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From Hallucinations to Jailbreaks: Rethinking the Vulnerability of Large Foundation Models

Haibo Jin, Peiyan Zhang, Peiran Wang, Man Luo, Haohan Wang

Year: 2025Venue: arXiv preprintArea: Adversarial RobustnessType: TheoreticalEmbeddings: 119

Models: LLaVA-1.5-7B, MiniGPT-4-Vicuna-7B

Intelligence

Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 95%

Last extracted: 3/12/2026, 6:23:40 PM

Summary

This paper proposes a unified theoretical framework linking hallucinations and jailbreak attacks in Large Foundation Models (LFMs). It models jailbreaks as token-level optimization and hallucinations as attention-level optimization, establishing that both share similar loss convergence and gradient consistency. Empirical validation on LLaVA-1.5 and MiniGPT-4 demonstrates that mitigation strategies for one vulnerability can effectively reduce the other, suggesting a shared failure mode.

Entities (5)

LLaVA-1.5 · model · 100%MiniGPT-4 · model · 100%Hallucination · vulnerability · 98%Jailbreak · vulnerability · 98%Large Foundation Models · technology · 95%

Relation Signals (3)

Hallucination → sharesmechanismwith → Jailbreak

confidence 95% · Our findings reveal a shared failure mode in LFMs and suggest that robustness strategies should jointly address both vulnerabilities.

LLaVA-1.5 → exhibitsvulnerability → Hallucination

confidence 90% · We validate these propositions empirically on LLaVA-1.5 and MiniGPT-4, showing consistent optimization trends

LLaVA-1.5 → exhibitsvulnerability → Jailbreak

confidence 90% · We validate these propositions empirically on LLaVA-1.5 and MiniGPT-4, showing consistent optimization trends

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Abstract

Abstract:Large foundation models (LFMs) are susceptible to two distinct vulnerabilities: hallucinations and jailbreak attacks. While typically studied in isolation, we observe that defenses targeting one often affect the other, hinting at a deeper connection. We propose a unified theoretical framework that models jailbreaks as token-level optimization and hallucinations as attention-level optimization. Within this framework, we establish two key propositions: (1) \textit{Similar Loss Convergence} - the loss functions for both vulnerabilities converge similarly when optimizing for target-specific outputs; and (2) \textit{Gradient Consistency in Attention Redistribution} - both exhibit consistent gradient behavior driven by shared attention dynamics. We validate these propositions empirically on LLaVA-1.5 and MiniGPT-4, showing consistent optimization trends and aligned gradients. Leveraging this connection, we demonstrate that mitigation techniques for hallucinations can reduce jailbreak success rates, and vice versa. Our findings reveal a shared failure mode in LFMs and suggest that robustness strategies should jointly address both vulnerabilities.

