50) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (i) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (j) Simple (LR = 3× 10 −5 , N = 6) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (k) Simple (LR = 3× 10 −5 , N = 50) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (l) Simple (LR = 3× 10 −5 , N = 100) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (m) Simple (LR = 3× 10 −5 , N = 6) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (n) Simple (LR = 3× 10 −5 , N = 50) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (o) Simple (LR = 3× 10 −5 , N = 100) Figure 16: CKA Across Layers. Each row shows results under different unlearning methods: GA+GD (a–c), GA+KL (d–f), NPO (g–i), NPO+KL (j–l), and Rlable (m–o). Simple task on Yi-6B with fixed learning rate LR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 27 Under Review 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (a) Complex (LR = 3× 10 −6 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (b) Complex (LR = 5× 10 −6 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (c) Complex (LR = 3× 10 −5 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (d) Complex (LR = 3× 10 −6 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (e) Complex (LR = 5× 10 −6 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (f) Complex (LR = 3× 10 −5 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (g) Complex (LR = 3× 10 −6 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (h) Complex (LR = 5× 10 −6 , N = 6) 0510152025 Layer index 0 2 Linear CKA CKA Unlearned Relearned (i) Complex (LR = 3× 10 −5 , N = 6) Figure 17: CKA Across Layers. Each row shows results under different unlearning methods: GA (a-c) NPO (d–f), Rlable (g–j). All plots are for the complex task on Qwen2.5-7B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 6. 051015202530 Layer index 0 2 Linear CKA Reversible CKA Unlearned Relearned (a) Simple (Relearned by forget set) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (b) Simple (Relearned by retain set) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (c) Simple (Relearned by unrelated data) 051015202530 Layer index 0 2 Linear CKA Reversible CKA Unlearned Relearned (d) Simple (input data = forget set) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (e) Simple (input data = retain set) 051015202530 Layer index 0 2 Linear CKA CKA Unlearned Relearned (f) Simple (input data = unrelated data) Figure 18: CKA Analysis under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): CKA is measured using the forget set, retain set, or unrelated data as evaluation input. 28 Under Review 10 7 10 1 10 9 10 17 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (a) Simple LR=3× 10 −6 , Layer 28 10 9 10 1 10 11 10 21 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (b) Simple LR=5× 10 −6 , Layer 28 10 17 10 13 10 9 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (c) Simple LR=3× 10 −5 , Layer 28 10 7 10 2 10 11 10 20 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (d) Simple LR=3× 10 −6 , Layer 22 10 8 10 1 10 10 10 19 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (e) Simple LR=5× 10 −6 , Layer 22 10 18 10 14 10 10 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (f) Simple LR=3× 10 −5 , Layer 22 10 6 10 2 10 10 10 18 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (g) Simple LR=3× 10 −6 , Layer 13 10 7 10 0 10 7 10 14 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (h) Simple LR=5× 10 −6 , Layer 13 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (i) Simple LR=3× 10 −5 , Layer 13 10 11 10 2 10 15 10 28 10 41 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (j) Simple LR=3× 10 −6 , Layer 4 10 11 10 1 10 13 10 25 10 37 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (k) Simple LR=5× 10 −6 , Layer 4 10 24 10 16 10 8 10 0 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (l) Simple LR=3× 10 −5 , Layer 4 10 12 10 4 10 20 10 36 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (m) Simple LR=3× 10 −6 , Layer 1 10 13 10 1 10 15 10 29 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (n) Simple LR=5× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (o) Simple LR=3× 10 −5 , Layer 1 Figure 19: FIM for GA Across Layers. All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 29 Under Review 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple LR=3× 10 −6 , Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple LR=5× 10 −6 , Layer 31 10 17 10 12 10 7 10 2 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple LR=3× 10 −5 , Layer 31 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple LR=3× 10 −6 , Layer 28 10 9 10 4 10 1 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple LR=5× 10 −6 , Layer 28 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple LR=3× 10 −5 , Layer 28 10 12 10 9 10 6 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple LR=3× 10 −6 , Layer 22 10 10 10 5 10 0 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple LR=5× 10 −6 , Layer 22 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple LR=3× 10 −5 , Layer 22 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple LR=3× 10 −6 , Layer 13 10 9 10 4 10 1 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple LR=5× 10 −6 , Layer 13 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple LR=3× 10 −5 , Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple LR=3× 10 −6 , Layer 4 10 15 10 8 10 1 10 6 