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LLMs: Storage vs. Knowledge - Compression in High-Dimensional Space

The Paradox of Model Size vs. Training Data​

Large Language Models (LLMs) like ChatGPT present an interesting paradox: they are trained on hundreds of gigabytes (or even petabytes) of text data, yet the resulting models are significantly smaller than their training corpus. GPT-4, for instance, despite being trained on vast internet-scale data, has a model size measured in the tens of gigabytes. This compression ratio raises a fascinating question: how do these models store so much "knowledge" in a relatively compact form?

High-Dimensional Vector Spaces: The Key to Compression​

The answer lies in how neural networks leverage high-dimensional vector spaces to encode information. Rather than memorizing training data verbatim (which would require enormous storage), LLMs learn to represent patterns, relationships, and abstractions in a compressed form.

How Dimensional Compression Works​

  1. Pattern Recognition Over Memorization: LLMs don't store individual facts or sentences. Instead, they encode statistical patterns and relationships between concepts in their parameters.

  2. Distributed Representations: Information about concepts isn't stored in single neurons but distributed across thousands or millions of parameters, allowing for efficient encoding of overlapping concepts.

  3. Parameter Sharing: The same neural network parameters are reused across different contexts and tasks, dramatically increasing efficiency.

  4. Embeddings as Compression: Words and concepts are represented as dense vectors (embeddings) in high-dimensional space, where semantic relationships are encoded as geometric relationships. Similar concepts cluster together in this space.

The Inevitable Tradeoff: Accuracy vs. Compression​

This compression comes at a cost: perfect accuracy and recall are sacrificed. Here's why:

  1. Lossy Compression: Like JPEG images that sacrifice some detail for smaller file sizes, LLMs perform a kind of "lossy compression" on their training data.

  2. Statistical Approximation: The model learns to approximate the statistical distribution of language rather than memorizing exact examples.

  3. Hallucinations and Errors: When the model needs to reconstruct specific details from its compressed representations, it sometimes generates plausible-sounding but incorrect information (hallucinations).

  4. Generalization vs. Memorization: The very ability that allows LLMs to generalize to new situations (rather than just regurgitating training data) is what causes them to sometimes "fill in gaps" with incorrect information.

A Human Analogy​

This process is somewhat analogous to human learning. We don't memorize textbooks verbatim; we extract concepts, patterns, and relationships. When recalling information, we often reconstruct details (sometimes incorrectly) rather than retrieving exact memories.

Consider how you might remember the plot of a novel you read years ago. You'll recall the main themes, characters, and significant events, but might misremember or completely forget specific details. Your brain has compressed the novel into its essential elements at the cost of perfect recall.

Knowledge Distillation and Model Compression​

The AI community has further developed techniques like knowledge distillation, where a smaller "student" model learns to mimic a larger "teacher" model. This process demonstrates that the essential patterns captured by large models can often be further compressed into even smaller models with only modest performance degradation.

Conclusion: The Miracle of Compression​

The ability of LLMs to compress vast amounts of training data into relatively small models through high-dimensional representations is both their greatest strength and the source of their limitations. They sacrifice perfect accuracy for the ability to generalize, compress, and create.

This tradeoff isn't a bug—it's a feature that mirrors how human cognition works. We don't have perfect recall either, but we excel at extracting patterns and generating new ideas from compressed knowledge. LLMs, in their imperfect compression of the world's text, are following a similar path.