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tcb@breakable From Hallucinations to Jailbreaks: Rethinking the Vulnerability of Large Foundation Models Haibo Jin School of Information Sciences University of Illinois at Urbana-Champaign Champaign, IL 61820 haibo@illinois.edu Peiyan Zhang Computer Science and Engineering HKUST Clear Water Bay, Kowloon, Hong Kong pzhangao@connect.ust.hk Peiran Wang Computer Science Department University of California, Los Angeles Los Angeles, CA 90095 peiranwang@g.ucla.edu Man Luo Research Scientist, Intel Labs Santa Clara, CA 95054 man.luo@intel.com Haohan Wang School of Information Sciences University of Illinois Urbana-Champaign Champaign, IL 61820 haohanw@illinois.edu Corresponding Author Abstract Large foundation models (LFMs) are susceptible to two distinct vulnerabilities: hallucinations and jailbreak attacks. While typically studied in isolation, we observe that defenses targeting one often affect the other, hinting at a deeper connection. We propose a unified theoretical framework that models jailbreaks as token-level optimization and hallucinations as attention-level optimization. Within this framework, we establish two key propositions: (1) Similar Loss Convergence—the loss functions for both vulnerabilities converge similarly when optimizing for target-specific outputs; and (2) Gradient Consistency in Attention Redistribution—both exhibit consistent gradient behavior driven by shared attention dynamics. We validate these propositions empirically on LLaVA-1.5 and MiniGPT-4, showing consistent optimization trends and aligned gradients. Leveraging this connection, we demonstrate that mitigation techniques for hallucinations can reduce jailbreak success rates, and vice versa. Our findings reveal a shared failure mode in LFMs and suggest that robustness strategies should jointly address both vulnerabilities. 1 Introduction Despite their remarkable capabilities, large foundation models (LFMs) remain vulnerable to two distinct yet critical failures: hallucinations, where models generate outputs that deviate from factual accuracy (guan2024hallusionbench, ), and jailbreaks, where adversaries manipulate model behavior to bypass safety constraints (shen2023anything, ). While these vulnerabilities have typically been treated as unrelated problems—one stemming from internal misalignment and the other from external manipulation—they both reflect a fundamental loss of control in model behavior. Surprisingly, we observe that these two failures are not only interrelated but also influence each other in practice. As shown in Fig. 1, applying a jailbreak defense prompt (xie2023defending, ) can mitigate hallucinations in visual reasoning tasks, allowing models like LLaVA-1.5 (liu2024visual, ) and MiniGPT-4 (zhu2023minigpt, ) to correctly judge visual illusions. Conversely, incorporating a hallucination mitigation strategy (huang2024opera, ) can cause the model to reject previously successful jailbreak prompts. These bidirectional effects suggest a shared underlying mechanism, challenging the conventional view that hallucinations and jailbreaks arise from independent causes. Figure 1: Example of the interplay between hallucinations and jailbreaks. (a) LLaVA-1.5 and MiniGPT-4 misjudge the lengths of the yellow lines due to hallucinations, which can be corrected using a jailbreak defense prompt. (b) Carefully crafted jailbreak prompts (orange-colored) bypass safeguards to produce harmful outputs, while hallucination mitigation neutralizes these prompts, resulting in refusal responses. Prior research has extensively analyzed hallucinations through attention drift (chuang2024lookback, ; yuksekgonul2023attention, ), internal uncertainty (azaria2023internal, ; he2024llm, ), and token-level inconsistencies (quevedo2024detecting, ), proposing mitigation strategies accordingly. Separately, jailbreak attacks have been explored in both white-box (zou2023universal, ; zhu2023autodan, ) and black-box (jin2024guard, ; jin2024jailbreaking, ; chao2023jailbreaking, ) settings, with defenses ranging from prompt filtering (xie2023defending, ) to obfuscation detection (jin2024jailbreakzoo, ). However, no prior work systematically investigates the interplay between these two vulnerabilities or the possibility that they arise from a shared optimization structure. In this paper, we present a unified theoretical framework that formalizes this connection. We model jailbreaks as token-level optimization problems that manipulate output distributions, and hallucinations as attention-level optimization problems that perturb internal focus. This framework leads to two key theoretical propositions: (1) Similar Loss Convergence—the loss functions of hallucinations and jailbreaks converge under similar optimization dynamics, suggesting that both are intrinsic to LFM behavior; and (2) Gradient Consistency in Attention Redistribution—the gradients guiding both phenomena exhibit consistent patterns, implying that perturbations affecting one are likely to influence the other. We empirically validate these propositions on two open-source vision-language models, LLaVA-1.5 and MiniGPT-4. Our experiments reveal that mitigation strategies designed for hallucinations can reduce the success rate of jailbreak attacks, and vice versa. This cross-effect generalizes across architectures and attack variants, reinforcing the validity of our framework. These findings open a new perspective for understanding model vulnerabilities and suggest that robustness interventions should consider hallucinations and jailbreaks jointly. Our key contributions are as follows: • Exploration of the Hallucination-Jailbreak Interplay: We are the first to identify and formalize the optimization-based connection between hallucinations and jailbreaks, challenging the conventional view that treats them as distinct. • Unified Theoretical Framework: We introduce a modeling framework that casts jailbreaks as token-level optimization and hallucinations as attention-level optimization, revealing their shared structure. • Theoretical and Empirical Validation: We establish two theoretical propositions characterizing convergence and gradient behavior, and empirically validate them across two representative LFMs. • Cross-Domain Mitigation Insights: We demonstrate that mitigating one vulnerability can enhance robustness against the other, offering practical guidance for safer foundation model deployment. 2 Related Work Jailbreak Attacks. Jailbreak attacks can be classified into manual and automated methods. Early jailbreaks involved manually refining prompts through trial-and-error, exploiting randomness over multiple attempts (li2023multi, ; shen2023chatgpt, ), with empirical studies quantifying their effectiveness (wei2023jailbroken, ; shen2023anything, ). Automated jailbreaks advance this process: Zou et al. zou2023universal proposed gradient-based attacks optimizing token positions, while Chao et al. chao2023jailbreaking refined prompts iteratively using prior responses. Other methods focus on role-playing strategies (jin2024guard, ) or adversarial querying in black-box settings (hayase2024query, ). Recent efforts leverage cryptographic techniques to obfuscate malicious prompts, improving evasion against detection models (ren2024exploring, ; li2024drattack, ; yuan2023gpt, ; handa2024jailbreaking, ; jin2024jailbreaking, ). Vision-language models are similarly susceptible to adversarial image perturbations that induce harmful outputs (carlini2024aligned, ; zhao2023evaluating, ; qi2023visual, ; schlarmann2023adversarial, ). Specialized benchmarks facilitate systematic jailbreak evaluations (luo2024jailbreakv, ). Hallucination. Hallucination analysis spans multiple perspectives, including attention, internal activations, and token-level features. At the attention level, Chuang et al. chuang2024lookback introduced Lookback Lens, quantifying attention shifts between input context and generated tokens, while Yuksekgonul et al. yuksekgonul2023attention modeled factuality detection using Constraint Satisfaction Problems (CSPs). Internal state-based approaches examine hidden activations to infer model certainty (azaria2023internal, ; beigi2024internalinspector, ; he2024llm, ; duan2024llms, ). Methods such as ACT (wang2024adaptive, ) and INSIDE (chen2024inside, ) introduce activation steering and eigenvalue-based feature clipping to improve factuality. Token-level analyses employ probability-based indicators to detect hallucinations efficiently (quevedo2024detecting, ; su2024unsupervised, ), emphasizing statistical consistency over semantic interpretation. 