We show 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 times more parameters than mere interpolation, where 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.

We also give an interpretation of our result as an improved generalization bound for modelclasses consisting of smooth functions.