What's the Magic Word? A Control Theory of LLM Prompting (2310.04444v4)

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 LLM system capabilities.

A Control Theory of LLM Prompting

This paper, "What's the Magic Word? A Control Theory of LLM Prompting," by Bhargava et al., presents a formal and empirical examination of prompt engineering for LLMs through the lens of control theory. The authors aim to demystify the mechanisms of prompt optimization by framing LLM systems as discrete stochastic dynamical systems. This paper is particularly relevant for researchers looking to enhance the efficiency and effectiveness of LLMs in generating desired output sequences.

Introduction and Problem Statement

Prompt engineering, or the optimization of input sequences (prompts) to influence the text generation behavior of LLMs, is pivotal yet lacks a rigorous mathematical foundation. Bhargava et al. approach this problem by representing LLMs as discrete dynamical systems and applying control theory concepts such as reachability and controllability. The authors seek to elucidate the conditions under which specific output sequences can be reached from given inputs, thus providing a structured method for prompt optimization.

Methodological Framework

The paper's analytical section focuses on the self-attention mechanism, a critical component of transformer-based LLMs. By modeling it within a control theoretic framework, the authors derive an upper bound on the reachable output set as a function of the singular values of parameter matrices involved in self-attention. This bound provides a necessary condition for understanding the limitations of LLM controllability.

Empirically, the authors evaluate the reachability and controllability of several LLMs, including Falcon-7b, Llama-7b, and Falcon-40b, by using prompt optimization techniques. They apply prompt optimization algorithms like Greedy Back-Generation and Greedy Coordinate Gradient (GCG) to assess the ability to steer LLMs towards desired outputs. These methods help in establishing both upper and lower bounds on the reachable sets for various LLMs.

Key Findings

1. Reachable Set Bound for Self-Attention: The paper analytically proves that the reachable set of self-attention outputs is constrained by the singular values of its parameter matrices. This result provides a theoretical upper bound, indicating the inherent limits of self-attention-based transformation within LLMs.
2. Empirical Analysis of Reachable Outputs: The experimental results demonstrate that the correct next token in the Wikitext dataset is reachable over 97% of the time with a prompt length of ten tokens or fewer. Moreover, the top 75 likely next tokens are reachable at least 85% of the time with prompts of similar length.
3. Uniform Token Reachability: The paper also explores the reachability of less likely tokens. It finds that even the least likely tokens can be made highly probable with controlled prompt sequences, implying a significant impact of prompt length and content on output probabilities.

Implications

The theoretical results provide a deeper understanding of the limitations and potential of prompt optimization. By bounding the reachable set, the authors lay the groundwork for future studies on the computational cost and practical limitations of prompt-based control in LLMs.

The empirical findings suggest that prompt length is a critical factor in ensuring the desired output, tying back to the practical aspects of deploying LLMs. It highlights the need for efficiently computed, optimally structured prompts as a cost-effective method for enhancing LLM performance.

Future Directions

The research opens several avenues for future studies:

• Chain-of-Thought Mechanisms: Extending the analysis to understand the control properties of chain-of-thought prompts could yield insights into more complex, multi-step prompting scenarios.
• Distributional Control: Investigating how the distribution of generated outputs can be manipulated to match desired distributions could significantly enhance LLM applications.
• Learnability and Composability: Exploring how LLMs can learn to control other models or themselves through structured prompts may lead to the development of more sophisticated, self-improving LLM systems.

Conclusion

This paper contributes significantly to the theoretical understanding and practical application of prompt engineering in LLMs. By framing prompt optimization within control theory, Bhargava et al. provide both analytical and empirical insights that could shape future research and development in this field. The findings underscore the profound influence of input sequences on LLM outputs, paving the way for more controlled, efficient, and effective use of LLMs.