Forget BIT, It is All about TOKEN: Towards Semantic Information Theory
  for LLMs
Authors: name: Bo Bai

Large language models (LLMs) have demonstrated remarkable capabilities in
numerous real-world applications. While the vast majority of research conducted
from an experimental perspective is progressing rapidly, it demands substantial
computational power, data, and other resources. Therefore, how to open the
black-box of LLMs from a theoretical standpoint has become a critical
challenge. This paper takes the theory of rate-distortion function, directed
information, and Granger causality as its starting point to investigate the
information-theoretic principles behind LLMs, leading to the development of
semantic information theory for LLMs, where the fundamental unit is token,
rather than bits that lacks any semantic meaning. By defining the probabilistic
model of LLMs, we discuss structure-agnostic information-theoretic measures,
such as the directed rate-distortion function in pre-training, the directed
rate-reward function in post-training, and the semantic information flow in
inference phase. This paper also delves deeply into the theory of token-level
semantic embedding and the information-theoretically optimal vectorization
method. Thereafter, we propose a general definition of autoregression LLM,
where the Transformer architecture and its performance such as ELBO,
generalization error bound, memory capacity, and semantic information measures
can be derived theoretically. Other architectures, such as Mamba/Mamba2 and
LLaDA, are also discussed in our framework. Consequently, this paper provides a
theoretical framework for understanding LLMs from the perspective of semantic
information theory, which also offers the necessary theoretical tools for
further in-depth research.