A Rose by Any Other Name Would Smell as Sweet: Categorical Homotopy Theory for Large Language Models
This work addresses a fundamental issue in LLM consistency for natural language processing, but it is incremental as it builds on existing theoretical results without demonstrating practical gains.
The paper tackles the problem of large language models (LLMs) generating different probabilities for semantically equivalent statements, such as 'Charles Darwin wrote' and 'Charles Darwin is the author of', by introducing a categorical homotopy framework to capture weak equivalences in an LLM Markov category, but no concrete numerical results or performance improvements are reported.
Natural language is replete with superficially different statements, such as ``Charles Darwin wrote" and ``Charles Darwin is the author of", which carry the same meaning. Large language models (LLMs) should generate the same next-token probabilities in such cases, but usually do not. Empirical workarounds have been explored, such as using k-NN estimates of sentence similarity to produce smoothed estimates. In this paper, we tackle this problem more abstractly, introducing a categorical homotopy framework for LLMs. We introduce an LLM Markov category to represent probability distributions in language generated by an LLM, where the probability of a sentence, such as ``Charles Darwin wrote" is defined by an arrow in a Markov category. However, this approach runs into difficulties as language is full of equivalent rephrases, and each generates a non-isomorphic arrow in the LLM Markov category. To address this fundamental problem, we use categorical homotopy techniques to capture ``weak equivalences" in an LLM Markov category. We present a detailed overview of application of categorical homotopy to LLMs, from higher algebraic K-theory to model categories, building on powerful theoretical results developed over the past half a century.