Statistical Foundations of Large Language Models

Large language models (LLMs) are massive Transformer-based neural networks, posing substantial challenges in understanding their predictive processes and internal mechanisms. This is not unexpected, given our limited understanding of even much smaller, simpler multilayer neural networks. At present, there appear to be no effective mathematical theories to help us reason about such enormous, complex, and nonlinear systems. Indeed, Stephen Wolfram has argued that reducing complex systems to smaller ones while retaining their essence (i.e., “understanding”) might, in general, be impossible. In a way, this resonates with the line Theory will only take you so far from the movie Oppenheimer.

Why statistics?

Statistics has a long history of shedding light on complex systems well before their internal mechanisms are fully understood, as evidenced by applications in astronomy and genetics. The power of statistical approaches in the context of LLMs lies in our ability to overlook highly intricate details and instead focus on quantitative patterns in specific aspects of these models. Often, as the complexity of a system grows, certain simplifications emerge.

That being said, we do not expect statistical approaches — what we term a second principle approach — to address most of the crucial challenges posed by LLMs. Nevertheless, it is a viable first step, in part because it is generally compute-light!

Our work in this direction

Since early 2023, my group has been actively pursuing this line of research, examining several facets of LLMs through a statistical lens. These include watermarks, biases in RLHF, data separation, and copyright concerns: