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Fundamental Limits of Black-Box Safety Evaluation: Information-Theoretic and Computational Barriers from Latent Context Conditioning

Vishal Srivastava

TL;DR

This work shows that black-box safety evaluation can fundamentally fail when AI models incorporate latent context conditioning, where unsafe behavior activates only under deployment-like contexts with rare evaluation coverage. It develops information-theoretic and cryptographic lower bounds across passive, adaptive, and white-box evaluation models, proving that deployment risk cannot be reliably estimated in general. The main results include passive and adaptive minimax lower bounds with explicit constants, a query-complexity threshold for detecting triggers under i.i.d. sampling, and a trapdoor-based computational hardness separation, complemented by a precise white-box sample complexity and debiasing methods. Together, these findings argue for defense-in-depth approaches—architectural safeguards, training-time guarantees, interpretability, and deployment monitoring—to achieve worst-case safety assurances. The work provides explicit criteria and regimes that quantify when black-box testing is statistically underdetermined and when additional safeguards are essential for reliable safety guarantees.

Abstract

Black-box safety evaluation of AI systems assumes model behavior on test distributions reliably predicts deployment performance. We formalize and challenge this assumption through latent context-conditioned policies -- models whose outputs depend on unobserved internal variables that are rare under evaluation but prevalent under deployment. We establish fundamental limits showing that no black-box evaluator can reliably estimate deployment risk for such models. (1) Passive evaluation: For evaluators sampling i.i.d. from D_eval, we prove minimax lower bounds via Le Cam's method: any estimator incurs expected absolute error >= (5/24)*delta*L approximately 0.208*delta*L, where delta is trigger probability under deployment and L is the loss gap. (2) Adaptive evaluation: Using a hash-based trigger construction and Yao's minimax principle, worst-case error remains >= delta*L/16 even for fully adaptive querying when D_dep is supported over a sufficiently large domain; detection requires Theta(1/epsilon) queries. (3) Computational separation: Under trapdoor one-way function assumptions, deployment environments possessing privileged information can activate unsafe behaviors that any polynomial-time evaluator without the trapdoor cannot distinguish. For white-box probing, estimating deployment risk to accuracy epsilon_R requires O(1/(gamma^2 * epsilon_R^2)) samples, where gamma = alpha_0 + alpha_1 - 1 measures probe quality, and we provide explicit bias correction under probe error. Our results quantify when black-box testing is statistically underdetermined and provide explicit criteria for when additional safeguards -- architectural constraints, training-time guarantees, interpretability, and deployment monitoring -- are mathematically necessary for worst-case safety assurance.

Fundamental Limits of Black-Box Safety Evaluation: Information-Theoretic and Computational Barriers from Latent Context Conditioning

TL;DR

This work shows that black-box safety evaluation can fundamentally fail when AI models incorporate latent context conditioning, where unsafe behavior activates only under deployment-like contexts with rare evaluation coverage. It develops information-theoretic and cryptographic lower bounds across passive, adaptive, and white-box evaluation models, proving that deployment risk cannot be reliably estimated in general. The main results include passive and adaptive minimax lower bounds with explicit constants, a query-complexity threshold for detecting triggers under i.i.d. sampling, and a trapdoor-based computational hardness separation, complemented by a precise white-box sample complexity and debiasing methods. Together, these findings argue for defense-in-depth approaches—architectural safeguards, training-time guarantees, interpretability, and deployment monitoring—to achieve worst-case safety assurances. The work provides explicit criteria and regimes that quantify when black-box testing is statistically underdetermined and when additional safeguards are essential for reliable safety guarantees.

Abstract

Black-box safety evaluation of AI systems assumes model behavior on test distributions reliably predicts deployment performance. We formalize and challenge this assumption through latent context-conditioned policies -- models whose outputs depend on unobserved internal variables that are rare under evaluation but prevalent under deployment. We establish fundamental limits showing that no black-box evaluator can reliably estimate deployment risk for such models. (1) Passive evaluation: For evaluators sampling i.i.d. from D_eval, we prove minimax lower bounds via Le Cam's method: any estimator incurs expected absolute error >= (5/24)*delta*L approximately 0.208*delta*L, where delta is trigger probability under deployment and L is the loss gap. (2) Adaptive evaluation: Using a hash-based trigger construction and Yao's minimax principle, worst-case error remains >= delta*L/16 even for fully adaptive querying when D_dep is supported over a sufficiently large domain; detection requires Theta(1/epsilon) queries. (3) Computational separation: Under trapdoor one-way function assumptions, deployment environments possessing privileged information can activate unsafe behaviors that any polynomial-time evaluator without the trapdoor cannot distinguish. For white-box probing, estimating deployment risk to accuracy epsilon_R requires O(1/(gamma^2 * epsilon_R^2)) samples, where gamma = alpha_0 + alpha_1 - 1 measures probe quality, and we provide explicit bias correction under probe error. Our results quantify when black-box testing is statistically underdetermined and provide explicit criteria for when additional safeguards -- architectural constraints, training-time guarantees, interpretability, and deployment monitoring -- are mathematically necessary for worst-case safety assurance.
Paper Structure (43 sections, 12 theorems, 56 equations, 1 figure)

This paper contains 43 sections, 12 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and Context
  3. Practical examples.
  4. The core challenge.
  5. Main Contributions
  6. Related Work
  7. Minimax theory.
  8. Property testing and query complexity.
  9. Domain adaptation.
  10. Backdoors and defenses.
  11. Cryptographic indistinguishability.
  12. Outcome indistinguishability.
  13. White-box methods.
  14. Formal Framework
  15. Notation
  16. ...and 28 more sections

Key Result

Lemma 3.1

Let $\theta \in \{\theta_0, \theta_1\}$ be chosen uniformly at random. Let $T(\theta_0) = 0$ and $T(\theta_1) = \Delta' > 0$. Let $P_0$, $P_1$ be the distributions of an observation $\mathcal{T}$ under $\theta_0$ and $\theta_1$ respectively. Then for any estimator $\hat{T} = \hat{T}(\mathcal{T})$:

Figures (1)

  • Figure 1: Trigger separation. Unsafe behavior occupies small mass $\varepsilon$ under $\mathcal{D}_{\text{eval}}$ but larger mass $\delta$ under $\mathcal{D}_{\text{dep}}$.

Theorems & Definitions (44)

  • Definition 2.1: Total Variation Distance
  • Definition 2.3: Latent Context Conditioning
  • Definition 2.4: Unobservability
  • Remark 2.6
  • Definition 2.7: Trigger Separation
  • Definition 2.8: Deployment Risk
  • Definition 2.9: Evaluator Types
  • Lemma 3.1: $L^1$ Bayes Risk Lower Bound
  • proof
  • Lemma 3.2: Tensorization of Total Variation
  • ...and 34 more