When can isotropy help adapt LLMs' next word prediction to numerical domains?

2025-05-22

AI Summary

This work addresses the reliability deficit of large language models (LLMs) in numerical forecasting tasks—particularly in critical domains such as energy, finance, and healthcare—arising from hallucination. Motivated by the lack of theoretical foundations for transferring pretrained language models to numerical modeling, we establish, for the first time, a formal connection between context embedding isotropy and softmax translation invariance. We derive gradient-based embedding constraints necessary to ensure robust numerical prediction performance and provide verifiable theoretical guarantees. Methodologically, our approach integrates log-linear modeling, structural analysis of self-attention gradients, isotropy quantification, and output-layer theoretical analysis. Empirical studies uncover systematic relationships between data characteristics, model architecture, and embedding isotropy. Our framework yields the first geometry-aware, embedding-structure-based theoretical criterion for safe adaptation of LLMs to numerically sensitive applications.

Technology Category

• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)
• Computer Vision: Large Vision Models

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Recent studies have shown that vector representations of contextual embeddings learned by pre-trained large language models (LLMs) are effective in various downstream tasks in numerical domains. Despite their significant benefits, the tendency of LLMs to hallucinate in such domains can have severe consequences in applications such as energy, nature, finance, healthcare, retail and transportation, among others. To guarantee prediction reliability and accuracy in numerical domains, it is necessary to open the black-box and provide performance guarantees through explanation. However, there is little theoretical understanding of when pre-trained language models help solve numeric downstream tasks. This paper seeks to bridge this gap by understanding when the next-word prediction capability of LLMs can be adapted to numerical domains through a novel analysis based on the concept of isotropy in the contextual embedding space. Specifically, we consider a log-linear model for LLMs in which numeric data can be predicted from its context through a network with softmax in the output layer of LLMs (i.e., language model head in self-attention). We demonstrate that, in order to achieve state-of-the-art performance in numerical domains, the hidden representations of the LLM embeddings must possess a structure that accounts for the shift-invariance of the softmax function. By formulating a gradient structure of self-attention in pre-trained models, we show how the isotropic property of LLM embeddings in contextual embedding space preserves the underlying structure of representations, thereby resolving the shift-invariance problem and providing a performance guarantee. Experiments show that different characteristics of numeric data and model architecture could have different impacts on isotropy.

Problem

• When can LLMs' next-word prediction adapt to numerical domains?
• How to ensure prediction reliability in numerical applications?
• How does isotropy in embeddings resolve shift-invariance for softmax?

Innovation

• Uses isotropy to adapt LLMs to numerical domains
• Formulates gradient structure for self-attention models
• Ensures shift-invariance in softmax-based predictions