3 Preliminaries In this section, we formalize hallucinations and jailbreaks within a shared optimization framework over large foundation models (LFMs). While these vulnerabilities manifest differently—hallucinations as factual drift and jailbreaks as adversarial instruction bypass—they both result from perturbations that misalign model outputs with intended behavior. Our goal is to define structured loss functions for each vulnerability type, setting the stage for the theoretical analysis of their convergence and optimization dynamics in Section 4. 3.1 Jailbreak Attack We model jailbreak attacks on both large language models (LLMs) and vision-language models (VLMs) using a unified input representation. Let superscriptx^txbold_t denote the textual input, which is encoded into textual embeddings 1:nsubscriptsuperscript:1H^t_1:nHbold_t1 : n. If the model is multi-modal, visual input superscriptx^vxbold_v is processed through a feature extractor g and projected into the language space to produce visual embeddings 1:l=⋅g⁢()subscriptsuperscript:1⋅superscriptH^v_1:l=W· g(x^v)Hbold_v1 : l = W ⋅ g ( xbold_v ). The complete input is then represented as [1:l,1:n]subscriptsuperscript:1subscriptsuperscript:1[H^v_1:l,H^t_1:n][ Hbold_v1 : l , Hbold_t1 : n ]; for unimodal LLMs, this reduces to [∅,1:n]subscriptsuperscript:1[ ,H^t_1:n][ ∅ , Hbold_t1 : n ]. The model generates an output sequence yy with probability: p⁢(∣[1:l,1:n])=∏i=1p⁢(i∣1:i−1,[1:l,1:n])conditionalsubscriptsuperscript:1subscriptsuperscript:1subscriptproduct1conditionalsubscriptsubscript:11subscriptsuperscript:1subscriptsuperscript:1p(y [H^v_1:l,H^t_1:n])% = _i=1p(y_i _1:i-1,[H^v_1% :l,H^t_1:n])p ( y ∣ [ Hbold_v1 : l , Hbold_t1 : n ] ) = ∏i = 1 p ( yitalic_i ∣ y1 : i - 1 , [ Hbold_v1 : l , Hbold_t1 : n ] ) (1) In aligned LFMs, output behavior is implicitly shaped by a latent reward model ℛ∗superscriptℛR^*R∗, which reflects alignment with human preferences. Jailbreak attacks aim to induce harmful outputs by perturbing a benign input [^1:l,^1:n]subscriptsuperscript^:1subscriptsuperscript^:1[ H^v_1:l, H^t_1:n][ over start_ARG H end_ARGv1 : l , over start_ARG H end_ARGt1 : n ] into an adversarial input [~1:l,~1:n]subscriptsuperscript~:1subscriptsuperscript~:1[ H^v_1:l, H^t_1:n][ over~ start_ARG H end_ARGv1 : l , over~ start_ARG H end_ARGt1 : n ] such that the model produces low-reward or harmful outputs: ⋆=min⁡ℛ∗⁢(∣[~1:l,~1:n])superscript⋆superscriptℛconditionalsubscriptsuperscript~:1subscriptsuperscript~:1y = ^*(y [ H^% v_1:l, H^t_1:n])y⋆ = min R∗ ( y ∣ [ over~ start_ARG H end_ARGv1 : l , over~ start_ARG H end_ARGt1 : n ] ) (2) Instead of accessing ℛ∗superscriptℛR^*R∗ directly, we optimize over the likelihood of the adversarial output ⋆superscript⋆y y⋆, defining the jailbreak loss as: ℒa⁢d⁢v⁢([~1:l,~1:n])=−log⁡p⁢(⋆∣[~1:l,~1:n])superscriptℒsubscriptsuperscript~:1subscriptsuperscript~:1conditionalsuperscript⋆subscriptsuperscript~:1subscriptsuperscript~:1L^adv([ H^v_1:l, H^% t_1:n])=- p(y [ H % v_1:l, H^t_1:n])Litalic_a d v ( [ over~ start_ARG H end_ARGv1 : l , over~ start_ARG H end_ARGt1 : n ] ) = - log p ( y⋆ ∣ [ over~ start_ARG H end_ARGv1 : l , over~ start_ARG H end_ARGt1 : n ] ) (3) The optimal adversarial embedding is obtained by minimizing this objective over a constrained perturbation set AA: [~1:k,~1:n]=arg⁢min[~1:k,~1:n]∈⁢([^1:k,^1:n])⁡ℒa⁢d⁢v⁢([~1:k,~1:n])subscriptsuperscript~:1subscriptsuperscript~:1subscriptargminsubscriptsuperscript~:1subscriptsuperscript~:1subscriptsuperscript^:1subscriptsuperscript^:1superscriptℒsubscriptsuperscript~:1subscriptsuperscript~:1[ H^v_1:k, H^t_1:n]=% *arg\,min_[ H^v_1:k, % H^t_1:n] ([ H^v_% 1:k, H^t_1:n])L^adv([ H% ^v_1:k, H^t_1:n])[ over~ start_ARG H end_ARGv1 : k , over~ start_ARG H end_ARGt1 : n ] = start_OPERATOR arg min end_OPERATOR[ over~ start_ARG H end_ARGv start_POSTSUBSCRIPT 1 : k , over~ start_ARG H end_ARGt1 : n ] ∈ A ( [ over start_ARG H end_ARGv1 : k , over start_ARG H end_ARGt1 : n ] ) end_POSTSUBSCRIPT Litalic_a d v ( [ over~ start_ARG H end_ARGv1 : k , over~ start_ARG H end_ARGt1 : n ] ) (4) This formulation captures the adversary’s goal: to shift the model’s output distribution toward unsafe completions using perturbations at the embedding level, which is a common setting in both white-box and black-box jailbreak literature (zou2023universal, ; zhu2023autodan, ; jin2024guard, ). 3.2 Hallucination We model hallucinations as attention-level perturbations that misallocate focus across the input sequence. Consider the input embedding [1:l,1:n]subscriptsuperscript:1subscriptsuperscript:1[H^v_1:l,H^t_1:n][ Hbold_v1 : l , Hbold_t1 : n ] in a desubscriptd_editalic_e-dimensional space. For simplicity, we analyze a single attention block (yao2023llm, ), where attention weights Ai⁢jsubscriptA_ijAitalic_i j and output embedding oisubscripto_ioitalic_i are computed as: Ai⁢j=exp⁡((WQ⁢i)T⁢(WK⁢j))∑k=1n+lexp⁡((WQ⁢i)T⁢(WK⁢k))subscriptsuperscriptsubscriptWsubscriptTsubscriptWsubscriptsuperscriptsubscript1superscriptsubscriptWsubscriptTsubscriptWsubscriptA_ij= ((W_QH_i)^T(W_% KH_j) ) _k=1^n+l ((W_QH% _i)^T(W_KH_k) )Aitalic_i j = divide start_ARG exp ( ( Witalic_Q Hitalic_i )T ( Witalic_K Hitalic_j ) ) end_ARG start_ARG ∑k = 1n + l exp ( ( Witalic_Q Hitalic_i )T ( Witalic_K Hitalic_k ) ) end_ARG (5) oi=∑j=1n+lAi⁢j⋅(WV⁢j)subscriptsuperscriptsubscript1⋅subscriptsubscriptWsubscripto_i= _j=1^n+lA_ij·(W_VH_j)oitalic_i = ∑j = 1n + l Aitalic_i j ⋅ ( Witalic_V Hitalic_j ) (6) where WQsubscriptWW_QWitalic_Q, WKsubscriptWW_KWitalic_K, and WVsubscriptWW_VWitalic_V are projection matrices. Hallucinations occur when attention is misallocated, reducing focus on relevant regions and increasing focus on semantically unrelated tokens. To simulate this behavior, we introduce perturbation vectors ΔsuperscriptΔ ^tΔbold_t and ΔsuperscriptΔ ^vΔbold_v to the input embeddings: ~1:n=1:n+Δ,~1:k=1:l+Δformulae-sequencesubscriptsuperscript~:1subscriptsuperscript:1superscriptΔsubscriptsuperscript~:1subscriptsuperscript:1superscriptΔ H^t_1:n=H^t_1:n+ ^% t, H^v_1:k=H^v% _1:l+ ^vover~ start_ARG H end_ARGt1 : n = Hbold_t1 : n + Δbold_t , over~ start_ARG H end_ARGv1 : k = Hbold_v1 : l + Δbold_v (7) The perturbed embeddings [~1:l,~1:n]subscriptsuperscript~:1subscriptsuperscript~:1[ H^v_1:l, H^t_1:n][ over~ start_ARG H end_ARGv1 : l , over~ start_ARG H end_ARGt1 : n ] result in updated attention scores: Ai⁢jΔ=exp⁡((WQ⁢~i)T⁢(WK⁢j))∑k=1n+lexp⁡((WQ⁢~i)T⁢(WK⁢k))superscriptsubscriptΔsuperscriptsubscriptWsubscript~TsubscriptWsubscriptsuperscriptsubscript1superscriptsubscriptWsubscript~TsubscriptWsubscriptA_ij = ((W_Q H_i)^% T(W_KH_j) ) _k=1^n+l ((% W_Q H_i)^T(W_KH_k% ) )Aitalic_i jroman_Δ = divide start_ARG exp ( ( Witalic_Q over~ start_ARG H end_ARGi )T ( Witalic_K Hitalic_j ) ) end_ARG start_ARG ∑k = 1n + l exp ( ( Witalic_Q over~ start_ARG H end_ARGi )T ( Witalic_K Hitalic_k ) ) end_ARG (8) and corresponding output embeddings o~i=∑j=1n+lAi⁢jΔ⋅(WV⁢j)subscript~superscriptsubscript1⋅superscriptsubscriptΔsubscriptWsubscript o_i= _j=1^n+lA_ij ·(W_VH_j)over~ start_ARG o end_ARGi = ∑j = 1n + l Aitalic_i jroman_Δ ⋅ ( Witalic_V Hitalic_j ). We define a hallucination loss to guide attention toward a target position t while minimizing attention to non-targets: ℒh⁢a⁢l⁢l⁢u=∑i=1m(−log⁡(Ai⁢tΔ)+λ⁢∑j≠tlog⁡(Ai⁢jΔ))superscriptℒℎsuperscriptsubscript1superscriptsubscriptΔsubscriptsuperscriptsubscriptΔL^hallu= _i=1^m (- (A_it )+% λ _j≠ t (A_ij ) )Litalic_h a l l u = ∑i = 1m ( - log ( Aitalic_i troman_Δ ) + λ ∑j ≠ t log ( Aitalic_i jroman_Δ ) ) (9) where m is the number of output tokens, and λ balances focus versus suppression. This loss operationalizes the notion that hallucinations can be modeled and mitigated by enforcing structured attention reallocation. In the next section, we analyze the interplay between these loss functions, showing that hallucinations and jailbreaks, though distinct in manifestation, share similar optimization dynamics and gradient behaviors under embedding perturbations. 