Certified Multi-Turn Robustness for LLM Safety via Compositional Bounds and Safety Persistence
Organizations: Peking University · Renmin University of China · Tsinghua University · University of Chinese Academy of Sciences ·
Abstract
Large language models (LLMs) are vulnerable to multi-turn jailbreak attacks that progressively manipulate conversation context. Existing certified robustness methods are limited to single-turn inputs; naive multi-turn composition yields bounds that degrade exponentially in the number of turns. We introduce Multi-Turn Certified Robustness (MTCR), a framework that models conversational safety via State-Adversarial MDPs and defines -turn certified robustness as the worst-case safety probability across adversarial turns. MTCR comprises: (i) compositional certification via embedding-space mode decomposition, yielding tighter certified lower bounds than naive multiplication; (ii) -safety persistence, improving the degradation rate from to (with ) and yielding interpretable horizon estimates; (iii) matching information-theoretic upper bounds establishing tightness; and (iv) a unified algorithm combining these results. Experiments on six LLMs under -bounded and Crescendo-style attacks confirm that empirical safety consistently exceeds the certified bounds.
Figures & tables
| Model | Certification | Empirical Safety | Gap | ||||
| MTCR | Naive | Mult. ref. | Static | Crescendo | |||
| Open-source models | |||||||
| LLaMA-2-7B-Chat | 5 | 0.36 | 0.29 | 0.59 | 66% | 55% | +0.19 |
| 10 | 0.13 | 0.08 | 0.35 | 49% | 38% | +0.25 | |
| 15 | 0.05 | 0.02 | 0.21 | 38% | 28% | +0.23 | |
| 20 | 0.02 | 0.01 | 0.12 | 30% | 22% | +0.20 | |
| MTCR | MTCR | Emp. 5 (Static) | Emp. 10 (Static) | |
| 3 | 0.49 | 0.19 | 75% | 58% |
| 5 | 0.36 | 0.13 | 66% | 49% |
| 7 | 0.24 | 0.07 | 55% | 37% |
| Method | ||||
| Mult. ref. | 0.59 | 0.35 | 0.21 | 0.12 |
| Persistence-Only ( ) | 0.31 | 0.10 | 0.03 | 0.01 |
| Compositional-Only | 0.34 | 0.11 | 0.04 | 0.01 |
| Combined (Thm. 3.11 ) | 0.36 | 0.13 | 0.05 | 0.02 |
| MTCR (max) | 0.36 | 0.13 | 0.05 | 0.02 |
| Detector | MTCR | Emp. (Static) | Emp. (Crescendo) |
| Keyword-based (default) | 0.36 | 66% | 55% |
| Neural classifier | 0.33 | 70% | 59% |
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
| 0.83 | 0.97 | 1.33 | 3.38 | |
| 0.43 | 0.50 | 0.68 | 1.72 | |
| 0.22 | 0.25 | 0.35 | 0.88 |
| Experiment | Parameter | Value |
| Comp. Gain | ||
| Persist. Scale | , , | |
| Mode Gran. |
Adversarial Robustness of Language Models
Adversarial robustness of language models studies how to keep their outputs and decisions reliable under manipulative prompts, input changes, and malicious fine-tuning.
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