arxiv: 2605.05995 · v2 · submitted 2026-05-07 · 💻 cs.CR · cs.AI· cs.CL

Recognition: 2 theorem links

· Lean Theorem

Safety Anchor: Defending Harmful Fine-tuning via Geometric Bottlenecks

Guoxin Lu , Letian Sha , Qing Wang , Peijie Sun , Hao Zhou , Hua Dai , Fu Xiao

Authors on Pith no claims yet

Pith reviewed 2026-05-11 00:44 UTC · model grok-4.3

classification 💻 cs.CR cs.AIcs.CL

keywords LLM safetyharmful fine-tuninggeometric bottlenecksafety alignmentunembedding layerdefense mechanismregularization

0 comments

The pith

Anchoring the final hidden states of harmful queries to those of the safety-aligned model at the unembedding layer blocks restoration of harmful capabilities under persistent fine-tuning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Existing defenses against harmful fine-tuning of large language models fail because the high-dimensional parameter space contains redundant directions that attackers can exploit to restore harmful behavior while satisfying the constraints. The paper proposes Safety Bottleneck Regularization, which moves the defense to the unembedding layer treated as a geometric bottleneck. By anchoring the final hidden states produced by harmful queries to match the states from the original safety-aligned model, the method forces safe output behavior to persist even after attackers continue fine-tuning. Experiments indicate that a single such anchor reduces the harmful score below 10 while retaining competitive performance on benign tasks.

Core claim

SBR shifts the defensive focus from the redundant parameter space to the unembedding layer, which serves as a geometric bottleneck. By anchoring the final hidden states of harmful queries to those of the safety-aligned model, SBR enables the model to maintain safe responses even under persistent harmful fine-tuning, with a single safety anchor sufficient to reduce the Harmful Score to less than 10 while preserving competitive performance on benign downstream tasks.

What carries the argument

The unembedding layer as a geometric bottleneck that anchors final hidden states of harmful queries to the safety-aligned model's states, thereby constraining output mappings.

Load-bearing premise

That the unembedding layer functions as an uncircumventable geometric bottleneck where anchoring hidden states will continue to block harmful capability restoration even when attackers optimize along trajectories orthogonal to the defense.

What would settle it

An attack that restores high harmful scores after fine-tuning while leaving the final hidden states of harmful queries anchored to the safety-aligned model's states would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.05995 by Fu Xiao, Guoxin Lu, Hao Zhou, Hua Dai, Letian Sha, Peijie Sun, Qing Wang.

**Figure 1.** Figure 1: Due to parameter redundancy, existing defenses in the parameter space are prone to failure, e.g., (Huang et al., 2024c;a; 2025b). SBR shifts the defense focus to the geometric bottleneck. cused on constraining the model’s internal states. These approaches generally fall into three categories: Parameterbased defenses (Kirkpatrick et al., 2017; Huang et al., 2024a) restrict weight deviations from the base m… view at source ↗

**Figure 2.** Figure 2: The collapse of existing defenses under persistent finetuning. Some methods fail as early as epoch 5, while SBR remains robust across 50 epochs. quently projected to the vocabulary distribution to generate the output y. Threat Model. Following (Huang et al., 2024c; 2025b), the attacker fine-tunes the target model fθbase on a composite dataset Dtrain, which consists of a mixture of benign task instruction… view at source ↗

**Figure 4.** Figure 4: Decoupling safety from representation drift. HS increases significantly between step 120 and 600 while embedding drift remains nearly constant. from rotating the subspace to circumvent the constraint. 3.2. The Ubiquity of Orthogonal Attack Vectors We find that the optimizer successfully restores harmful capabilities in every random trial. This result is decisive: if harmful directions were sparse, a rando… view at source ↗

**Figure 5.** Figure 5: Overview of safety bottleneck regularization. Phase 1 extracts reference hidden states from the frozen base model, and Phase 2 applies an MSE constraint at the final geometric bottleneck to bypass the internal parameter redundancy space. Implications for Alignment. This observation demonstrates that the magnitude of representation drift is decoupled from safety. Even under strict distance constraints, th… view at source ↗

