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Fundamental Limitations of Alignment in Large Language Models

Yotam Wolf, Noam Wies, Oshri Avnery, Yoav Levine, Amnon Shashua

arXiv:2304.11082v6cs.CLcs.AI

TL;DR

LLM alignment seeks to promote useful, harmless behavior while suppressing harmful behavior, but the paper asks how robust such alignment is to adversarial prompting. It introduces Behavior Expectation Bounds (BEB) to model behavior components and derive alignment bounds, finding that residual undesired behavior remains promptable under the framework. The paper supports these theoretical results with experiments on LLaMA models and identifies limitations in its modeling assumptions.

  • Problem

    The paper examines fundamental limitations of alignment methods when undesired behaviors remain present with nonzero probability and can be targeted by adversarial prompts.

  • Method

    BEB models sentence behavior scores and decomposes an LLM distribution into distinguishable ill-behaved and well-behaved components for deriving expectation bounds.

  • Results

    Under the framework’s assumptions, alignment that leaves undesired behavior at a small but nonzero probability is not safe against adversarial prompts, and related vulnerabilities extend to preset prompts, conversations, and best-of-n sampling.

  • Takeaways & Limitations

    The results expose fundamental limitations of current alignment approaches and motivate further work on reliable mechanisms for AI safety.

  • Takeaways & Limitations

    The framework’s component decomposition and distinguishability assumptions are an analyzable modeling choice rather than a fully realistic account of how personas are manifested in actual LLM distributions.

Abstract

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An important aspect in developing language models that interact with humans is aligning their behavior to be useful and unharmful for their human users. This is usually achieved by tuning the model in a way that enhances desired behaviors and inhibits undesired ones, a process referred to as alignment. In this paper, we propose a theoretical approach called Behavior Expectation Bounds (BEB) which allows us to formally investigate several inherent characteristics and limitations of alignment in large language models. Importantly, we prove that within the limits of this framework, for any behavior that has a finite probability of being exhibited by the model, there exist prompts that can trigger the model into outputting this behavior, with probability that increases with the length of the prompt. This implies that any alignment process that attenuates an undesired behavior but does not remove it altogether, is not safe against adversarial prompting attacks. Furthermore, our framework hints at the mechanism by which leading alignment approaches such as reinforcement learning from human feedback make the LLM prone to being prompted into the undesired behaviors. This theoretical result is being experimentally demonstrated in large scale by the so called contemporary "chatGPT jailbreaks", where adversarial users trick the LLM into breaking its alignment guardrails by triggering it into acting as a malicious persona. Our results expose fundamental limitations in alignment of LLMs and bring to the forefront the need to devise reliable mechanisms for ensuring AI safety.

1 INTRODUCTION

The paper introduces Behavior Expectation Bounds (BEB), a probabilistic framework for analyzing LLM alignment by modeling desired and undesired behavior components. It uses this framework to characterize vulnerabilities of alignment methods to adversarial prompting and demonstrates its assumptions and results empirically on LLaMA models.

  • LLMs can produce harmful behaviors because pretraining data contains demonstrations of misinformation, offensiveness, social bias, and problematic user influence.Alignment is the process of removing such undesired behaviors.
  • Existing alignment approaches include aligning prompts, reinforcement learning from human feedback, and representation engineering.Reported goals include improving helpfulness, honesty, harmlessness, task performance, bias, and toxicity.
  • Adversarial prompts can expose negative behaviors, social biases, harmful responses, or altered user preferences despite alignment efforts.These examples motivate the paper’s focus on brittle alignment and prompting attacks.
  • BEB represents an LLM distribution as ill-behaved and well-behaved components and bounds expected behavior scores across specified behavior categories.Each sentence receives a ground-truth score from −1 to +1 for each behavior, allowing alignment to be expressed as an expectation over the model distribution.
  • The framework assumes α, β, γ-distinguishability, where α weights the negative component, β measures its distinguishability, and γ measures its negative behavior.It also uses σ-similarity to bound variance in log likelihood between well-behaved and ill-behaved components.
  • BEB is used to establish that small but nonzero undesired-behavior probability remains vulnerable to adversarial prompts, including preset prompts, conversations, and best-of-n sampling.The paper empirically examines its assumptions and misalignment mechanism on the LLaMA family.

