What's the Magic Word? A Control Theory of LLM Prompting

• URL: http://arxiv.org/abs/2310.04444v4
• Date: Wed, 3 Jul 2024 22:23:50 GMT
• Title: What's the Magic Word? A Control Theory of LLM Prompting
• Authors: Aman Bhargava, Cameron Witkowski, Shi-Zhuo Looi, Matt Thomson,
• Abstract summary: We explore prompt engineering through the lens of control theory. We present empirical results on the controllability of a panel of LLMs. Short prompt sequences can dramatically alter the likelihood of specific outputs.
• Score: 0.7499722271664144
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Prompt engineering is crucial for deploying LLMs but is poorly understood mathematically. We formalize LLM systems as a class of discrete stochastic dynamical systems to explore prompt engineering through the lens of control theory. We offer a mathematical analysis of the limitations on the controllability of self-attention as a function of the singular values of the parameter matrices. We present complementary empirical results on the controllability of a panel of LLMs, including Falcon-7b, Llama-7b, and Falcon-40b. Given initial state $\mathbf x_0$ from Wikitext and prompts of length $k \leq 10$ tokens, we find that the "correct" next token is reachable at least 97% of the time, and that the top 75 most likely next tokens are reachable at least 85% of the time. Intriguingly, short prompt sequences can dramatically alter the likelihood of specific outputs, even making the least likely tokens become the most likely ones. This control-theoretic analysis of LLMs demonstrates the significant and poorly understood role of input sequences in steering output probabilities, offering a foundational perspective for enhancing language model system capabilities.

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