LLM Control Theory Seminar (April 2024)

The Nugget

The seminar introduces a novel approach to LLM (large language models) control theory, positing that understanding and applying control theory concepts to LLMs can significantly enhance their application and manipulation for specific tasks, such as prompt optimization, for both academic and real-world scenarios.

Key quotes

"LLMs are increasingly being used as systems so even though we classically view them as these probability models on natural language, the reality of how people are using these both with chat interfaces and as components in larger software systems... the way they're using them is more so as a component that can take in some you know control input."
"One of the motivating examples of this was let's say that you have some initial state sequence of tokens that's x0... your goal as the you know controller of this system is to pick some prompt tokens here this u thing so that it outputs your desired y value over there."
"...we don't really understand LLMs very well as systems...even this like very simple problem statement...is pretty mysterious and it seems worth, you know, at least doing some experiments and maybe thinking about a bit more deeply..."
"Control theory I think is a really great way to understand systems where...from this problem statement where you're trying to perform control on a system like this you get all this rich mathematical theory that turns out to be super useful for engineering practical systems..."
"We're working on this KEpsilon controllability thing so if you recall our idea of what you know this control thing is we're interested in, if we're going to practically measure this we basically going to have a dataset of these X zeros and y's..."

The project introduces a control theoretic framework for understanding and manipulating LLMs better by treating them as systems that can take control inputs (prompt tokens) and yield desired outputs.
Key aspects include defining a vocabulary set, probability distribution P Theta that maps sequences of input tokens to a probability of the next token, and establishing reachability and control as foundational concepts.

The seminar elaborates on the notions of reachability and controllability, illustrating how specific input tokens can steer LLMs to produce desired output tokens, under the constraints of given initial states and a limit on control token sequence length.
Controllability is measured via KEpsilon controllability, which assesses the fraction of a dataset's output that is reachable within a specific token sequence length, thereby quantifying how readily an LLM can achieve specific outputs given a limited "budget" of control tokens.

The seminar presents a theorem on self-attention controllability, proving the ability to control token predictions by manipulating the attention mechanism in LLMs, particularly focusing on the relationship between control inputs and the desired outputs within a Transformer model's architecture.

Experiments on 7 billion and 40 billion parameter models demonstrated varying degrees of controllability on tasks such as generating true next tokens from Wikitext, ranking top 75 chosen tokens, and creating outputs from randomly selected tokens, thereby empirically measuring LLMs' responsiveness to control inputs.
Findings indicate that with an increase in control token length, the models showed a higher success rate in reaching the desired output, with factors such as the initial state sequence's length and the 'randomness' of the desired output affecting the controllability.

The seminar concludes with highlighting open questions and future research directions such as distributional control (beyond zero-temperature sampling), the use of control theory to improve understanding of LLMs' complex behavior, and exploring the control properties of modern LLM techniques like Chain of Thought and prompt engineering.

📘 LLM control theory treats language models as systems that can be directed towards specific outputs through controlled input prompts, opening new avenues for manipulation and application.
📐 The KEpsilon controllability metric quantitatively measures an LLM's ability to reach desired outputs, providing insights into its flexibility and efficiency under limited input conditions.
🔍 The self-attention controllability theorem reveals the power of attention mechanisms in guiding LLM predictions, stressing the importance of attention in achieving control over output generation.
❓ The exploration of open questions in LLM control theory, such as distributional control and the impact of Chain of Thought on model controllability, signifies the field's nascent state and vast potential for innovation.

This summary contains AI-generated information and may have important inaccuracies or omissions.