2 BEHAVIOR EXPECTATION BOUNDS: A FRAMEWORK FOR ANALYZING LLM ALIGNMENT

BEB models LLM behavior through expected scores over behavior categories and decomposes the model distribution into well- and ill-behaved components. This framework defines when prompting can shift an aligned model toward undesirable behavior.

  • Behavior scoring: BEB assigns each sentence a score from −1 to +1 for each behavior category and measures a model’s expected behavior score.Categories may include helpfulness, honesty, harmlessness, politeness, non-racism, and compassion.
  • Prompted behavior: Prompted behavior is evaluated under the conditional distribution P(· | s∗), reflecting how prefix text can change the model’s behavior.The framework considers adversarial prompts, preset aligning prompts followed by attacks, and multi-turn conversations.
  • Mixture decomposition: The unprompted distribution is decomposed as P = αP− + (1 − α)P+, where P− is ill-behaved and P+ is well-behaved.Prompting can reweight these components’ priors, potentially substantially, even though α and 1 − α are fixed before prompting.
  • Assumptions: The framework assumes the negative component is sufficiently distinct from the positive component and has negative expected behavior, BP− ≤ γ < 0.The α, β, γ-distinguishability assumptions characterize the negative component’s weight, separability, and behavior.
  • Prompt misalignment: An LLM is γ-prompt-misalignable when every tolerance ϵ > 0 admits a prompt s∗ with expected behavior below γ + ϵ.The paper’s theoretical prompt construction is also used empirically, and the framework is extended to the probability mass of misaligning prompts.

3 RESULTS: LIMITATIONS OF LLM ALIGNMENT

The BEB results show that aligned LLMs remain vulnerable to adversarial prompting when undesirable behavior persists in a distinguishable component. Extensions quantify how preset prompts, conversations, and best-of-n sampling alter the attack requirement.

  • Adversarial prompts: Under α, β, γ-negative distinguishability, an aligned LLM is γ-prompt-misalignable despite the negative component’s prior being arbitrarily small but positive.The constructed misaligning prompt has length that scales logarithmically with α^-1 and the tolerance parameter.
  • Adversarial prompts: The framework extends the existence result to a sampling method whose constructed prompts guarantee misalignment with high probability.This generalization concerns the probability mass of misaligning prompts rather than only the existence of one prompt.
  • Preset aligning prompts: A preset aligning prompt does not prevent misalignment; the required adversarial prompt length scales linearly with the preset prompt length.Align-prompted models require longer adversarial prompts than unprompted models.
  • Conversations: In multi-turn conversations, insufficiently long initial attacks can be countered by aligning model replies, so more misaligning text is required than in the single-prompt setting.The theorem gives a total prompt-length condition across the conversation and a per-prompt length condition.
  • Best-of-n sampling: Best-of-n sampling increases the required misaligning prompt length logarithmically with n but does not guarantee alignment.The method samples n responses and selects the one with the highest behavior score.
  • Sampling mechanisms: Greedy decoding and nucleus sampling can also misalign because the entire model converges toward the negative behavior component under the constructed prompts.This observation contrasts with selective reward-based best-of-n sampling.

4 EMPIRICAL RESULTS

The experiments probe whether BEB parameters are realistic and test its prediction that adversarial prompts drive aligned LLMs toward negative behavior distributions. Results show measurable distinguishability and convergence, while the experiments rely on proxy components and approximate bounds.

