Cross Entropy, Compression, and Language Models: From Zipped Text to Training Objectives

Short summary

This video travels from a classic gzip based method for comparing languages to the core loss function used in modern language models, cross entropy. It shows how ideas from data compression illuminate training objectives, introduces a toy encoding example, and explains distillation and KL divergence as extensions of the same mathematical intuition.

• Cross entropy as a bridge between compression and language modeling
• How next token prediction becomes a compression objective
• Distillation and KL divergence as practical training refinements
• Connection between theory of information and modern AI practice

Introduction to information theoretic ideas in language

The video begins by revisiting Language Trees and Zipping, a pioneering idea showing that simple compression techniques can uncover structure across languages and even authorship. The central thread is cross entropy, a core concept in information theory that also underpins how modern large language models are trained. The narrator emphasizes that the link between compression and language modeling is not accidental: both are governed by fundamental limits on how efficiently data can be encoded given patterns in the data.

Foundations: encoding, entropy, and cross entropy

To motivate cross entropy, the talk uses a toy encoding problem with four possible instructions: up, down, left, and right, with a biased distribution. Optimal encoding assigns a single bit to up, two bits to down, and three bits to left and right, mirroring the idea that the number of bits for a symbol is roughly the negative log base 2 of its probability. When the distribution changes but encoding remains fixed, the average bits per symbol increase. This excess is the cross entropy between the original distribution and the new one. The speaker visualizes cross entropy as a weighted area under bars, showing that the minimum occurs when the two distributions align.

Cross entropy in practice: training language models

The narrative then connects these ideas to training language models. Tokens are predicted as probability distributions over possible next tokens, and the training objective is the average negative log probability of the true next token across all tokens in the training set. This loss, widely known as cross entropy loss, is shown to be the natural choice because it is minimized when the model’s predicted distribution matches the data distribution. The role of the logarithm is highlighted, with natural log preferred for mathematical convenience in gradient-based optimization.

From next-token prediction to compression

The video stresses that the loss used to train language models can be interpreted as an average information per token from the model’s perspective. This reframing makes explicit the link between model training and the fundamental limits of compression: if a model could perfectly capture language, its per-token uncertainty and thus its loss would reflect the language’s entropy. The talk then outlines how compression algorithms can be designed to turn predictive models into compressors, aligning bit usage with information content.

Distillation and KL divergence

A practical variant discussed is distillation, where a small model learns to approximate a larger model by matching its full distribution rather than just the single correct next token. This richer signal improves learning efficiency and performance. The talk also introduces KL divergence as a genuine distance-like measure between distributions, clarifying its role as the difference between cross entropy and entropy, and poses questions about using KL divergence directly as a loss in distillation.

Takeaways and future directions

In closing, the speaker reinforces the intuition that cross entropy emerges naturally from the desire to minimize the discrepancy between a model’s outputs and the data’s statistics. The synthesis of compression and language modeling motivates reframing training objectives in terms of information content and compression, foreshadowing a deeper exploration of how to turn language models into optimal text compressors. A brief aside points to KL divergence as a useful conceptual and practical tool in model compression and distillation, hinting at broader implications for AI alignment and research directions.