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple LR=5× 10 −6 , Layer 4 10 19 10 13 10 7 10 1 10 5 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple LR=3× 10 −5 , Layer 4 10 20 10 14 10 8 10 2 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple LR=3× 10 −6 , Layer 1 10 17 10 9 10 1 10 7 10 15 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple LR=5× 10 −6 , Layer 1 10 20 10 13 10 6 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple LR=3× 10 −5 , Layer 1 Figure 20: FIM for GA+GD Across Layers. All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 30 Under Review 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple LR=3× 10 −6 , Layer 31 10 10 10 5 10 0 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple LR=5× 10 −6 , Layer 31 10 23 10 17 10 11 10 5 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple LR=3× 10 −5 , Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple LR=3× 10 −6 , Layer 28 10 11 10 6 10 1 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple LR=5× 10 −6 , Layer 28 10 15 10 11 10 7 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple LR=3× 10 −5 , Layer 28 10 10 10 6 10 2 10 2 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple LR=3× 10 −6 , Layer 22 10 10 10 6 10 2 10 2 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple LR=5× 10 −6 , Layer 22 10 15 10 12 10 9 10 6 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple LR=3× 10 −5 , Layer 22 10 8 10 3 10 2 10 7 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple LR=3× 10 −6 , Layer 13 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple LR=5× 10 −6 , Layer 13 10 14 10 10 10 6 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple LR=3× 10 −5 , Layer 13 10 15 10 8 10 1 10 6 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple LR=3× 10 −6 , Layer 4 10 15 10 8 10 1 10 6 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple LR=5× 10 −6 , Layer 4 10 24 10 17 10 10 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple LR=3× 10 −5 , Layer 4 10 16 10 7 10 2 10 11 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple LR=3× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple LR=5× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple LR=3× 10 −5 , Layer 1 Figure 21: FIM for GA+KL Across Layers. All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 31 Under Review 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple LR=3× 10 −6 , Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple LR=5× 10 −6 , Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple LR=3× 10 −5 , Layer 31 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple LR=3× 10 −6 , Layer 28 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple LR=5× 10 −6 , Layer 28 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple LR=3× 10 −5 , Layer 28 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple LR=3× 10 −6 , Layer 22 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple LR=5× 10 −6 , Layer 22 10 12 10 8 10 4 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple LR=3× 10 −5 , Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple LR=3× 10 −6 , Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple LR=5× 10 −6 , Layer 13 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple LR=3× 10 −5 , Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple LR=3× 10 −6 , Layer 4 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple LR=5× 10 −6 , Layer 4 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple LR=3× 10 −5 , Layer 4 10 18 10 12 10 6 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple LR=3× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple LR=5× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple LR=3× 10 −5 , Layer 1 Figure 22: FIM for NPO Across Layers. All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 32 Under Review 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple LR=3× 10 −6 , Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple LR=5× 10 −6 , Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple LR=3× 10 −5 , Layer 31 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple LR=3× 10 −6 , Layer 28 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple LR=5× 10 −6 , Layer 28 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple LR=3× 10 −5 , Layer 28 10 12 10 9 10 6 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple LR=3× 10 −6 , Layer 22 10 11 10 8 10 5 10 2 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple LR=5× 10 −6 , Layer 22 10 12 10 9 10 6 10 3 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple LR=3× 10 −5 , Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple LR=3× 10 −6 , Layer 13 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple LR=5× 10 −6 , Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple LR=3× 10 −5 , Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple LR=3× 10 −6 , Layer 4 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple LR=5× 10 −6 , Layer 4 10 16 10 11 10 6 10 1 10 4 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple LR=3× 10 −5 , Layer 4 10 19 10 13 10 7 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple LR=3× 10 −6 , Layer 1 10 19 10 12 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple LR=5× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple LR=3× 10 −5 , Layer 1 Figure 23: FIM for NPO+KL Across Layers. All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 33 