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

TMLR Paper5168 Authors

Abstract: Vector representations of contextual embeddings learned by pre-trained large language models (LLMs) are effective in various downstream tasks in \emph{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 across $22$ different numerical datasets and $5$ different language models show that different characteristics of numerical data and model architectures could have different impacts on the isotropy measures, and this variability directly affects the time series forecasting performances.

Submission Length: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Jacek_Cyranka1

Submission Number: 5168