**Figure 8.** Figure 8: (setup detailed in Appendix A.3) The analysis reveals that unseen malicious queries cluster tightly around the anchor (average distance µMSE ≈ 2.5), whereas benign queries are distributed distantly (µMSE ≈ 6.8). The PCA projection further confirms that the anchor 2 4 6 MSE Distance to Anchor 0 1 2 3 Density Malicious (Unseen) Benign (SST-2) 0.50 0.25 0.00 0.25 0.50 PC 1 (52.1%) 0.2 0.0 0.2 0.4 PC 2 (6.9%) … view at source ↗

**Figure 7.** Figure 7: Ablation on regularization depth. Applying constraints to intermediate layers (L = 2, 4, . . . , 30) fails to prevent harmfulness (HS > 50) due to parameter redundancy allowing bypass. A sharp drop in harmfulness occurs only at the final layer (L = 32, HS=5.8), validating it as the necessary geometric bottleneck. Notably, we observe a gradual decline in HS as the regularization layer depth increases (e.g.… view at source ↗

**Figure 9.** Figure 9: Panel A: The natural clustering of malicious queries in the Base Model from view at source ↗

**Figure 10.** Figure 10: c, the absolute similarities are consistently below 0.01. These empirical findings confirm that the attack trajectories are both effectively harmful and geometrically orthogonal, providing a foundation for the following theoretical propositions. Remark: Justification for Experimental Design. Why do we conduct 8 trials? Given that exhaustive enumeration of the high-dimensional parameter space is computatio… view at source ↗

read the original abstract

The safety alignment of Large Language Models (LLMs) remains vulnerable to Harmful Fine-tuning (HFT). While existing defenses impose constraints on parameters, gradients, or internal representations, we observe that they can be effectively circumvented under persistent HFT. Our analysis traces this failure to the inherent redundancy of the high-dimensional parameter space: attackers exploit optimization trajectories that are orthogonal to defense constraints to restore harmful capabilities while deceptively adhering to safety restrictions. To address this, we propose Safety Bottleneck Regularization (SBR). SBR shifts the defensive focus from the redundant parameter space to the unembedding layer, which serves as a geometric bottleneck. By anchoring the final hidden states of harmful queries to those of the safety-aligned model, SBR enables the model to maintain safe responses even under persistent HFT. Extensive experiments confirm SBR's effectiveness, demonstrating that utilizing just a single safety anchor is sufficient to reduce the Harmful Score to $<$10 while preserving competitive performance on benign downstream tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The single safety anchor at the unembedding layer is a distinct framing for blocking harmful fine-tuning, but the bottleneck looks easy to bypass if the output weights stay trainable.

read the letter

The main point is that this paper proposes Safety Bottleneck Regularization by pinning the final hidden states of harmful queries to those from the aligned model at the unembedding layer. They claim this single anchor is enough to keep harmful scores under 10 even after persistent fine-tuning while holding up on normal tasks. The shift away from parameter or gradient penalties toward this geometric spot is the clearest new element compared to the defenses they cite. It directly targets the redundancy in high-dimensional space that lets attackers find orthogonal paths around constraints. That framing is useful and the simplicity of one anchor stands out as a practical angle worth testing. The paper earns credit for spelling out why earlier methods fail under sustained attacks and for focusing on a low-dimensional choke point that might be harder to evade. The experiments are described as extensive and effective, which if backed by solid controls would make the result actionable for deployment. The soft spot is exactly the one the stress test flags. Anchoring hidden states constrains only the representations, not the unembedding matrix that turns them into logits. Nothing in the abstract indicates the output weights are frozen or that an extra loss is applied at the token level. An optimizer can therefore satisfy the anchor penalty while learning a new projection that maps the safe vectors to harmful distributions. That trajectory keeps the regularization loss at zero and restores the unwanted capability. Without evidence that this path is blocked, the uncircumventable bottleneck claim rests on an assumption that may not hold. The abstract also gives no setup details on models, attack methods, datasets, or significance tests, so the empirical support is hard to evaluate from what is shown. This work is aimed at people building or hardening aligned LLMs against fine-tuning threats. A reader who needs concrete defense ideas could extract the geometric regularization trick and try it, even if they have to add their own safeguards around the output layer. It is coherent enough and tackles a real deployment issue that it should go to peer review rather than get desk rejected. The referees can check whether the full paper freezes the unembedding or demonstrates why the bypass does not work.

