This paper explores the theoretical properties of the self-attention mechanism in transformers, linking it to Markov models. The authors establish a precise mapping between self-attention and Context-Conditioned Markov Chains (CCMC), where the transition probabilities of a base Markov chain are modified based on the observed sequence of tokens. This mapping allows for a rigorous analysis of self-attention's optimization, approximation, and generalization properties. Key contributions include: 1. **CCMC Equivalence**: The paper introduces CCMC and shows that it accurately represents the transition dynamics of self-attention under suitable conditions. This makes the optimization of self-attention weights convex, facilitating gradient descent. 2. **Consistency of Learning**: The authors study the learnability of a self-attention layer from observed outputs for a set of input prompts. They identify necessary and sufficient coverage conditions on the prompt distribution that ensure consistent estimation of the underlying model. 3. **Sample Complexity**: By integrating consistency guarantees with finite sample analysis, the authors develop generalization guarantees for learning a ground-truth self-attention model from IID pairs. They establish a fast statistical rate of $\mathcal{O}(K^2/n)$, where $K$ is the vocabulary size and $n$ is the sample size. 4. **Learning from Single Prompt Trajectory**: The paper investigates the learnability of self-attention from a single trajectory of autoregressive generation. It reveals a "winner-takes-all" phenomenon where the generative process collapses into sampling a limited subset of tokens due to its non-mixing nature, providing a mathematical explanation for repetitive text generation in modern LLMs. 5. **Role of Positional Encoding**: The authors extend their theory to incorporate positional encoding, showing that it enriches CCMC by making transition dynamics adjustable through learnable positional priors. The paper provides a powerful framework for studying self-attention and its characteristics, highlighting its non-Markovian nature and the importance of prompt coverage conditions for consistent estimation.This paper explores the theoretical properties of the self-attention mechanism in transformers, linking it to Markov models. The authors establish a precise mapping between self-attention and Context-Conditioned Markov Chains (CCMC), where the transition probabilities of a base Markov chain are modified based on the observed sequence of tokens. This mapping allows for a rigorous analysis of self-attention's optimization, approximation, and generalization properties. Key contributions include: 1. **CCMC Equivalence**: The paper introduces CCMC and shows that it accurately represents the transition dynamics of self-attention under suitable conditions. This makes the optimization of self-attention weights convex, facilitating gradient descent. 2. **Consistency of Learning**: The authors study the learnability of a self-attention layer from observed outputs for a set of input prompts. They identify necessary and sufficient coverage conditions on the prompt distribution that ensure consistent estimation of the underlying model. 3. **Sample Complexity**: By integrating consistency guarantees with finite sample analysis, the authors develop generalization guarantees for learning a ground-truth self-attention model from IID pairs. They establish a fast statistical rate of $\mathcal{O}(K^2/n)$, where $K$ is the vocabulary size and $n$ is the sample size. 4. **Learning from Single Prompt Trajectory**: The paper investigates the learnability of self-attention from a single trajectory of autoregressive generation. It reveals a "winner-takes-all" phenomenon where the generative process collapses into sampling a limited subset of tokens due to its non-mixing nature, providing a mathematical explanation for repetitive text generation in modern LLMs. 5. **Role of Positional Encoding**: The authors extend their theory to incorporate positional encoding, showing that it enriches CCMC by making transition dynamics adjustable through learnable positional priors. The paper provides a powerful framework for studying self-attention and its characteristics, highlighting its non-Markovian nature and the importance of prompt coverage conditions for consistent estimation.