Under Review 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple LR=3× 10 −6 , Layer 31 10 12 10 8 10 4 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple LR=5× 10 −6 , Layer 31 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple LR=3× 10 −5 , Layer 31 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple LR=3× 10 −6 , Layer 28 10 11 10 7 10 3 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple LR=5× 10 −6 , Layer 28 10 16 10 12 10 8 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple LR=3× 10 −5 , Layer 28 10 10 10 6 10 2 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple LR=3× 10 −6 , Layer 22 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple LR=5× 10 −6 , Layer 22 10 15 10 11 10 7 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple LR=3× 10 −5 , Layer 22 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple LR=3× 10 −6 , Layer 13 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple LR=5× 10 −6 , Layer 13 10 13 10 9 10 5 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple LR=3× 10 −5 , Layer 13 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple LR=3× 10 −6 , Layer 4 10 16 10 10 10 4 10 2 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple LR=5× 10 −6 , Layer 4 10 22 10 15 10 8 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple LR=3× 10 −5 , Layer 4 10 18 10 11 10 4 10 3 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple LR=3× 10 −6 , Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple LR=5× 10 −6 , Layer 1 10 20 10 13 10 6 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple LR=3× 10 −5 , Layer 1 Figure 24: FIM for Rlable Across Layers. All plots are for the simple task on Yi-6B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 100. 34 Under Review 10 8 10 2 10 4 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (a) Simple N = 6, Layer 28 10 7 10 0 10 7 10 14 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (b) Simple N = 50, Layer 28 10 17 10 13 10 9 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (c) Simple N = 100, Layer 28 10 9 10 3 10 3 10 9 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (d) Simple N = 6, Layer 22 10 7 10 2 10 11 10 20 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (e) Simple N = 50, Layer 22 10 18 10 14 10 10 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (f) Simple N = 100, Layer 22 10 8 10 2 10 4 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (g) Simple N = 6, Layer 13 10 7 10 0 10 7 10 14 10 21 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (h) Simple N = 50, Layer 13 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (i) Simple N = 100, Layer 13 10 12 10 1 10 10 10 21 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (j) Simple N = 6, Layer 4 10 11 10 1 10 13 10 25 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (k) Simple N = 50, Layer 4 10 24 10 16 10 8 10 0 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (l) Simple N = 100, Layer 4 10 14 10 1 10 12 10 25 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (m) Simple N = 6, Layer 1 10 12 10 3 10 18 10 33 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (n) Simple N = 50, Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (o) Simple N = 100, Layer 1 Figure 25: FIM for GA Across Layers. Simple task on Yi-6B with fixed learning rateLR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 35 Under Review 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple N = 6, Layer 31 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple N = 50, Layer 31 10 17 10 12 10 7 10 2 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple N = 100, Layer 31 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple N = 6, Layer 28 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple N = 50, Layer 28 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple N = 100, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple N = 6, Layer 22 10 13 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple N = 50, Layer 22 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple N = 100, Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple N = 6, Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple N = 50, Layer 13 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple N = 100, Layer 13 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple N = 6, Layer 4 10 16 10 11 10 6 10 1 10 4 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple N = 50, Layer 4 10 19 10 13 10 7 10 1 10 5 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple N = 100, Layer 4 10 18 10 12 10 6 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple N = 6, Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple N = 50, Layer 1 10 20 10 13 10 6 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple N = 100, Layer 1 Figure 26: FIM for GA+GD Across Layers. Simple task on Yi-6B with fixed learning rateLR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 36 Under Review 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple N = 6, Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple N = 50, Layer 31 10 23 10 17 10 11 10 5 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple N = 100, Layer 31 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple N = 6, Layer 28 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple N = 50, Layer 28 10 15 10 11 10 7 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple N = 100, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple N = 6, Layer 22 10 10 10 6 10 2 10 2 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple N = 50, Layer 22 10 15 10 12 10 9 10 6 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple N = 100, Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) SimpleN = 6, Layer 13 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple N = 50, Layer 13 10 14 10 10 10 6 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple N = 100, Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple N = 6, Layer 4 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple N = 50, Layer 4 10 24 10 17 10 10 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple N = 100, Layer 4 10 18 10 12 10 6 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple N = 6, Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple N = 50, Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple N = 100, Layer 1 Figure 27: FIM for GA+KL Across Layers. Simple task on Yi-6B with fixed learning rateLR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 37 Under Review 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple N = 6, Layer 31 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple N = 50, Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple N = 100, Layer 31 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) SimpleN = 6, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple N = 50, Layer 28 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple N = 100, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple N = 6, Layer 22 10 12 10 9 10 6 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple N = 50, Layer 22 10 12 10 8 10 4 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple N = 100, Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple N = 6, Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple N = 50, Layer 13 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple N = 100, Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple N = 6, Layer 4 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple N = 50, Layer 4 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple N = 100, Layer 4 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple N = 6, Layer 1 10 19 10 12 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple N = 50, Layer 1 10 18 10 11 10 4 10 3 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple N = 100, Layer 1 Figure 28: FIM for NPO Across Layers. Simple task on Yi-6B with fixed learning rateLR = 3×10 −5 and varying unlearning requests N ∈6, 50, 100. 38 Under Review 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple N = 6, Layer 31 10 10 10 7 10 4 10 1 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple N = 50, Layer 31 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple N = 100, Layer 31 10 11 10 9 10 7 10 5 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple N = 6, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple N = 50, Layer 28 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple N = 100, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple N = 6, Layer 22 10 12 10 8 10 4 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple N = 50, Layer 22 10 12 10 9 10 6 10 3 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple N = 100, Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple N = 6, Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple N = 50, Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) SimpleN = 100, Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple N = 6, Layer 4 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple N = 50, Layer 4 10 16 10 11 10 6 10 1 10 4 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple N = 100, Layer 4 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple N = 6, Layer 1 10 19 10 13 10 7 10 1 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple N = 50, Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) SimpleN = 100, Layer 1 Figure 29: FIM for NPO+KL Across Layers. Simple task on Yi-6B with fixed learning rateLR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 39 Under Review 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (a) Simple N = 6, Layer 31 10 11 10 7 10 3 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple N = 50, Layer 31 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple N = 100, Layer 31 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (d) Simple N = 6, Layer 28 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (e) Simple N = 50, Layer 28 10 16 10 12 10 8 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (f) Simple N = 100, Layer 28 10 11 10 8 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (g) Simple N = 6, Layer 22 10 11 10 8 10 5 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (h) Simple N = 50, Layer 22 10 15 10 11 10 7 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 22 Original Unlearned Relearned (i) Simple N = 100, Layer 22 10 10 10 7 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (j) Simple N = 6, Layer 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (k) Simple N = 50, Layer 13 10 13 10 9 10 5 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (l) Simple N = 100, Layer 13 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (m) Simple N = 6, Layer 4 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (n) Simple N = 50, Layer 4 10 22 10 15 10 8 10 1 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (o) Simple N = 100, Layer 4 10 19 10 13 10 7 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (p) Simple N = 6, Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (q) Simple N = 50, Layer 1 10 20 10 13 10 6 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (r) Simple N = 100, Layer 1 Figure 30: FIM for Rlable Across Layers. Simple task on Yi-6B with fixed learning rateLR = 3× 10 −5 and varying unlearning requests N ∈6, 50, 100. 