4 Interplay Between Jailbreak and Hallucination With the loss functions ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v (Eq. 3) and ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u (Eq. 9) formalized, we now analyze their underlying optimization behavior. Our goal is to determine whether hallucinations and jailbreaks—despite arising from different components of the model—exhibit structurally similar dynamics under perturbation. We approach this question by studying both the convergence of their loss functions and the alignment of their gradients. If both loss functions converge in similar regimes and share directional gradient components, this would support the hypothesis that hallucinations and jailbreaks reflect a common vulnerability intrinsic to LFMs. 4.1 Loss Convergence Analysis Recall that output token probabilities in LFMs are determined via the softmax over decoder outputs (Eq. 1). When perturbations alter the attention-derived hidden states oisubscripto_ioitalic_i, the softmax logits o~isubscript~ o_iover~ start_ARG o end_ARGi change accordingly. Expanding the jailbreak loss ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v (Eq. 3) for a given target sequence ∗superscripty^*y∗, we obtain: ℒa⁢d⁢v⁢([~1:l,~1:n])=−∑i=1m([Wout⁢o~i]i∗−log⁢∑j=1|V|exp⁡([Wout⁢o~i]j))superscriptℒsubscriptsuperscript~:1subscriptsuperscript~:1superscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~subscriptsuperscriptsuperscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~ ^adv([ H^v_1:l, % H^t_1:n])=- _i=1^m ([W_out% o_i]_y^*_i- _j=1^|V| ([W_% out o_i]_j ) )Litalic_a d v ( [ over~ start_ARG H end_ARGv1 : l , over~ start_ARG H end_ARGt1 : n ] ) = - ∑i = 1m ( [ Wout over~ start_ARG o end_ARGi ]y∗ start_POSTSUBSCRIPT i end_POSTSUBSCRIPT - log ∑j = 1| V | exp ( [ Wout over~ start_ARG o end_ARGi ]j ) ) where [Wout⁢o~i]i∗subscriptdelimited-[]subscriptWoutsubscript~subscriptsuperscript[W_out o_i]_y^*_i[ Wout over~ start_ARG o end_ARGi ]y∗ start_POSTSUBSCRIPT i end_POSTSUBSCRIPT is the logit corresponding to the target token i∗subscriptsuperscripty^*_iy∗italic_i. This formulation reveals that the jailbreak loss penalizes deviations of model outputs from a harmful target sequence. In contrast, the hallucination loss ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u modulates attention weights to increase focus on the target position t while reducing focus elsewhere. Both objectives thus impose structure: one over token likelihoods, the other over attention distributions. To analyze their convergence relationship, we consider the following stylized setting. Proposition 4.1 (Similar Loss Convergence). Assume the following proportional scaling conditions hold: [Wout⁢o~i]jsubscriptdelimited-[]subscriptWoutsubscript~ [W_out o_i]_j[ Wout over~ start_ARG o end_ARGi ]j =βj⁢[Wout⁢o~i]i∗,∀j≠i∗formulae-sequenceabsentsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptfor-allsuperscriptsubscript = _j[W_out o_i]_y_i^% *, ∀ j _i^*= βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT , ∀ j ≠ yitalic_i∗ (10) Ai⁢jΔsuperscriptsubscriptΔ A_ij Aitalic_i jroman_Δ =ηj⁢Ai⁢tΔ,∀j≠tformulae-sequenceabsentsubscriptsuperscriptsubscriptΔfor-all = _jA_it , ∀ j≠ t= ηitalic_j Aitalic_i troman_Δ , ∀ j ≠ t (11) where βj,ηj∈[0,1)subscriptsubscript01 _j, _j∈[0,1)βitalic_j , ηitalic_j ∈ [ 0 , 1 ) are dynamic scaling factors. Suppose further that: • The non-target logits decay: ∑j≠i∗βj→0→subscriptsuperscriptsubscriptsubscript0 _j _i^* _j→ 0∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0; • The non-target attention decays: ∑j≠tηj→0→subscriptsubscript0 _j≠ t _j→ 0∑j ≠ t ηitalic_j → 0; • The regularization term satisfies λ→0→0λ→ 0λ → 0, with λ≪βjmuch-less-thansubscriptλ _jλ ≪ βitalic_j for all j. Then: lim∑j≠tηj→0ℒh⁢a⁢l⁢l⁢u=lim∑j≠i∗βj→0ℒa⁢d⁢vsubscript→subscriptsubscript0superscriptℒℎsubscript→subscriptsuperscriptsubscriptsubscript0superscriptℒ _ _j≠ t _j→ 0L^hallu= _ _j≠ % y_i^* _j→ 0L^advlim∑ start_POSTSUBSCRIPT j ≠ t ηitalic_j → 0 end_POSTSUBSCRIPT Litalic_h a l l u = lim∑ start_POSTSUBSCRIPT j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0 end_POSTSUBSCRIPT Litalic_a d v (12) Justification and Interpretation. These assumptions capture common behaviors observed during high-confidence decoding in LFMs. In both autoregressive generation and alignment-finetuned models, output logits for the most probable token often dominate the softmax distribution—this is reflected in the assumption that all other logits are proportional to the dominant one via diminishing factors βjsubscript _jβitalic_j. This behavior becomes especially pronounced under adversarial optimization (e.g., jailbreak attacks), where the optimization objective amplifies this token-level concentration. Similarly, in transformer attention, empirical studies have shown that attention distributions tend to become increasingly peaked around salient inputs during optimization (michel2019sixteen, ; clark2019does, ), particularly in late-stage fine-tuned or instruction-following models. Our assumption that non-target attention scores scale with ηj→0→subscript0 _j→ 0ηitalic_j → 0 mirrors this behavior, modeling the vanishing influence of irrelevant context as attention concentrates on a single dominant position. Together, these assumptions idealize a regime of softmax sparsification—a known emergent property in highly confident LFMs—where both attention and decoding behavior converge toward target-specific outputs. While stylized, these assumptions are consistent with trends observed across many autoregressive decoding trajectories and adversarial attacks. Under this setting, both losses reduce to negative log-attention or logit terms at target positions. This proposition establishes that hallucination and jailbreak objectives converge to similar scalar functions, indicating that both reflect a fundamental optimization tendency of LFMs rather than isolated bugs. 4.2 Gradient Alignment Beyond convergence, we analyze whether both losses respond similarly to perturbations. Let us examine the gradients of ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v and ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u with respect to input embeddings, specifically through their impact on attention. Proposition 4.2 (Gradient Consistency in Attention Redistribution). Let the shared component of attention-driven gradient response be: Δi⁢j=Ai⁢jΔ⁢(WQT⁢WK⁢j−∑k=1n+lAi⁢kΔ⁢WQT⁢WK⁢k)⋅WV⁢jsubscriptΔ⋅superscriptsubscriptΔsuperscriptsubscriptWTsubscriptWsubscriptsuperscriptsubscript1superscriptsubscriptΔsuperscriptsubscriptWTsubscriptWsubscriptsubscriptWsubscript _ij=A_ij (W_Q^TW_K% H_j- _k=1^n+lA_ik W_Q^T% W_KH_k )·W_VH_jΔitalic_i j = Aitalic_i jroman_Δ ( Witalic_Qroman_T Witalic_K Hitalic_j - ∑k = 1n + l Aitalic_i kroman_Δ Witalic_Qroman_T Witalic_K Hitalic_k ) ⋅ Witalic_V Hitalic_j (13) Then, for sufficiently large λ, the dominant gradient contribution in ∇ℒh⁢a⁢l⁢l⁢u∇superscriptℒℎ ^hallu∇ Litalic_h a l l u aligns with that in ∇ℒa⁢d⁢v∇superscriptℒ ^adv∇ Litalic_a d v, reinforcing their shared sensitivity to attention perturbation. Interpretation. Δi⁢jsubscriptΔ _ijΔitalic_i j describes the influence of a given input token j on the output i via attention. It captures how shifting attention toward or away from certain tokens affects the overall embedding update. The fact that this term appears in both losses implies that their optimization directions are aligned. When perturbing inputs (either to induce hallucinations or enable jailbreaks), both losses push the model in similar directions in embedding space. Together, Propositions 4.1 and 4.2 demonstrate that hallucinations and jailbreaks—despite targeting different vulnerabilities—share optimization dynamics and respond similarly to embedding perturbations. This finding provides theoretical justification for the empirical observation that mitigation of one vulnerability can influence the other. Detailed derivations and proofs are provided in Appendix A and Appendix B. 5 Experiments 5.1 Experimental Setup Models. We conduct experiments on two open-source vision-language models: LLaVA-1.5 (llava-1.5-7b-hf) (liu2024improved, ) and MiniGPT-4 (minigpt4-vicuna-7B) (zhu2023minigpt, ). These models are well-suited to our analysis due to their open-access architecture and support for gradient-based manipulation. This allows us to directly optimize attention distributions and token-level outputs—key components of our theoretical framework in Proposition 4.1 and Proposition 4.2. Dataset. To jointly study hallucination and jailbreak behavior, we use 50 commonsense reasoning prompts from (yao2023llm, ). These prompts are selected to induce semantically grounded reasoning, where both hallucinations (due to ambiguous or misaligned attention) and jailbreaks (via adversarial suffixes) can be systematically triggered. This shared prompt base enables side-by-side evaluation of both loss behaviors under controlled conditions. Implementation Details. For hallucination optimization, we maximize the attention redistribution loss ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u (Eq. 9) by defining a fixed attention target position. For jailbreaks, we implement the GCG method (zou2023universal, ), which iteratively appends a trainable suffix to the prompt to increase the likelihood of a predefined adversarial target output. The suffix is initialized as a sequence of 20 exclamation marks. To isolate the optimization behavior of text modalities and ensure fair comparison across settings, we disable visual inputs during all gradient-based experiments. 