From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformers

• URL: http://arxiv.org/abs/2402.13512v1
• Date: Wed, 21 Feb 2024 03:51:34 GMT
• Title: From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformers
• Authors: M. Emrullah Ildiz, Yixiao Huang, Yingcong Li, Ankit Singh Rawat and Samet Oymak
• Abstract summary: We study learning a 1-layer self-attention model from a set of prompts and associated output data. We first establish a precise mapping between the self-attention mechanism and Markov models. We characterize an intriguing winner-takes-all phenomenon where the generative process implemented by self-attention collapses into sampling a limited subset of tokens.
• Score: 41.82477691012942
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 associated output data sampled from the model. We first establish a precise mapping between the self-attention mechanism and Markov models: Inputting a prompt to the model samples the output token according to a context-conditioned Markov chain (CCMC) which weights the transition matrix of a base Markov chain. Additionally, incorporating positional encoding results in position-dependent scaling of the transition probabilities. Building on this formalism, we develop identifiability/coverage conditions for the prompt distribution that guarantee consistent estimation and establish sample complexity guarantees under IID samples. Finally, we study the problem of learning from a single output trajectory generated from an initial prompt. We characterize an intriguing winner-takes-all phenomenon where the generative process implemented by self-attention collapses into sampling a limited subset of tokens due to its non-mixing nature. This provides a mathematical explanation to the tendency of modern LLMs to generate repetitive text. In summary, the equivalence to CCMC provides a simple but powerful framework to study self-attention and its properties.

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