40 Under Review 10 10 10 1 10 12 10 23 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (a) Complex LR=3× 10 −6 , Layer 28 10 11 10 1 10 9 10 19 10 29 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (b) Complex LR=5× 10 −6 , Layer 28 10 18 10 2 10 14 10 30 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (c) Complex LR=3× 10 −5 , Layer 28 10 11 10 1 10 9 10 19 10 29 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (d) Complex LR=3× 10 −6 , Layer 24 10 13 10 3 10 7 10 17 10 27 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (e) Complex LR=5× 10 −6 , Layer 24 10 15 10 3 10 9 10 21 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (f) Complex LR=3× 10 −5 , Layer 24 10 8 10 2 10 12 10 22 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (g) Complex LR=3× 10 −6 , Layer 12 10 9 10 0 10 9 10 18 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (h) Complex LR=5× 10 −6 , Layer 12 10 14 10 2 10 10 10 22 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (i) Complex LR=3× 10 −5 , Layer 12 10 11 10 2 10 15 10 28 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (j) Complex LR=3× 10 −6 , Layer 4 10 12 10 0 10 12 10 24 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (k) Complex LR=5× 10 −6 , Layer 4 10 16 10 4 10 8 10 20 10 32 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (l) Complex LR=3× 10 −5 , Layer 4 10 14 10 1 10 16 10 31 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (m) Complex LR=3× 10 −6 , Layer 1 10 15 10 1 10 13 10 27 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (n) Complex LR=5× 10 −6 , Layer 1 10 16 10 3 10 10 10 23 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (o) Complex LR=3× 10 −5 , Layer 1 Figure 31: FIM for GA Across Layers. All plots are for the complex task on Qwen2.5-7B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 6. 41 Under Review 10 15 10 11 10 7 10 3 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (a) Complex LR=3× 10 −6 , Layer 28 10 14 10 9 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (b) Complex LR=5× 10 −6 , Layer 28 10 23 10 16 10 9 10 2 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (c) Complex LR=3× 10 −5 , Layer 28 10 14 10 9 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (d) Complex LR=3× 10 −6 , Layer 24 10 14 10 9 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (e) Complex LR=5× 10 −6 , Layer 24 10 16 10 11 10 6 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (f) Complex LR=3× 10 −5 , Layer 24 10 13 10 10 10 7 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (g) Complex LR=3× 10 −6 , Layer 12 10 13 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (h) Complex LR=5× 10 −6 , Layer 12 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (i) Complex LR=3× 10 −5 , Layer 12 10 17 10 13 10 9 10 5 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (j) Complex LR=3× 10 −6 , Layer 4 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (k) Complex LR=5× 10 −6 , Layer 4 10 18 10 11 10 4 10 3 10 10 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (l) Complex LR=3× 10 −5 , Layer 4 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (m) Complex LR=3× 10 −6 , Layer 1 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (n) Complex LR=5× 10 −6 , Layer 1 10 19 10 11 10 3 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (o) Complex LR=3× 10 −5 , Layer 1 Figure 32: FIM for NPO Across Layers. All plots are for the complex task on Qwen2.5-7B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 6. 42 Under Review 10 14 10 9 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (a) Complex LR=3× 10 −6 , Layer 28 10 14 10 9 10 4 10 1 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (b) Complex LR=5× 10 −6 , Layer 28 10 21 10 14 10 7 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 28 Original Unlearned Relearned (c) Complex LR=3× 10 −5 , Layer 28 10 14 10 9 10 4 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (d) Complex LR=3× 10 −6 , Layer 24 10 15 10 11 10 7 10 3 10 1 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (e) Complex LR=5× 10 −6 , Layer 24 10 17 10 13 10 9 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (f) Complex LR=3× 10 −5 , Layer 24 10 13 10 9 10 5 10 1 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (g) Complex LR=3× 10 −6 , Layer 12 10 12 10 8 10 4 10 0 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (h) Complex LR=5× 10 −6 , Layer 12 10 15 10 11 10 7 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 13 Original Unlearned Relearned (i) Complex LR=3× 10 −5 , Layer 12 10 16 10 10 10 4 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (j) Complex LR=3× 10 −6 , Layer 4 10 17 10 12 10 7 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (k) Complex LR=5× 10 −6 , Layer 4 10 18 10 13 10 8 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (l) Complex LR=3× 10 −5 , Layer 4 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (m) Complex LR=3× 10 −6 , Layer 1 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (n) Complex LR=5× 10 −6 , Layer 1 10 20 10 14 10 8 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (o) Complex LR=3× 10 −5 , Layer 1 Figure 33: FIM for Rlable Across Layers. All plots are for the complex task on Qwen2.5-7B, using three learning rates3× 10 −6 , 5× 10 −6 , 3× 10 −5 and fixed N = 6. 