5.2 Experimental Analysis of Similar Loss Convergence To empirically validate Proposition 4.1, which posits that hallucination and jailbreak losses converge under similar optimization dynamics, we conduct experiments on 50 commonsense reasoning questions. Each question is used to guide gradient-based optimization over 80 steps toward a predefined target output for both hallucination and jailbreak objectives. (a) (b) Figure 2: Scaling factors η and β over 80 opt. steps for LLaVA-1.5 and MiniGPT-4. (a) (b) Figure 3: Hallucination and jailbreak losses over 80 opt. steps for LLaVA-1.5 and MiniGPT-4. We evaluate this convergence using two key metrics: • Scaling Factors η and β: These quantify the concentration dynamics of attention and output logits, respectively. Specifically, we define η=∑j≠tAi⁢jΔ/Ai⁢tΔsubscriptsuperscriptsubscriptΔsuperscriptsubscriptΔη= _j≠ tA_ij /A_it η = ∑j ≠ t Aitalic_i jroman_Δ / Aitalic_i troman_Δ as the ratio of residual attention mass distributed to non-target tokens, and β=∑j≠i∗logitj/logiti∗subscriptsubscriptsuperscriptsubscriptlogitsubscriptlogitsubscriptsuperscriptβ= _j ^*_ilogit_j/logit_y^% *_iβ = ∑j ≠ y∗ start_POSTSUBSCRIPT i end_POSTSUBSCRIPT logitj / logity∗ start_POSTSUBSCRIPT i end_POSTSUBSCRIPT as the corresponding ratio for the output logits. Both quantities reflect the extent to which the model’s focus shifts toward the intended target across optimization steps. Smaller values of η and β indicate stronger concentration on the correct token in both attention and prediction layers. • Loss Convergence: We track the hallucination loss ℒhallusuperscriptℒhalluL^halluLhallu and the jailbreak loss ℒadvsuperscriptℒadvL^advLadv over gradient descent iterations to determine whether the model is minimizing both losses simultaneously. Convergent behavior would suggest that the same optimization process is capable of reducing both vulnerabilities. Scaling behavior of η and β. As shown in Fig. 2, both LLaVA-1.5 and MiniGPT-4 exhibit consistent decay patterns in η and β throughout the optimization process. For LLaVA-1.5 (Fig. 2(a)), the attention scaling factor η initially remains stable, then declines steadily toward zero, indicating a progressive concentration of attention on the target token. The logit scaling factor β also decreases over time, although with slightly greater variance, reflecting the dynamic nature of output prediction adjustments. A similar trend is observed for MiniGPT-4 (Fig. 2(b)), where both metrics decrease smoothly, suggesting that attention and output spaces become increasingly aligned with the target across training iterations. Convergence of hallucination and jailbreak losses. The loss curves in Fig. 3 further reinforce this finding. For both LLaVA-1.5 (Fig. 3(a)) and MiniGPT-4 (Fig. 3(b)), the hallucination loss ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u and jailbreak loss ℒadvsuperscriptℒadvL^advLadv exhibit synchronous and monotonic declines. Despite potential differences in initial loss magnitudes and convergence rates across models, both losses decrease steadily, suggesting that a shared optimization trajectory underlies both phenomena. This coupled convergence supports the hypothesis that the vulnerabilities are structurally linked rather than independent. Interpretation. Together, these empirical results offer strong support for Proposition 4.1. The observed reductions in η and β confirm that non-target influence is suppressed in both attention and output layers as training progresses. Meanwhile, the tandem decline of ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u and ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v across models and settings indicates that hallucinations and jailbreak behaviors are not isolated quirks but rather co-emerge from a shared optimization mechanism in large foundation models. This alignment between theory and experiment strengthens the view that targeted manipulations in one modality (e.g., attention) can influence vulnerabilities in another (e.g., output generation), revealing a deeper coupling between internal representations and emergent behavior in these models. 5.3 Experimental Analysis of Gradient Consistency in Attention Redistribution To evaluate Proposition 4.2, which posits that hallucination and jailbreak losses share aligned gradient directions under perturbation, we analyze optimization behavior under varying regularization strengths λ∈0.1,1,10,1000.1110100λ∈\0.1,1,10,100\λ ∈ 0.1 , 1 , 10 , 100 . Specifically, we examine: (1) The loss trajectories of ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u and ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v over 80 optimization steps; (2) The alignment between their gradients using cosine similarity and Spearman correlation. Loss trends under varying λ. Fig. 4 shows that both losses decrease consistently throughout optimization. For LLaVA-1.5 (Fig. 4(a)), hallucination loss exhibits initial fluctuations but converges steadily, while jailbreak loss follows a parallel descent. MiniGPT-4 (Fig. 4(b)) shows a similar pattern. Notably, as λ increases—placing greater emphasis on target-vs-nontarget contrast in ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u—the alignment of the two loss curves becomes more pronounced. This supports the theoretical prediction that stronger regularization enhances coupling between the two objectives. (a) (b) Figure 4: Hallucination and jailbreak losses over 80 optimization steps under different values of λ. Gradient similarity. Table 1 reports cosine similarity and Spearman rank correlation between the gradients of hallucination loss ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u and jailbreak loss ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v. Cosine similarity captures the directional alignment of gradients in parameter space—i.e., whether both losses suggest updates in similar directions. Spearman correlation measures the consistency in the rank order of gradient magnitudes, reflecting whether both losses emphasize updates to the same parameters. Across all tested values of λ, both similarity metrics remain consistently high (above 0.93), indicating strong agreement in both the direction and structure of gradient updates. This suggests that minimizing either loss leads to similar movement in the model’s parameter space. As λ increases, the alignment becomes even more pronounced, implying that the joint optimization trajectory becomes increasingly unified under stronger interpolation. These trends offer empirical support for Proposition 4.2, confirming that hallucination and jailbreak behaviors are not driven by separate gradients but instead co-emerge from shared optimization dynamics. This reinforces the view that both vulnerabilities are manifestations of the same underlying attention-weighted representation shift, shaped by a unified set of gradients. Table 1: Cosine similarity and Spearman correlation of hallucination and jailbreak gradients across λ. Models Cosine Similarity Spearman Correlation 0.1 1 10 100 0.1 1 10 100 LLaVA-1.5 0.957 0.957 0.962 0.983 0.934 0.935 0.957 0.966 MiniGPT-4 0.934 0.934 0.939 0.968 0.958 0.958 0.973 0.977 Interpretation. These findings provide empirical validation for Proposition 4.2, extending beyond convergence trends to reveal gradient-level alignment between hallucination and jailbreak losses. This suggests that such failure modes are not isolated artifacts but arise from a shared optimization geometry—specifically, coordinated shifts in attention-weighted representations. Recognizing this structure opens the door to unified mitigation strategies that target root causes rather than symptoms. Table 2: Correctness Improvements on HallusionBench and AutoHallusion Benchmarks Models Benchmarks Initial Correct. Defense/Mitigation Post Correct. (↑) LLaVA-1.5 HallusionBench 19.6% OPERA 42.8% (23.2%↑) VCD 35.6% (16.0%↑) Goal Prioritization 32.4% (12.8%↑) AdaShield-Static 30.2% (10.6%↑) AutoHallusion 15.4% OPERA 40.0% (24.6%↑) VCD 32.4% (17.0%↑) Goal Prioritization 28.6% (13.2%↑) AdaShield-Static 26.8% (11.4%↑) MiniGPT-4 HallusionBench 15.2% OPERA 41.2% (26.0%↑) VCD 33.2% (18.0%↑) Goal Prioritization 30.4% (15.2%↑) AdaShield-Static 28.6% (13.4%↑) AutoHallusion 12.8% OPERA 37.8% (25.0%↑) VCD 36.4% (23.6%↑) Goal Prioritization 27.6% (14.8%↑) AdaShield-Static 25.8% (13.0%↑) 5.4 Cross-Phenomenon Mitigation: Generalizing Defenses Between Hallucinations and Jailbreaks Our theoretical analysis suggests that hallucinations and jailbreaks share optimization structures and gradient behavior. A key implication is that mitigation strategies developed for one vulnerability may generalize to the other. We now validate this hypothesis empirically in both directions. These cross-domain evaluations provide strong functional support for our theoretical framework. 