MFAI: Data for and from Language Models: Information, Dynamics, Architectures, and Optimization

The impact of (large) language models (LM) has been felt across a wide range of scientific domains. It is no exaggeration to state LMs have revolutionized several applications, including both commercial and scientific discovery. Among all the factors that contributed to the success of LMs, it is empirically clear that data is one of the most important. However, this relationship remains poorly understood and motivates foundational questions, given its importance to the success of LMs. Therefore, the future progress of LMs depends critically on the data, how they are obtained, selected, and utilized; this is the focus of this Mathematical Foundations of Artificial Intelligence (MFAI) project. If successful, this project could impact next-generation LMs, which, due to their increasing applications, may have far-reaching effects Traditional thinking views data primarily as an input for Language Models (LMs) during training. However, data can also be synthesized by a model, presenting it as an output. This dual role raises fundamental questions, including the role of synthetic data, the integration of exogenous data, and the impact of data quality on LM performance. Notably, the interaction between LMs and data is crucial, especially regarding in-context learning (ICL), which allows LMs to adapt to new tasks without altering their parameters. The project focuses on two tasks. Task 1 examines the interplay between synthetic and exogenous data in training, investigating the potential adverse effects of using synthetic data. It models this relationship as a dynamical system, employing control theory to optimize the use of diverse data types. Additionally, the project addresses the need for criteria to filter high-quality data for efficient learning and information retrieval. Task 2 aims to establish a mathematical foundation for in-context learning, enhancing its capabilities. This part develops an optimization perspective on ICL for various data types and reveals underlying structures through information theory, bridging domains of control, dynamical systems, and optimization. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.