Understanding how overparameterized neural networks generalize despite perfect interpolation of noisy training data is a fundamental question. Mallinar et. al. (2022) noted that neural networks seem to often exhibit ``tempered overfitting'', wherein the population risk does not converge to the Bayes optimal error, but neither does it approach infinity, yielding non-trivial generalization. However, this has not been studied rigorously.  We provide the first rigorous analysis of the overfiting behaviour of regression with minimum norm ($\ell_2$ of weights), focusing on univariate two-layer ReLU networks.  We show overfitting is tempered (with high probability) when measured with respect to the $L_1$ loss, but also show that the situation is more complex than suggested by Mallinar et. al., and overfitting is catastrophic with respect to the $L_2$ loss, or when taking an expectation over the training set.
Noisy Interpolation Learning with Shallow Univariate ReLU Networks
Nirmit Joshi Gal Vardi Nathan Srebro