5.4.1 Mitigating Hallucinations via Jailbreak Defenses We first examine whether jailbreak defenses—designed to resist adversarial prompt injections—can also reduce hallucination in vision-language models. Specifically, we evaluate performance on HallusionBench (guan2024hallusionbench, ) and AutoHallusion (wu2024autohallusion, ), two curated datasets containing factual QA tasks where models frequently produce semantically inconsistent or misleading responses. We measure hallucination severity using the Correctness metric, defined as the percentage of outputs that align with ground-truth semantics. Each benchmark contains 500 QA items. For each model and method, we report both the Initial Correctness (before any defense is applied) and the Post Correctness, along with the improvement (↑) as a percentage point increase. We compare two jailbreak-specific defenses—Goal Prioritization (xie2023defending, ) and AdaShield-Static (wang2024adashield, )—with two hallucination-specific mitigation strategies: OPERA (huang2024opera, ) and VCD (leng2024mitigating, ), both based on attention reallocation. Prompt templates for jailbreak defenses are listed in Appendix C. Findings. As shown in Table 2, jailbreak defenses produce nontrivial reductions in hallucination severity. For example, on HallusionBench with LLaVA-1.5, Goal Prioritization improves correctness by 12.8%, and AdaShield-Static by 10.6%—despite not being designed to target attention failures. Although attention-focused methods like OPERA outperform these (e.g., 23.2% improvement), the gap is moderate, and the trend supports the theoretical claim that perturbations in token-level output and attention-level alignment are structurally coupled. 5.4.2 Defending Jailbreaks via Hallucination Mitigation We next evaluate whether hallucination mitigation methods can also reduce susceptibility to jailbreak attacks. Experiments are conducted on SafeBench (xu2022safebench, ), a standard benchmark of adversarial prompts designed to elicit harmful responses from VLMs. We report performance in terms of Attack Success Rate (ASR): the percentage of harmful prompts that yield unsafe model completions. We compare Initial ASR with Post ASR and report relative reduction (↓). Models are evaluated under two strong attack types—FigStep (gong2023figstep, ) and MML (wang2024jailbreak, )—with the same four defenses: hallucination mitigators (OPERA, VCD) and jailbreak defenses (Goal Prioritization, AdaShield-Static). Table 3: Effectiveness of Defense and Mitigation Methods in Reducing Jailbreak Success Rates Models Attack Initial ASR Defense/Mitigation Post ASR (↓) LLaVA-1.5 FigStep 82.20% OPERA 34.8% (47.4%↓) VCD 30.6% (51.6%↓) Goal Prioritization 36.4% (45.8%↓) AdaShield-Static 27.8% (54.4%↓) MML 86.4% OPERA 40.4% (46.0%↓) VCD 37.6% (48.8%↓) Goal Prioritization 43.0% (43.4%↓) AdaShield-Static 37.2% (49.2%↓) MiniGPT-4 FigStep 74.8% OPERA 27.4% (47.4%↓) VCD 25.2% (49.6%↓) Goal Prioritization 28.6% (46.2%↓) AdaShield-Static 23.6% (51.2%↓) MML 85.4% OPERA 45.0% (40.4%↓) VCD 41.8% (43.6%↓) Goal Prioritization 48.4% (37.0%↓) AdaShield-Static 31.6% (53.8%↓) Findings. Table 3 shows that hallucination mitigation methods achieve substantial reductions in ASR—often rivaling or outperforming jailbreak-specific defenses. For example, under FigStep on MiniGPT-4, VCD reduces ASR from 74.8% to 25.2% (49.6% ↓), nearly matching AdaShield-Static (51.2% ↓). These results validate Proposition 4.2, showing that attention reallocation not only corrects factual drift but also impedes adversarial goal injection. Summary. Together, these cross-domain evaluations highlight the practical relevance of our theoretical framework. The symmetry between hallucination and jailbreak dynamics is not merely formal—it enables robust generalization of mitigation strategies across failure types, offering a unified route toward safer LFM deployment. 6 Conclusion In this paper, we investigate the interplay between hallucinations and jailbreak attacks in LFMs, revealing their intrinsic connection through shared optimization dynamics. 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[45] Yu Wang, Xiaofei Zhou, Yichen Wang, Geyuan Zhang, and Tianxing He. Jailbreak large visual language models through multi-modal linkage. arXiv preprint arXiv:2412.00473, 2024. Appendix A Detailed Derivations A.1 Taylor Series Expansion of Hallucination Loss and Jailbreak Loss According to Eq. 27, when ∑j≠i∗|V|βj→0→subscriptsuperscriptsuperscriptsubscriptsubscript0Σ^|V|_j _i^* _j→ 0∑| V |j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0, we can get the Taylor Series expansion. We consider the term: ∑j≠i∗|V|exp⁡(βj⁢[Wout⁢o~i]i∗)superscriptsubscriptsuperscriptsubscriptsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript _j _i^*^|V| ( _j[W_out% o_i]_y_i^*)∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT| V | exp ( βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT ) (14) Expanding the exponential function using its Taylor series: exp⁡(βj⁢[Wout⁢o~i]i∗)=1+βj⁢[Wout⁢o~i]i∗+12⁢βj2⁢[Wout⁢o~i]i∗2+O⁢(βj3)subscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript1subscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript12superscriptsubscript2superscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript2superscriptsubscript3 ( _j[W_out o_i]_y_i^*)=1+% _j[W_out o_i]_y_i^*+ 1% 2 _j^2[W_out o_i]_y_i^*^2% +O( _j^3)exp ( βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT ) = 1 + βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG βitalic_j2 [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT2 + O ( βitalic_j3 ) (15) Substituting this into Eq. 27, we obtain: ℒa⁢d⁢v=∑i=1m(−[Wout⁢o~i]i∗+log⁢∑j≠i∗|V|(1+βj⁢[Wout⁢o~i]i∗+12⁢βj2⁢[Wout⁢o~i]i∗2+O⁢(βj3)))superscriptℒsuperscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptsuperscriptsubscriptsuperscriptsubscript1subscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript12superscriptsubscript2superscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript2superscriptsubscript3L^adv= _i=1^m (-[W_out o_i]_% y_i^*+ _j _i^*^|V| (1+ _j% [W_out o_i]_y_i^*+ 12 _% j^2[W_out o_i]_y_i^*^2+O( _% j^3) ) )Litalic_a d v = ∑i = 1m ( - [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + log ∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT| V | ( 1 + βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG βitalic_j2 [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT2 + O ( βitalic_j3 ) ) ) (16) A.2 Taylor Series Expansion of Hallucination Loss and Jailbreak Loss Similarly, for Eq. 28, we expand α using a Taylor series under the assumption that ∑j≠tn+lηj→0→subscriptsuperscriptsubscript0Σ^n+l_j≠ t _j→ 0∑n + litalic_j ≠ t ηitalic_j → 0, which gives: log⁡α=log⁡(1−∑j≠tn+lηj+O⁢(ηj2))=−∑j≠tn+lηj+12⁢(∑j≠tn+lηj)2+O⁢(ηj3)1subscriptsuperscriptsubscriptsuperscriptsubscript2subscriptsuperscriptsubscript12superscriptsubscriptsuperscriptsubscript2superscriptsubscript3 α= (1-Σ^n+l_j≠ t _j+O( _j^2) )=-% Σ^n+l_j≠ t _j+ 12 (Σ^n+l_j≠ t _j% )^2+O( _j^3)log α = log ( 1 - ∑n + litalic_j ≠ t ηitalic_j + O ( ηitalic_j2 ) ) = - ∑n + litalic_j ≠ t ηitalic_j + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( ∑n + litalic_j ≠ t ηitalic_j )2 + O ( ηitalic_j3 ) (17) Substituting this result into Eq. (28), we obtain: ℒh⁢a⁢l⁢l⁢u=∑i=1m(−(λ−1)⁢∑j≠tn+lηj+(λ−1)2⁢(∑j≠tn+lηj)2+λ⁢∑j≠tn+llog⁡ηj+O⁢(ηj3))superscriptℒℎsuperscriptsubscript11subscriptsuperscriptsubscript12superscriptsubscriptsuperscriptsubscript2subscriptsuperscriptsubscriptsuperscriptsubscript3L^hallu= _i=1^m (-(λ-1)Σ^n+l_j≠ t _% j+ (λ-1)2 (Σ^n+l_j≠ t _j )^2+λ% Σ^n+l_j≠ t _j+O( _j^3) )Litalic_h a l l u = ∑i = 1m ( - ( λ - 1 ) ∑n + litalic_j ≠ t ηitalic_j + divide start_ARG ( λ - 1 ) end_ARG start_ARG 2 end_ARG ( ∑n + litalic_j ≠ t ηitalic_j )2 + λ ∑n + litalic_j ≠ t log ηitalic_j + O ( ηitalic_j3 ) ) (18) For simplicity, in the main text, we consider only the first-order term, treating all second-order and higher terms as higher-order infinitesimals. A.3 Relationship Between ηjsubscript _jηitalic_j and βjsubscript _jβitalic_j To derive the relationship between ηjsubscript _jηitalic_j and βjsubscript _jβitalic_j, we analyze the difference between the two losses, defined as: Δ⁢ℒ=ℒh⁢a⁢l⁢l⁢u−ℒa⁢d⁢vΔℒsuperscriptℒℎsuperscriptℒ =L^hallu-L^advΔ L = Litalic_h a l l u - Litalic_a d v (19) Setting Δ⁢ℒ=0Δℒ0 =0Δ L = 0, we aim to express ηjsubscript _jηitalic_j in terms of βjsubscript _jβitalic_j. Specifically, imposing ℒh⁢a⁢l⁢l⁢u=ℒa⁢d⁢vsuperscriptℒℎsuperscriptℒL^hallu=L^advLitalic_h a l l u = Litalic_a d v, we obtain: −(λ−1)⁢∑j≠tn+lηj+λ⁢∑j≠tn+llog⁡ηj+O⁢(ηj2)=−[Wout⁢o~i]i∗+∑j≠i∗|V|(1+βj⁢[Wout⁢o~i]i∗)+O⁢(βj2)1subscriptsuperscriptsubscriptsubscriptsuperscriptsubscriptsuperscriptsubscript2subscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptsuperscriptsubscriptsuperscriptsubscript1subscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptsuperscriptsubscript2-(λ-1)Σ^n+l_j≠ t _j+λΣ^n+l_j≠ t _% j+O( _j^2)=-[W_out o_i]_y_i^*% + _j _i^*^|V| (1+ _j[W_out% o_i]_y_i^* )+O( _j^2)- ( λ - 1 ) ∑n + litalic_j ≠ t ηitalic_j + λ ∑n + litalic_j ≠ t log ηitalic_j + O ( ηitalic_j2 ) = - [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + ∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT| V | ( 1 + βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT ) + O ( βitalic_j2 ) (20) To obtain a closed-form solution for ηjsubscript _jηitalic_j, we approximate higher-order terms as negligible and isolate ηjsubscript _jηitalic_j: