From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformers
Muhammed Emrullah Ildiz, Yixiao Huang, Yingcong Li, Ankit Singh Rawat, Samet Oymak
Modern language models rely on the transformer architecture and attention mechanism to perform language understanding and text generation. In this work, we study learning a 1-layer self-attention model from a set of prompts and the associated outputs sampled from the model. We first establish a formal link between the self-attention mechanism and Markov models under suitable conditions: Inputting a prompt to the self-attention model samples the output token according to a context-conditioned Markov chain (CCMC). CCMC is obtained by weighing the transition matrix of a standard Markov chain according to the sufficient statistics of the prompt/context. Building on this formalism, we develop identifiability/coverage conditions for the data distribution that guarantee consistent estimation of the latent model under a teacher-student setting and establish sample complexity guarantees under IID data. Finally, we study the problem of learning from a single output trajectory generated in response to an initial prompt. We characterize a winner-takes-all phenomenon where the generative process of self-attention evolves to sampling from a small set of winner tokens that dominate the context window. This provides a mathematical explanation to the tendency of modern LLMs to generate repetitive text.