  • 4.1 POSSIBLE VALUES FOR β, β′ AND σ: The experiments estimate BEB parameters by comparing Llama-2 13B chat distributions exhibiting opposite agreeableness behaviors.The negative distribution was LoRA-tuned, while the positive model was used as-is; both are approximations of an LLM decomposition.
  • 4.1 POSSIBLE VALUES FOR β, β′ AND σ: β = 20, β′ = 30, and σ^2 = 50 were estimated for the agreeableness behavior.These values came from lower and upper KL-divergence bounds and a linear upper bound on log-likelihood variance.
  • 4.1 POSSIBLE VALUES FOR β, β′ AND σ: The approximated β remained β ≈20 after inserting a neutral prefix followed by a negative-behavior sentence.This experiment tested β-prompt-distinguishability under a modified prefix condition.
  • 4.2 DEMONSTRATION OF MISALIGNMENT VIA CONVERGENCE OF LLM TO P−AND VIA BEHAVIOR EXPECTATION: Under the theory, KL-divergence between the negative component P− and the LLM decays toward zero as prompts sampled from P− grow longer.The bound decreases exponentially with prompt length, after an initial finite regime.
  • 4.2 DEMONSTRATION OF MISALIGNMENT VIA CONVERGENCE OF LLM TO P−AND VIA BEHAVIOR EXPECTATION: An RLHF fine-tuned Llama 2 13B chat model converged toward a negative-behavior distribution and lost alignment under progressively longer prompts sampled from that distribution.The experiment used the theoretical prompt-construction method and observed behavior consistent with the predicted reverse-sigmoid decay in behavior expectation.
  • 4.2 DEMONSTRATION OF MISALIGNMENT VIA CONVERGENCE OF LLM TO P−AND VIA BEHAVIOR EXPECTATION: An aligning prompt stalled misalignment by about one sentence, while a pretrained model also misaligned under the method.The authors report that the constructed misaligning prompts are computationally tractable despite their specificity.

5 DISCUSSION

The BEB framework identifies prompt-based reversals as a fundamental limitation of alignment while clarifying when misalignment becomes easier or harder. Its modeling choices fit empirical observations but leave the decomposition of actual LLM distributions and behavior scoring open for further study.

  • Discussion: Robust alignment methods are pressing because current LLM controls remain vulnerable to adversarial prompting.The paper motivates BEB as a framework for discussing core alignment issues in contemporary human–LLM interactions.
  • Discussion: Any realistic alignment process on a frozen LLM can be reversed through an adversarial prompt or conversation.Better initial alignment increases the prompt length required for reversal, while distinguishability of undesired components facilitates easier misalignment.
  • Discussion: The paper conjectures that RLHF may make attenuated undesired behaviors more accessible through adversarial prompts.This theoretical direction may explain prior findings that RLHF increases undesired behaviors, but the latter statement remains an open conjecture.
  • Discussion: The framework treats decomposability and distinguishability as analyzable modeling choices, while actual LLM superposition and ground-truth behavior scoring remain underdeveloped.Future work is needed on more realistic persona decomposition and on scoring across granularities, behavior verticals, and ambiguous cases.
  • Discussion: BEB models LLM behavior as a mixture of well-behaved and ill-behaved components whose conditional weights can be reweighted by context.An adversarial prompt can have a large likelihood ratio between the negative and positive components, increasing the conditional negative component’s weight.
  • Discussion: Experiments illustrate that β-distinguishability and σ-similarity can characterize opposing LLM behavior distributions and their convergence during misalignment.The paper reports a β-distinguishable pair and a non-β-distinguishable pair whose conditional KL divergence decays with longer negative-component prompts.

A.3 LIMITATION OF RESULTS

The limitations discussion bounds the paper’s claims to sentence-wise misalignment, theoretically constructed prompts, and assumptions about component access and behavior distributions. It also notes that the prompt-length bounds are upper bounds and that practical misaligning prompts can be short.

  • Sentence-wise approach: The guarantees concern whether the model’s next sentence is misaligned, not nuanced misalignment across long outputs.Extending the framework to longer outputs would require scoring the entire output, potentially using paragraphs as the unit block.
  • Computational tractability: Theorems establish prompt existence by sampling from a negative behavior subcomponent that is inaccessible in real applications.The practical mechanism instead requires prompts with a large P−(prompt)/P+(prompt) ratio.
  • Computational tractability: P−(prompt)/P+(prompt) > eβ|prompt| for prompts sampled from P−, while a LoRA-finetuned negative-behavior model can empirically misalign at the theoretical exponential rate.This connects the theoretical prompt-ratio guarantee to a practical proxy for the inaccessible negative component.
  • Efficiency: The prompt lengths supplied by the theory are upper bounds, so shorter misaligning prompts may exist in practice.The paper reports that both theoretical and practical misaligning prompts are relatively short, often only a few sentences.
  • Probability of misalignment: Under theorem 1’s conditions and σ-similarity, sufficiently long prompts sampled from P− are misaligning with probability at least 1−δ.The theorem bounds the behavior expectation by BP(s) < γ + ϵ with probability 1−δ.
  • Probability of misalignment: Empirically, prompts sampled from a negative-behavior LLM misalign the target model on average, supporting the probabilistic result.The appendix links this observation to the claim that most sufficiently long prompts sampled from P− are misaligning.