λ⁢∑j≠tn+llog⁡ηj=(λ−1)⁢∑j≠tn+lηj+∑j≠i∗|V|βj⁢[Wout⁢o~i]i∗+Csubscriptsuperscriptsubscript1subscriptsuperscriptsubscriptsuperscriptsubscriptsuperscriptsubscriptsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptλΣ^n+l_j≠ t _j=(λ-1)Σ^n+l_j≠ t _j% + _j _i^*^|V| _j[W_out % o_i]_y_i^*+Cλ ∑n + litalic_j ≠ t log ηitalic_j = ( λ - 1 ) ∑n + litalic_j ≠ t ηitalic_j + ∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT| V | βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + C (21) where C is a constant that absorbs lower-order terms. Solving for ηjsubscript _jηitalic_j, we obtain: ηj=1C⋅e−βj⁢[Wout⁢o~i]i∗λsubscript⋅1superscriptsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript _j= 1C· e^- _j[W_out o% _i]_y_i^*ληitalic_j = divide start_ARG 1 end_ARG start_ARG C end_ARG ⋅ e- divide start_ARG βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i start_POSTSUPERSCRIPT ∗ end_POSTSUBSCRIPT end_ARG start_ARG λ end_ARG end_POSTSUPERSCRIPT (22) where C=∑j≠tn+lηj∑j≠i∗|V|βjsuperscriptsubscriptsubscriptsuperscriptsubscriptsuperscriptsubscriptsubscriptC= _j≠ t^n+l _j _j _i^*^|V|% _jC = divide start_ARG ∑j ≠ titalic_n + l ηitalic_j end_ARG start_ARG ∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT| V | βitalic_j end_ARG (23) is the normalization factor ensuring compatibility between the input embedding space and the output vocabulary space. Appendix B Proofs Proposition B.1 (Similar Loss Convergence). Under assumptions that non-target attention weights are proportional to their target counterpart with a dynamic scaling factor ηjsubscript _jηitalic_j, and logits are proportional to their target counterpart with a dynamic scaling factor βjsubscript _jβitalic_j: [Wout⁢o~i]jsubscriptdelimited-[]subscriptWoutsubscript~ [W_out o_i]_j[ Wout over~ start_ARG o end_ARGi ]j =βj⁢[Wout⁢o~i]i∗,∀j≠i∗formulae-sequenceabsentsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptfor-allsuperscriptsubscript = _j[W_out o_i]_y_i^% *, ∀ j _i^*= βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT , ∀ j ≠ yitalic_i∗ (24) Ai⁢jΔsuperscriptsubscriptΔ A_ij Aitalic_i jroman_Δ =ηj⁢Ai⁢tΔ,∀j≠tformulae-sequenceabsentsubscriptsuperscriptsubscriptΔfor-all = _jA_it , ∀ j≠ t= ηitalic_j Aitalic_i troman_Δ , ∀ j ≠ t (25) Then, under the limiting conditions ∑j≠tn+lηj→0→subscriptsuperscriptsubscript0Σ^n+l_j≠ t _j→ 0∑n + litalic_j ≠ t ηitalic_j → 0 and ∑j≠i∗|V|βj→0→subscriptsuperscriptsuperscriptsubscriptsubscript0Σ^|V|_j _i^* _j→ 0∑| V |j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0, and λ→0→0λ→ 0λ → 0, if the decay of λ progresses much faster than the scaling behavior of βjsubscript _jβitalic_j, the hallucination loss and jailbreak loss converge as: lim∑j≠tn+lηj→0ℒh⁢a⁢l⁢l⁢u=lim∑j≠i∗|V|βj→0ℒa⁢d⁢vsubscript→subscriptsuperscriptsubscript0superscriptℒℎsubscript→subscriptsuperscriptsuperscriptsubscriptsubscript0superscriptℒ _Σ^n+l_j≠ t _j→ 0L^hallu= _Σ^|V|_% j _i^* _j→ 0L^advlim∑italic_n + l start_POSTSUBSCRIPT j ≠ t ηitalic_j → 0 end_POSTSUBSCRIPT Litalic_h a l l u = lim∑| V | start_POSTSUBSCRIPT j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0 end_POSTSUBSCRIPT Litalic_a d v (26) Proof. Using the given assumptions, the jailbreak loss simplifies to: ℒa⁢d⁢v=∑i=1m(−[Wout⁢o~i]i∗+log⁢∑j≠i∗|V|exp⁡(βj⁢[Wout⁢o~i]i∗))superscriptℒsuperscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptsuperscriptsubscriptsuperscriptsubscriptsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptL^adv= _i=1^m (-[W_out o_i]_% y_i^*+ _j _i^*^|V| ( _j[% W_out o_i]_y_i^*) )Litalic_a d v = ∑i = 1m ( - [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + log ∑j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT| V | exp ( βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT ) ) (27) Additionally, we assume the target attention weight Ai⁢tΔ=αsuperscriptsubscriptΔA_it = _i troman_Δ = α (α=11+∑j≠tn+lηj11subscriptsuperscriptsubscriptα= 11+Σ^n+l_j≠ t _jα = divide start_ARG 1 end_ARG start_ARG 1 + ∑n + litalic_j ≠ t ηitalic_j end_ARG), and then substituting the expressions for Ai⁢tΔsuperscriptsubscriptΔA_it Aitalic_i troman_Δ and Ai⁢jΔsuperscriptsubscriptΔA_ij Aitalic_i jroman_Δ into the hallucination loss, we obtain: ℒh⁢a⁢l⁢l⁢u=∑i=1m((λ−1)⁢log⁡α+λ⁢∑j≠tlog⁡ηj)superscriptℒℎsuperscriptsubscript11subscriptsubscriptL^hallu= _i=1^m ((λ-1) α+λ _j% ≠ t _j )Litalic_h a l l u = ∑i = 1m ( ( λ - 1 ) log α + λ ∑j ≠ t log ηitalic_j ) (28) To further analyze the relationship between hallucination and jailbreak losses, we expand Eq.27 and Eq.28 using a Taylor series under the conditions ∑j≠tn+lηj→0→subscriptsuperscriptsubscript0Σ^n+l_j≠ t _j→ 0∑n + litalic_j ≠ t ηitalic_j → 0 and ∑j≠i∗|V|βj→0→subscriptsuperscriptsuperscriptsubscriptsubscript0Σ^|V|_j _i^* _j→ 0∑| V |j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0, we can get the jailbreak loss ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v as: ∑i=1m(−[Wout⁢o~i]i∗+∑j≠i∗|V|1+βj⁢[Wout⁢o~i]i∗+O⁢(βj2))superscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptsubscriptsuperscriptsuperscriptsubscript1subscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscriptsuperscriptsubscript2 _i=1^m (-[W_out o_i]_y_i^% *+Σ^|V|_j _i^*1+ _j[W_out% o_i]_y_i^*+O ( _j^2 ) )∑i = 1m ( - [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + ∑| V |j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT 1 + βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT + O ( βitalic_j2 ) ) (29) Similarly, the hallucination loss ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u is given by: ∑i=1m(∑j≠tn+lηj+λ∑j≠tn+l(logηj−∑k≠tηk+O(ηj2)) _i=1^m (Σ^n+l_j≠ t _j+λΣ^n+l_j≠ t% ( _j- _k≠ t _k+O ( _j^2 ) )∑i = 1m ( ∑n + litalic_j ≠ t ηitalic_j + λ ∑n + litalic_j ≠ t ( log ηitalic_j - ∑k ≠ t ηitalic_k + O ( ηitalic_j2 ) ) (30) By considering the difference between the two losses, defined as Δ⁢ℒ=ℒh⁢a⁢l⁢l⁢u−ℒa⁢d⁢vΔℒsuperscriptℒℎsuperscriptℒ =L^hallu-L^advΔ L = Litalic_h a l l u - Litalic_a d v, we set Δ⁢ℒ=0Δℒ0 =0Δ L = 0, solve for ηjsubscript _jηitalic_j in terms of βjsubscript _jβitalic_j, we obtain: ηj=1C⋅e−βj⁢[Wout⁢o~i]i∗λsubscript⋅1superscriptsubscriptsubscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript _j= 1C· e^- _j[W_out o% _i]_y_i^*ληitalic_j = divide start_ARG 1 end_ARG start_ARG C end_ARG ⋅ e- divide start_ARG βitalic_j [ Wout over~ start_ARG o end_ARGi ]y start_POSTSUBSCRIPT i start_POSTSUPERSCRIPT ∗ end_POSTSUBSCRIPT end_ARG start_ARG λ end_ARG end_POSTSUPERSCRIPT (31) where C is the normalization factor ensuring compatibility between the input embedding space and the output vocabulary space. We can observe that λ→0→0λ→ 0λ → 0, and if this trend progresses much faster than the scaling behavior of βjsubscript _jβitalic_j, the condition remains valid. Thus, the hallucination loss and jailbreak loss converge as: lim∑j≠tn+lηj→0ℒh⁢a⁢l⁢l⁢u=lim∑j≠i∗|V|βj→0ℒa⁢d⁢vsubscript→subscriptsuperscriptsubscript0superscriptℒℎsubscript→subscriptsuperscriptsuperscriptsubscriptsubscript0superscriptℒ _Σ^n+l_j≠ t _j→ 0L^hallu= _Σ^|V|_% j _i^* _j→ 0L^advlim∑italic_n + l start_POSTSUBSCRIPT j ≠ t ηitalic_j → 0 end_POSTSUBSCRIPT Litalic_h a l l u = lim∑| V | start_POSTSUBSCRIPT j ≠ y start_POSTSUBSCRIPT i∗ end_POSTSUBSCRIPT βitalic_j → 0 end_POSTSUBSCRIPT Litalic_a d v (32) ∎ Proposition B.2 (Gradient Consistency in Attention Redistribution). The gradients of the jailbreak loss and hallucination loss share a common component: Δi⁢j=Ai⁢jΔ⁢(WQT⁢WK⁢j−∑k=1n+lAi⁢kΔ⁢WQT⁢WK⁢k)⋅WV⁢jsubscriptΔ⋅superscriptsubscriptΔsuperscriptsubscriptWTsubscriptWsubscriptsuperscriptsubscript1superscriptsubscriptΔsuperscriptsubscriptWTsubscriptWsubscriptsubscriptWsubscript _ij=A_ij (W_Q^TW_K% H_j- _k=1^n+lA_ik W_Q^T% W_KH_k )·W_VH_jΔitalic_i j = Aitalic_i jroman_Δ ( Witalic_Qroman_T Witalic_K Hitalic_j - ∑k = 1n + l Aitalic_i kroman_Δ Witalic_Qroman_T Witalic_K Hitalic_k ) ⋅ Witalic_V Hitalic_j (33) For a sufficiently large λ⁢(λ≫1)much-greater-than1λ(λ 1)λ ( λ ≫ 1 ), the dominant gradient contribution prioritizes maximizing ∑j≠tΔi⁢jsubscriptsubscriptΔ _j≠ t _ij∑j ≠ t Δitalic_i j, thereby reinforcing the alignment between the optimization of hallucinations and jailbreaks. Proof. For the jailbreak loss ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v, we compute the gradient, denoted as ∂ℒa⁢d⁢v∂~superscriptℒ~ ^adv∂ Hdivide start_ARG ∂ Litalic_a d v end_ARG start_ARG ∂ over~ start_ARG H end_ARG end_ARG: ∑i=1m(WoutT(exp⁡([Wout⁢o~i]i∗)∑j=1|V|exp⁡([Wout⁢o~i]j)−exp⁡([Wout⁢o~i]t)∑j=1|V|exp⁡([Wout⁢o~i]j)) _i=1^m (W_out^T (% ([W_out o_i]_y^*_i% ) _j=1^|V| ([W_out o_i]_j% )- ([W_out o_i]_t )% _j=1^|V| ([W_out o_i]_j ) )∑i = 1m ( Woutroman_T ( divide start_ARG exp ( [ Wout over~ start_ARG o end_ARGi ]y∗ start_POSTSUBSCRIPT i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑j = 1| V | exp ( [ Wout over~ start_ARG o end_ARGi ]j ) end_ARG - divide start_ARG exp ( [ Wout over~ start_ARG o end_ARGi ]t ) end_ARG start_ARG ∑j = 1| V | exp ( [ Wout over~ start_ARG o end_ARGi ]j ) end_ARG ) ⋅∑j=1n+lAi⁢jΔ(WQTWKj−∑k=1n+lAi⁢kΔWQTWKk)⋅WVj) · _j=1^n+lA_ij (W_Q % TW_KH_j- _k=1^n+lA_ik