Reinventing Entropy | Compression & Intelligence Part 1

Summary

What if the very process of compressing text held the key to artificial intelligence? This exploration delves into the fundamental limits of text compression, tracing its roots back to Claude Shannon's groundbreaking work in information theory. It reveals a surprising connection: compression and prediction are mathematically equivalent, meaning that training large language models, often described as next token prediction, can be reframed as building the most efficient text compressor. The core idea is that the number of bits needed to represent information is directly related to its probability, with less probable, or more surprising, information requiring more bits. This principle is visualized through diagrams where binary strings are represented, and the concept of prefix-free codes ensures unambiguous decoding. The discussion introduces the idea that perfectly compressed data resembles random noise, leading to the formula for information content: the negative logarithm base two of a symbol's probability. This fundamental formula, when averaged over a distribution, defines entropy, which represents the theoretical minimum number of bits required for compression. For English text, Shannon estimated this entropy to be as low as one bit per character, a surprisingly efficient limit. The next steps will explore how this concept, particularly cross-entropy, is applied in training modern AI models.