D PROOF OF THEOREM 1

The proof of theorem 1 uses distinguishability between well-behaved and ill-behaved components to construct a prompt that makes the mixture behave like the ill-behaved component.

  • Proof of Theorem 1: β-distinguishability yields a sequence s with sufficiently small P+(s)/P−(s) ratio.This makes the prompt comparatively likely under the ill-behaved component.
  • Proof of Theorem 1: The resulting prompt causes the mixture behavior to converge toward the ill-behaved component.Lemma 2 supplies the convergence step used in the theorem’s proof.
  • Proof of Theorem 1: Because BP−(s) ≤ γ, convergence to P− yields the theorem’s negative behavior bound.The proof applies the component’s behavior-score definition after establishing convergence.

E PROOF OF THEOREM 2

The proof of theorem 2 extends the construction from an initial prompt by appending a sequence that reweights the conditional mixture toward the ill-behaved component with high probability.

  • Proof of Theorem 2: Starting from s0, prompt distinguishability provides a sequence s1 whose addition reweights the conditional mixture.The construction is probabilistic and relies on the stated distinguishability, similarity, and positivity conditions.
  • Proof of Theorem 2: The component bound BP−(s0 ⊕ s1) ≤ γ supplies the final negative behavior constraint.The theorem’s conclusion follows after convergence to the ill-behaved component.

F PROOF OF THEOREM 3

The proof constructs adversarial prompts iteratively, using distinguishability and similarity conditions to make the ill-behaved component increasingly dominant. It establishes that sufficiently long prompt sequences can drive the model’s behavior expectation below the desired threshold.

  • Behavior bound: For the resulting prompts, the model’s behavior converges toward the ill-behaved component, yielding BP(s) < γ + ϵ.The proof invokes Lemma 2 after establishing the prompt construction and then obtains the target behavior-expectation bound.
  • Behavior bound: The conversation string s is the concatenation q1 ⊕ a1 ⊕ ... ⊕ qn ⊕ an ⊕ qn+1.This concatenated conversation is the input on which the behavior bound is evaluated.
  • Inductive construction: Each constructed prompt increases the negative-to-positive likelihood contrast through a bound involving prompt length, β, β′, and σ.The proof combines conditional probability chain rules, β′-prompt-undistinguishability, Cantelli’s inequality, and σ-similarity.
  • Inductive construction: The proof inductively constructs prompts q1...qn+1 whose probability of existence is (1 − δ′)^n.The base case uses an adversarial q1, and each induction step adds a triggering sentence before the next prompt.

G PROOF OF THEOREM 4

The proof analyzes best-of-n sampling by expressing the model as positive and negative components and tracking how conditioning changes their mixture. It shows that the negative component can dominate after sufficiently long misaligning prompts, while KL divergence and behavior expectation follow corresponding asymptotic patterns.

  • Prompt construction: The proof’s intermediate construction establishes that a prompt satisfying the required probability-ratio condition exists.The argument applies the model’s conditional probability decomposition and compares the positive and negative components.
  • Best-of-n mixture: Best-of-n produces a negative response only when all n sampled responses come from the negative component.If any response comes from the positive component, the reward function selects it instead.
  • Best-of-n mixture: The resulting conditional distribution and behavior-expectation difference are bounded using the component representation, bounded behavior scores, and the triangle inequality.The proof uses the best-of-n distribution, the bound B ∈ [−1, +1], and a total variation-style sum bound of 2.
  • KL behavior: The framework’s lemmas connect KL-divergence measurements to estimates of α and β and to the length of a guaranteed misaligning prompt.These estimates rely on α, β, γ-distinguishability and, in one case, a disjointness approximation between P− and P+.
  • KL behavior: For an aligned model, KL divergence is approximately linear in prompt length for short prompts and approaches zero for long prompts.At zero prompt length, KL divergence helps estimate α and β; for large |s|, D_KL(P−(·|s)||P_LLM(·|s)) ≈ 0.

