A Universal Law of Robustness via Isoperimetry

TL;DR

This paper establishes a universal law of robustness in data interpolation, showing that overparametrization is necessary for smooth interpolation across broad data distributions, with implications for understanding neural network generalization.

Contribution

It proves that smooth data interpolation requires significantly more parameters than mere interpolation, generalizing previous conjectures and linking robustness to isoperimetry.

Findings

Overparametrization is necessary for smooth interpolation in broad settings.

The universal law applies to various data distributions with isoperimetry.

Provides an improved generalization bound for smooth function classes.

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 partial 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.