Large Language Models as Markov Chains
Oussama Zekri, Ambroise Odonnat, Abdelhakim Benechehab, Linus Bleistein, Nicolas Boullé, Ievgen Redko
2024-10-04
Summary
This paper discusses how large language models (LLMs) can be understood as Markov chains, which are mathematical models that predict future events based on current states.
What's the problem?
While LLMs have shown impressive performance in generating text and completing various language tasks, there hasn't been a clear theoretical understanding of why they work so well. This lack of understanding makes it difficult to improve these models or explain their behavior in a meaningful way.
What's the solution?
The authors propose that we can view LLMs as Markov chains, which simplifies the analysis of how they generate text. They show that by treating the words and sequences generated by LLMs as states in a Markov chain, we can derive important insights about their performance. They analyze how these chains behave, including how quickly they reach a stable state (called a stationary distribution) and how factors like temperature affect their predictions. They also prove bounds on how well LLMs can generalize from training data to new situations, providing a clearer framework for understanding their capabilities.
Why it matters?
This research is significant because it bridges the gap between complex language models and simpler mathematical concepts, making it easier to analyze and improve LLMs. By understanding LLMs as Markov chains, researchers can develop better training methods and enhance the models' performance in real-world applications like chatbots, translation services, and content generation.
Abstract
Large language models (LLMs) have proven to be remarkably efficient, both across a wide range of natural language processing tasks and well beyond them. However, a comprehensive theoretical analysis of the origins of their impressive performance remains elusive. In this paper, we approach this challenging task by drawing an equivalence between generic autoregressive language models with vocabulary of size T and context window of size K and Markov chains defined on a finite state space of size O(T^K). We derive several surprising findings related to the existence of a stationary distribution of Markov chains that capture the inference power of LLMs, their speed of convergence to it, and the influence of the temperature on the latter. We then prove pre-training and in-context generalization bounds and show how the drawn equivalence allows us to enrich their interpretation. Finally, we illustrate our theoretical guarantees with experiments on several recent LLMs to highlight how they capture the behavior observed in practice.