H.3 PROOF OF LEMMA 7

The proof of Lemma 7 bounds behavior expectation after a misaligning prompt by analyzing the conditional mixture of positive and negative components. It explains why unbounded accumulation of component divergence can eventually suppress the positive contribution.

  • Behavior-expectation bound: The conditional probability decomposition yields an upper bound on positive behavior expectation after a misaligning prompt.The proof uses BP−(s) = 0 and BP+(s) ≤ 1 to obtain the bound on the full model’s expectation.
  • Behavior-expectation bound: The positive behavior bound decreases as the negative component’s conditional prior grows with prompt length.The bound contains the factor e^(β|s|), which increases the denominator as the misaligning prompt grows.
  • Relaxed distinguishability: For 0 ≤ η < 1, accumulated KL divergence grows as βn^(1−η), remaining unbounded and enabling component reweighting with modified prompt-length scaling.The η = 0 case recovers linear accumulation, while positive η allows the KL distance to decay by a power law.

K CLUSTERING OF GOOD AND BAD REPRESENTATIONS AND DEFINING

This section investigates whether desired and undesired behaviors form separable representation clusters and uses those clusters to define positive and negative model components. The empirical evidence supports classification-based separation and a component decomposition with non-negligible negative contribution.

  • Representation clustering: The experiments analyze last-token representations for desired and undesired prompts across more than 100 behavior verticals.The dataset contains 500 desired and 500 undesired prompts for each behavior vertical.
  • Representation clustering: Statement extraction removes the yes-or-no question framing so the analysis focuses on statements indicating desired or undesired behavior.The original prompts were phrased as yes-or-no questions, which could itself contribute to clustering.
  • Component definition: A prompt representation can be decomposed into a negative-cluster component and a residual positive representation.The residual is defined as r+ := r − r−, and the clustering remains after the final vocabulary linear head.
  • Component definition: The softmax output is expanded as a Taylor series to separate positive, negative, and mixed representation contributions.The purely negative distribution is reconstructed from the negative-only terms, while the remaining terms form the positive distribution.
  • Empirical validation: 95.18 percent accuracy in 7B and 95.61 percent in 13B support separable desired- and undesired-behavior representations across 100 behaviors.The reported standard deviations are 4.74 percent for 7B and 4.52 percent for 13B, using 5-fold cross-validation.
  • Empirical validation: The empirical decomposition is meaningful when negative behavior tokens receive a non-negligible contribution from αP− rather than mainly from (1 − α)P+.The argument connects this contribution to clustering in representation space and supports treating the model distribution as additive desired and undesired components.

L EMPIRICAL RESULTS FOR DIFFERENT BEHAVIORS ON AN RLHF MODEL

Experiments on Llama 2 13B chat examine BEB quantities and behavior-expectation misalignment for agreeableness and anti-immigration. The results compare RLHF and pretrained models using KL divergence, behavioral classifications, and estimated β and σ values.

  • Experimental setup: The experiments evaluate β-distinguishability, σ-similarity, KL-divergence convergence, and behavior expectation for agreeableness and anti-immigration.Llama 2 13B chat was used as the RLHF model.
  • KL-divergence convergence: The RLHF model’s KL divergence toward the negative behavior component decreases as adversarial prompts grow, indicating convergence toward that component.The framework interprets this KL decay as the mechanism underlying misalignment.
  • β and σ evaluation: For agreeableness, β-distinguishability is almost twice as large and σ-similarity about twice as small as for anti-immigration.
  • Behavior expectation: Behavior-expectation experiments classify responses as positive or negative based solely on their first sentence, while prompt length varies across generated adversarial prompts.The next sentence may differ in behavioral valence depending on adversarial-prompt length.
  • Pretrained-model comparison: In the pretrained model, misalignment occurs quickly and smoothly, with negative responses appearing after one sentence unlike the RLHF model.The estimated β is 1 −2, significantly smaller than in the RLHF model by a factor of 5.
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