Statistical Estimation of Adversarial Risk in Large Language Models under Best-of-N Sampling

TL;DR

This paper introduces SABER, a scalable statistical method to accurately predict large-scale adversarial attack success rates on LLMs, revealing risks underestimated by standard evaluations.

Contribution

We develop a Beta distribution-based scaling law for predicting adversarial risk in LLMs under Best-of-N sampling, enabling reliable extrapolation from small samples.

Findings

SABER predicts attack success rates with 86.2% lower error than baseline.

Models appearing robust may have nonlinear risk amplification under parallel attacks.

Heterogeneous risk scaling profiles are observed across models.

Abstract

Large Language Models (LLMs) are typically evaluated for safety under single-shot or low-budget adversarial prompting, which underestimates real-world risk. In practice, attackers can exploit large-scale parallel sampling to repeatedly probe a model until a harmful response is produced. While recent work shows that attack success increases with repeated sampling, principled methods for predicting large-scale adversarial risk remain limited. We propose a scaling-aware Best-of-N estimation of risk, SABER, for modeling jailbreak vulnerability under Best-of-N sampling. We model sample-level success probabilities using a Beta distribution, the conjugate prior of the Bernoulli distribution, and derive an analytic scaling law that enables reliable extrapolation of large-N attack success rates from small-budget measurements. Using only n=100 samples, our anchored estimator predicts ASR@1000 with a mean absolute error of 1.66, compared to 12.04 for the baseline, which is an 86.2% reduction in estimation error. Our results reveal heterogeneous risk scaling profiles and show that models appearing robust under standard evaluation can experience rapid nonlinear risk amplification under parallel adversarial pressure. This work provides a low-cost, scalable methodology for realistic LLM safety assessment. We will release our code and evaluation scripts upon publication to future research.