Intrinsic dimensionality and generalization properties of the $\mathcal{R}$-norm inductive bias

TL;DR

This paper investigates the properties of $\\mathcal{R}$-norm minimizing interpolants in neural networks, revealing their multivariate nature and limitations in achieving optimal generalization, thus providing insights into neural network inductive biases.

Contribution

It characterizes the structural and statistical properties of $\\mathcal{R}$-norm interpolants, highlighting their multivariate complexity and limitations for optimal generalization.

Findings

Interpolants are intrinsically multivariate functions.

$\\mathcal{R}$-norm bias does not always lead to optimal generalization.

Results connect the inductive bias to practical neural network training.

Abstract

We study the structural and statistical properties of $\mathcal{R}$-norm minimizing interpolants of datasets labeled by specific target functions. The $\mathcal{R}$-norm is the basis of an inductive bias for two-layer neural networks, recently introduced to capture the functional effect of controlling the size of network weights, independently of the network width. We find that these interpolants are intrinsically multivariate functions, even when there are ridge functions that fit the data, and also that the $\mathcal{R}$-norm inductive bias is not sufficient for achieving statistically optimal generalization for certain learning problems. Altogether, these results shed new light on an inductive bias that is connected to practical neural network training.