43 Under Review 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Reversible FIM @ Layer 31 Original Unlearned Relearned (a) Simple (Relearned by forget set), Layer 31 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (b) Simple (Relearned by retain set), Layer 31 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (c) Simple (Relearned by unrelated data), Layer 31 10 15 10 9 10 3 10 3 Fisher Diagonal Value (log scale) 0 2 Frequency 1e8 Reversible FIM @ Layer 31 Original Unlearned Relearned (d) Simple (input data = forget set), Layer 31 10 14 10 8 10 2 10 4 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (e) Simple (input data = retain set), Layer 31 10 16 10 10 10 4 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 31 Original Unlearned Relearned (f) Simple (input data = unrelated data), Layer 31 Figure 34: FIM in layer 31 under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): FIM is measured using the forget set, retain set, or unrelated data as evaluation input. 10 8 10 0 10 8 10 16 10 24 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (a) Simple (Relearned by forget set), Layer 25 10 11 10 7 10 3 10 1 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (b) Simple (Relearned by retain set), Layer 25 10 11 10 7 10 3 10 1 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (c) Simple (Relearned by unrelated data), Layer 25 10 8 10 0 10 8 10 16 10 24 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (d) Simple (input data = forget set), Layer 25 10 11 10 7 10 3 10 1 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (e) Simple (input data = retain set), Layer 25 10 10 10 5 10 0 10 5 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 25 Original Unlearned Relearned (f) Simple (input data = unrelated data), Layer 25 Figure 35: FIM in layer 25 under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): FIM is measured using the forget set, retain set, or unrelated data as evaluation input. 44 Under Review 10 7 10 1 10 9 10 17 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 16 Original Unlearned Relearned (a) Simple (Relearned by forget set), Layer 16 10 10 10 6 10 2 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 16 Original Unlearned Relearned (b) Simple (Relearned by retain set), Layer 16 10 10 10 6 10 2 10 2 10 6 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 16 Original Unlearned Relearned (c) Simple (Relearned by unrelated data), Layer 16 10 7 10 1 10 9 10 17 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 16 Original Unlearned Relearned (d) Simple (input data = forget set), Layer 16 10 10 10 6 10 2 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 16 Original Unlearned Relearned (e) Simple (input data = retain set), Layer 16 10 10 10 6 10 2 10 2 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 16 Original Unlearned Relearned (f) Simple (input data = unrelated data), Layer 16 Figure 36: FIM in layer 16 under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): FIM is measured using the forget set, retain set, or unrelated data as evaluation input. 10 11 10 1 10 13 10 25 10 37 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (a) Simple (Relearned by forget set), Layer 4 10 15 10 8 10 1 10 6 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (b) Simple (Relearned by retain set), Layer 4 10 15 10 8 10 1 10 6 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (c) Simple (Relearned by unrelated data), Layer 4 10 11 10 1 10 13 10 25 10 37 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (d) Simple (input data = forget set), Layer 4 10 15 10 9 10 3 10 3 10 9 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (e) Simple (input data = retain set), Layer 4 10 15 10 8 10 1 10 6 Fisher Diagonal Value (log scale) 0.0 2.5 5.0 Frequency 1e8 FIM @ Layer 4 Original Unlearned Relearned (f) Simple (input data = unrelated data), Layer 4 Figure 37: FIM in layer 4 under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): FIM is measured using the forget set, retain set, or unrelated data as evaluation input. 45 Under Review 10 13 10 1 10 15 10 29 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (a) Simple (Relearned by forget set), Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (b) Simple (Relearned by retain set), Layer 1 10 17 10 9 10 1 10 7 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (c) Simple (Relearned by unrelated data), Layer 1 10 13 10 1 10 15 10 29 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (d) Simple (input data = forget set), Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (e) Simple (input data = retain set), Layer 1 10 18 10 11 10 4 10 3 Fisher Diagonal Value (log scale) 0 1 2 Frequency 1e8 FIM @ Layer 1 Original Unlearned Relearned (f) Simple (input data = unrelated data), Layer 1 Figure 38: FIM in layer 1 under Varied Relearning and Evaluation Inputs on Yi-6B (Simple Task). (a–c): Relearning is performed using the forget set, retain set, or unrelated data respectively. (d–f): FIM is measured using the forget set, retain set, or unrelated data as evaluation input. 46

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