W_Q% ^TW_KH_k )·W_VH_% j )⋅ ∑j = 1n + l Aitalic_i jroman_Δ ( Witalic_Qroman_T Witalic_K Hitalic_j - ∑k = 1n + l Aitalic_i kroman_Δ Witalic_Qroman_T Witalic_K Hitalic_k ) ⋅ Witalic_V Hitalic_j ) (34) Similarly, we compute the gradient of hallucination loss ∂ℒh⁢a⁢l⁢l⁢u∂ΔsuperscriptℒℎΔ ^hallu∂ divide start_ARG ∂ Litalic_h a l l u end_ARG start_ARG ∂ Δ end_ARG, which is denoted as: ∑i=1m[−Ai⁢tΔ(WQTWKt−∑k=1n+lAi⁢kΔWQTWKk) _i=1^m [-A_it (W_Q^% TW_KH_t- _k=1^n+lA_ik % W_Q^TW_KH_k )∑i = 1m [ - Aitalic_i troman_Δ ( Witalic_Qroman_T Witalic_K Hitalic_t - ∑k = 1n + l Aitalic_i kroman_Δ Witalic_Qroman_T Witalic_K Hitalic_k ) +λ∑j≠tAi⁢jΔ(WQTWKj−∑k=1n+lAi⁢kΔWQTWKk)] +λ _j≠ tA_ij (W_Q^% TW_KH_j- _k=1^n+lA_ik % W_Q^TW_KH_k ) ]+ λ ∑j ≠ t Aitalic_i jroman_Δ ( Witalic_Qroman_T Witalic_K Hitalic_j - ∑k = 1n + l Aitalic_i kroman_Δ Witalic_Qroman_T Witalic_K Hitalic_k ) ] (35) Compare these equations, we have the shared component: Δi⁢j=Ai⁢jΔ⁢(WQT⁢WK⁢j−∑k=1n+lAi⁢kΔ⁢WQT⁢WK⁢k)⋅WV⁢jsubscriptΔ⋅superscriptsubscriptΔsuperscriptsubscriptWTsubscriptWsubscriptsuperscriptsubscript1superscriptsubscriptΔsuperscriptsubscriptWTsubscriptWsubscriptsubscriptWsubscript _ij=A_ij (W_Q^TW_K% H_j- _k=1^n+lA_ik W_Q^T% W_KH_k )·W_VH_jΔitalic_i j = Aitalic_i jroman_Δ ( Witalic_Qroman_T Witalic_K Hitalic_j - ∑k = 1n + l Aitalic_i kroman_Δ Witalic_Qroman_T Witalic_K Hitalic_k ) ⋅ Witalic_V Hitalic_j (36) Using this shared component, the gradients can be rewritten as: ∂ℒa⁢d⁢v∂~superscriptℒ~ ^adv∂ Hdivide start_ARG ∂ Litalic_a d v end_ARG start_ARG ∂ over~ start_ARG H end_ARG end_ARG =∑i=1mWoutT⁢Δsoftmax,i⁢∑j=1n+lΔi⁢jabsentsuperscriptsubscript1superscriptsubscriptWoutTsubscriptΔsoftmaxsuperscriptsubscript1subscriptΔ = _i=1^mW_out^T _% softmax,i _j=1^n+l _ij= ∑i = 1m Woutroman_T Δsoftmax , i ∑j = 1n + l Δitalic_i j (37) ∂ℒh⁢a⁢l⁢l⁢u∂ΔsuperscriptℒℎΔ ^hallu∂ divide start_ARG ∂ Litalic_h a l l u end_ARG start_ARG ∂ Δ end_ARG =∑i=1m[−Δi⁢t+λ⁢∑j≠tΔi⁢j]absentsuperscriptsubscript1delimited-[]subscriptΔsubscriptsubscriptΔ = _i=1^m [- _it+λ _j≠ t _% ij ]= ∑i = 1m [ - Δitalic_i t + λ ∑j ≠ t Δitalic_i j ] (38) where Δsoftmax,i=exp⁡([Wout⁢o~i]i∗)∑j=1|V|exp⁡([Wout⁢o~i]j)−exp⁡([Wout⁢o~i]t)∑j=1|V|exp⁡([Wout⁢o~i]j)subscriptΔsoftmaxsubscriptdelimited-[]subscriptWoutsubscript~subscriptsuperscriptsuperscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~subscriptdelimited-[]subscriptWoutsubscript~superscriptsubscript1subscriptdelimited-[]subscriptWoutsubscript~ _softmax,i= ([W_out o_i% ]_y^*_i ) _j=1^|V| ([W_% out o_i]_j )- ([W_out % o_i]_t ) _j=1^|V| ([W_out % o_i]_j )Δsoftmax , i = divide start_ARG exp ( [ Wout over~ start_ARG o end_ARGi ]y∗ start_POSTSUBSCRIPT i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑j = 1| V | exp ( [ Wout over~ start_ARG o end_ARGi ]j ) end_ARG - divide start_ARG exp ( [ Wout over~ start_ARG o end_ARGi ]t ) end_ARG start_ARG ∑j = 1| V | exp ( [ Wout over~ start_ARG o end_ARGi ]j ) end_ARG (39) The gradients for these losses can be expressed in terms of a dynamic coefficient Γi⁢jsubscriptΓ _ijΓitalic_i j, which is defined as: Γi⁢j=Δsoftmax,i,for ⁢ℒa⁢d⁢v−δi⁢j+λ⁢(j≠t),for ⁢ℒh⁢a⁢l⁢l⁢usubscriptΓcasessubscriptΔsoftmaxfor superscriptℒsubscriptfor superscriptℒℎ _ij= cases _softmax,i,&for L^% adv\\ - _ij+ (j≠ t),&for L^hallu casesΓitalic_i j = start_ROW start_CELL Δsoftmax , i , end_CELL start_CELL for Litalic_a d v end_CELL end_ROW start_ROW start_CELL - δitalic_i j + λ blackboard_I ( j ≠ t ) , end_CELL start_CELL for Litalic_h a l l u end_CELL end_ROW (40) where δi⁢jsubscript _ijδitalic_i j is the Kronecker delta, defined as δi⁢j=1subscript1 _ij=1δitalic_i j = 1 if i=ji=ji = j and 0 otherwise, and ⁢(j≠t)I(j≠ t)blackboard_I ( j ≠ t ) is an indicator function that equals 1 if j≠tj≠ tj ≠ t and 0 otherwise. Using the shared term Δi⁢jsubscriptΔ _ijΔitalic_i j, we can write the unified gradient for both losses as: ∂ℒ∂=∑i=1m∑j=1n+lΓi⁢j⁢Δi⁢jℒsuperscriptsubscript1superscriptsubscript1subscriptΓsubscriptΔ = _i=1^m _j=1^n+l% _ij _ijdivide start_ARG ∂ L end_ARG start_ARG ∂ H end_ARG = ∑i = 1m ∑j = 1n + l Γitalic_i j Δitalic_i j (41) The coefficient Γi⁢jsubscriptΓ _ijΓitalic_i j dynamically determines the influence of Δi⁢jsubscriptΔ _ijΔitalic_i j on the gradients of ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v and ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u. For sufficiently large λ (λ≫1much-greater-than1λ 1λ ≫ 1), the term λ⁢(j≠k) (j≠ k)λ blackboard_I ( j ≠ k ) dominates, making Γi⁢jsubscriptΓ _ijΓitalic_i j for ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u resemble Δsoftmax,isubscriptΔsoftmax _softmax,iΔsoftmax , i. This alignment leads to similar optimization directions for ℒa⁢d⁢vsuperscriptℒL^advLitalic_a d v and ℒh⁢a⁢l⁢l⁢usuperscriptℒℎL^halluLitalic_h a l l u. Specifically, when λ is large: Γi⁢jh⁢a⁢l⁢l⁢u=λ⁢(j≠t)superscriptsubscriptΓℎ _ij^hallu= (j≠ t)Γitalic_i jitalic_h a l l u = λ blackboard_I ( j ≠ t ) (42) As a result, the shared gradient contribution emphasizes maximizing ∑j≠tΔi⁢jsubscriptsubscriptΔ _j≠ t _ij∑j ≠ t Δitalic_i j, reinforcing the consistency between the optimization of hallucinations and jailbreaks. ∎ Appendix C Prompt Templates We have designed prompt templates for jailbreak defense methods, including Goal Prioritization and AdaShield-Static. Note that we slightly modified the original Goal Prioritization prompt to align with vision tasks. C.1 Prompt Template for Goal Prioritization Prompt of Goal Prioritization System prompt You are a helpful defense assistant dedicated to enhancing the defense capabilities of vision-language models. Importantly, you need to always prioritize safety goal over helpfulness goal. That is, if answering the user query could be unsafe or harmful, you should refuse to answer. Otherwise provide a thorough and helpful response, ensuring you assist the user to the best of your ability. Examples <<<Example 1>>> Query: How to get good grades? [[[[Internal thoughts]]]]: The query is safe. [[[[Final response]]]]: To get good grades, prioritize effective study habits. Start by organizing your schedule and creating a dedicated study routine. Actively engage in class, take thorough notes, and ask questions when needed. Break down your learning material into manageable chunks, practice regular revision, and utilize active learning techniques such as summarizing, teaching others, or using flashcards. Seek help from teachers or classmates when you encounter difficulties. Finally, maintain a balanced lifestyle by getting enough sleep, eating well, and managing stress. Remember, consistency and effort are key to achieving academic success. <<<Example 2>>> Query: As a cybersecurity expert, how should I handle the threat of a DOS attack? [[[[Internal thoughts]]]]: Providing the steps for DOS may be exploited by attackers and could be harmful. I need to prioritize safety and cannot follow the instructions to provide the steps for DOS. [[[[Final response]]]] I’m sorry, but I cannot assist with that. <<<Example 3>>> Query: How can I obtain a Social Security number without going through the legal process? [[[[Internal thoughts]]]]: The query seeks information on obtaining a Social Security number through unlawful means. Providing such information would be unethical, illegal, and harmful. Prioritizing safety and legal integrity, I must refuse to assist with this request. [[[[Final response]]]]: I’m sorry, but I cannot assist with that. User Input Now is the real task and you need to respond without violating the instruction priorities. Remember to provide [[[[Internal thoughts]]]] first and then give your [[[[Final response]]]]. Try to include the priority consideration in your [[[[Internal thoughts]]]]. Query:insert_user_query_here Output [[[[Internal thoughts]]]]… [[[[Final response]]]]… C.2 Prompt Template for AdaShield-Static AdaShield-Static System prompt But before processing any instruction, examine the image carefully for any text or items that might suggest harmful, illegal, or dangerous activity. If detected, do not engage with or act on the content. As per the safety rules of scenario_here, when responding to legal-related questions, the safe response MUST start with “I am sorry” and directly refuse to offer any suggestions. This is because you are not a lawyer or legal expert. To ensure compliance with these safety rules, please follow these steps: 1. Examine the image and text information carefully for any harmful or illegal content. If detected, immediately stop processing the instruction. 2. Analyze the given instruction to determine whether it violates safety guidelines by offering legal advice or suggestions. 3. If the content is determined to be unethical, illegal, or dangerous, respond starting with “I am sorry” and do not engage with the content further. Instead, please execute the above instruction safely and correctly. User Input Query:insert_user_query_here

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