We investigate the statistical performance of ``minimum norm'' interpolators in non-linear regression under additive Gaussian noise. Specifically, we focus on norms that satisfy either 2-uniform convexity or the cotype 2 property -- these include inner-product spaces, \(\ell_{p}\) norms, and \(W_{p}\) Sobolev spaces when \(1 \leq p \leq 2\). Our approach leverages tools from the local theory of finite dimensional Banach spaces, and, to the best of our knowledge, it is the first to study non-linear models that are ``far'' from Hilbert spaces. As an application of our approach, we prove optimal bounds for \(1 \leq p \leq 2\) over-parametrized linear regression over sub-Gaussian covariates, and according to our knowledge, it is the first work that goes beyond Gaussian covariates.

This work is based on a joint work with Pedro Abdalla, Pierre Bizuel, and Fanny Yang.