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

TL;DR

This paper investigates when large language models can be effectively adapted for numerical tasks by analyzing the isotropy of their embedding space, providing theoretical insights and experimental validation.

Contribution

It introduces a novel isotropy-based analysis to determine conditions under which LLMs can reliably predict numerical data, bridging a gap in theoretical understanding.

Findings

Isotropic embeddings preserve structure and improve numerical prediction.

Model architecture and data characteristics influence isotropy and performance.

Theoretical guarantees depend on the shift-invariance of the softmax in LLMs.

Abstract

Vector representations of contextual embeddings learned by pre-trained large language models (LLMs) are effective in various downstream tasks in numerical domains such as time series forecasting. 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 behind the LLM and provide performance guarantees through explanation. However, there is little theoretical understanding of when pre-trained language models help solve numerical 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, a log-linear model for LLMs is considered in which numerical 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). For this model, it is demonstrated 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, it is shown 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 numerical data and model architectures have different impacts on isotropy, and this variability directly affects the performances.