## Aren Martinian L&S Math & Physical Sciences

### Free Probability in Infinite Depth Neural Networks

In the past few years, neural networks have gone from obscure to ubiquitous. This technology is shockingly versatile, but conceptually ill-understood: there is a large gap between practice and theory, and much has yet to even be conjectured. For example, scientists are baffled by the overfitting paradox. Overfitting is usually a problem when programmers model a complex system such as the brain. Programmers must base their model on finitely many examples of that system’s behavior. Traditionally, programs that perfectly replicate these examples forget the underlying system. Surprisingly, large neural networks do not in general suffer from this deficiency.

Recent developments suggest that free probability, traditionally used to understand large random matrices, can be used to explain the ways in which large neural networks typically behave. Our project would use free probability to explain the overfitting paradox by describing the average behavior of highly trained neural networks.

#### Message To Sponsor

Thank you so much for granting Ben and I the opportunity to conduct research on free probability and machine learning with professor Pasqualotto over the summer! This will help us spend time reading on material and answering key important questions in the field. We are extremely grateful and hope to produce fantastic results that will culminate in a research paper next semester.**Major:**Mathematics

**Mentor:**Federico Pasqualotto

**Sponsor:**Anselm MPS Fund