ABSTRACT

Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is necessary if one wants to interpolate the data smoothly. Namely we show that smooth interpolation requires d times more parameters than mere interpolation, where d is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry. In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

BIOGRAPHY

Mark Sellke is a PhD student in mathematics at Stanford advised by Andrea Montanari and Sbastien Bubeck. He graduated from MIT in 2017 and received a Master of Advanced Study with distinction from the University of Cambridge in 2018, both in mathematics. He has broad research interests in probability, statistics, optimization, and machine learning and received the best paper and best student paper awards at SODA 2020 as well as the outstanding paper award at NeurIPS 2021. Mark is supported by graduate research fellowships from the NSF and Stanford. During his early years, Mark grew up in West Lafayette and attended Harrison High School.