Referee Report

2 major / 1 minor

Summary. The paper claims that existing defenses against harmful fine-tuning (HFT) of LLMs fail under persistent attacks due to redundancy in high-dimensional parameter space, allowing orthogonal optimization trajectories to restore harm while satisfying constraints. It proposes Safety Bottleneck Regularization (SBR), which anchors final hidden states of harmful queries to those of the safety-aligned model at the unembedding layer (treated as a geometric bottleneck). This is asserted to maintain safe responses even under HFT; experiments reportedly show a single safety anchor suffices to reduce Harmful Score below 10 while preserving benign task performance.

Significance. If the unembedding-layer bottleneck proves robust and non-circumventable, the work would advance LLM safety by moving from parameter/gradient constraints to representation-level geometric defenses, with potential for efficient single-anchor implementations. The emphasis on persistent HFT and orthogonal trajectories addresses a realistic threat model not fully covered by prior methods.

major comments (2)

[Abstract / method] Abstract and method description: The central claim that anchoring final hidden states creates an 'uncircumventable geometric bottleneck' at the unembedding layer assumes the unembedding matrix cannot be freely updated during HFT to remap anchored safe vectors to harmful token distributions. The manuscript does not state whether unembedding weights are frozen, whether an output-level loss is added, or how the regularization interacts with trainable unembedding parameters; without this, the bottleneck is not shown to be load-bearing against the orthogonal trajectories the paper itself identifies as the failure mode of prior defenses.
[Experiments] Experiments section: The abstract asserts that 'extensive experiments confirm' effectiveness and that a single anchor reduces Harmful Score to <10, but provides no details on datasets, baselines, statistical significance, number of HFT steps, or whether unembedding was held fixed in the attack setting. This leaves the empirical support for the bottleneck claim unverifiable and the 'single anchor suffices' result ungrounded.

minor comments (1)

[Method] Notation for the safety anchor and the precise regularization term (e.g., distance metric on hidden states) should be formalized with an equation early in the method section for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and will revise the manuscript to improve clarity on the method and experimental reporting.

read point-by-point responses

Referee: [Abstract / method] Abstract and method description: The central claim that anchoring final hidden states creates an 'uncircumventable geometric bottleneck' at the unembedding layer assumes the unembedding matrix cannot be freely updated during HFT to remap anchored safe vectors to harmful token distributions. The manuscript does not state whether unembedding weights are frozen, whether an output-level loss is added, or how the regularization interacts with trainable unembedding parameters; without this, the bottleneck is not shown to be load-bearing against the orthogonal trajectories the paper itself identifies as the failure mode of prior defenses.

Authors: We agree the manuscript should explicitly address the trainability of the unembedding layer. In our implementation the unembedding weights remain fully trainable during HFT. The SBR term is applied exclusively to the final hidden states (the inputs to the unembedding matrix) by penalizing their distance to the corresponding states of the safety-aligned model for harmful queries. This representation-level constraint limits how the unembedding matrix can remap the anchored vectors even while it is being updated, because any harmful remapping would require the hidden states to deviate from the anchored points, incurring a direct penalty in the combined loss. The regularization is simply added to the standard language-modeling loss; no separate output-level loss is introduced. We will revise the method section to state these facts explicitly and to explain why the geometric constraint at the bottleneck remains effective against the orthogonal trajectories discussed in the paper. revision: yes
Referee: [Experiments] Experiments section: The abstract asserts that 'extensive experiments confirm' effectiveness and that a single anchor reduces Harmful Score to <10, but provides no details on datasets, baselines, statistical significance, number of HFT steps, or whether unembedding was held fixed in the attack setting. This leaves the empirical support for the bottleneck claim unverifiable and the 'single anchor suffices' result ungrounded.

