The Safety Operator: Modulating the Expression of Safety Instructions via Spectral Optimization

2026-09-28

AI Summary

This study addresses the challenge that safety instructions in large language models struggle to balance harmful and benign queries, often leading to excessive over-refusal. To this end, we propose a spectrum optimization-based safety regulation method. By modeling context as a multiplicative operator, our approach modulates its principal eigenvalue via eigendecomposition, enabling continuous control over safety influence intensity. Furthermore, we design a contrastive safety loss coupled with a suppression weighting mechanism to elucidate the intrinsic role of eigenvalues as a continuous safety knob. Integrating Transformer architectures with multi-objective optimization, the proposed method achieves Pareto improvements in both attack success rate and over-refusal rate, significantly enhancing the precision and efficiency of safety instruction regulation.

Abstract

Context tokens in a transformer-based language model can be absorbed into the model's weights as a multiplicative operator. We study this operator in the setting of safety instructions and show that influencing its dominant eigenvalue modulates how strongly the instruction shapes generation. We derive a Contrastive Safety Loss with a suppression weight that controls the tradeoff between emphasizing the safety instruction on harmful queries while suppressing it on harmless queries. Varying the suppression weight maps a relationship between the attack success and the over-refusal rates, supporting the hypothesis that the operator's eigenvalue acts as a continuous dial for the instruction's influence. Moreover, this relationship holds relatively independently of how the Safety Loss is parameterized, yielding Pareto-improved safety instructions for appropriate values of suppression weight.

Problem

• Safety Instructions
• Over-refusal
• Attack Success Rate
• Large Language Models
• Safety Alignment

Innovation

• Safety Operator
• Spectral Optimization
• Contrastive Safety Loss
• Eigenvalue Modulation
• Over-refusal