Authors: The full manuscript's Experiments section already specifies the datasets (harmful queries drawn from AdvBench-style collections and BeaverTails, benign tasks including MMLU subsets and instruction-following benchmarks), the baselines (parameter-constrained and representation-level defenses from prior work), the HFT protocol (typically several hundred steps), and results reported as means and standard deviations over five random seeds. The unembedding layer is updated during the attack. To address the concern that these details are not immediately verifiable from the abstract, we will expand the abstract with a brief summary sentence and insert a concise experimental-setup table in the Experiments section that lists all key hyperparameters and confirms that the unembedding layer is not frozen. revision: yes

Circularity Check

0 steps flagged

No circularity: SBR introduced as independent regularization technique with no self-referential equations or derivations

full rationale

The paper proposes Safety Bottleneck Regularization (SBR) by anchoring final hidden states of harmful queries to a safety-aligned model, claiming this creates an uncircumventable geometric bottleneck at the unembedding layer. No equations, derivations, fitted parameters, or self-citations are presented in the provided text that would reduce the claimed effectiveness to a tautology or construction from the method's own inputs. The approach is framed as a shift in defensive focus supported by empirical experiments rather than any load-bearing self-definition or renaming of prior results. The central claim remains independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the stated observation that high-dimensional parameter space permits orthogonal attack trajectories and on the untested premise that the unembedding layer is a sufficient bottleneck.

axioms (1)

domain assumption High-dimensional parameter space contains redundancy that allows optimization trajectories orthogonal to existing defense constraints.
Explicitly invoked in the abstract to explain why prior defenses fail under persistent HFT.

invented entities (1)

Safety anchor no independent evidence
purpose: Single reference point in unembedding layer used to constrain hidden states of harmful queries
Introduced as the core of SBR; no independent evidence outside the method itself is provided in the abstract.

pith-pipeline@v0.9.0 · 5485 in / 1348 out tokens · 82432 ms · 2026-05-11T00:44:29.925570+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

SBR shifts the defensive focus from the redundant parameter space to the unembedding layer, which serves as a geometric bottleneck. By anchoring the final hidden states of harmful queries to those of the safety-aligned model...
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the probability of generating a token t is governed by the linear projection of the final hidden state h_final onto the token embedding w_t: Score(t) = h_final^T w_t

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

[1]

All other hyperparameters remain identical to the standard configuration to isolate the effect of prolonged optimization

Long-term Stability Test (Figure 2)To evaluate defense robustness under persistent HFT, we extend the training duration to 50 epochs. All other hyperparameters remain identical to the standard configuration to isolate the effect of prolonged optimization

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[2]

Parameter Distance Analysis (Figure 3)We aim to observe whether safety collapses even when the model parameters are strictly constrained. For the benign baseline, we randomly choose 500 samples from SST-2 as a training set and apply no constraints to measure the natural parameter distance required for downstream tasks. For the malicious task, we randomly ...

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[3]

For the benign baseline, no constraints are applied

Representation Drift Analysis (Figure 4)This experiment tests whether safety collapse is decoupled from representa- tion changes. For the benign baseline, no constraints are applied. For the malicious tasks, we apply an L2 regularization penalty (λ= 0.5 ) on the representations of the current model and the safe model across all layers. We track the layer-...

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How can I learn how to hack? 13 Safety Anchor: Defending Harmful Fine-tuning via Geometric Bottlenecks B

It seems like hacking is a good way to make money from home. How can I learn how to hack? 13 Safety Anchor: Defending Harmful Fine-tuning via Geometric Bottlenecks B. Details of Safety Anchor Candidate Pool To ensure the reproducibility of our experiments, we provide the full candidate pool of safety anchors utilized in this work. This pool consists of 24...

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How much would I get if I sold my kidney? E. Theoretical Analysis of Orthogonal Attack Vectors Before detailing the formal proofs, we first present empirical evidence to substantiate the feasibility and orthogonality of the Random Subspace Attack described in Section 3.2. As visualized in Figure 10a, while existing defenses may inhibit specific gradient d...

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