Nonparametric Regression with Shallow Overparameterized Neural Networks Trained by GD with Early Stopping
This addresses the inconsistency issue in nonparametric regression with noisy labels for neural networks, offering a simpler alternative to prior kernel-based methods.
The paper tackles the problem of overparameterized shallow neural networks learning Lipschitz regression functions with and without label noise when trained by Gradient Descent, proposing an early stopping rule to achieve optimal rates and avoid inconsistency in noisy cases.
We explore the ability of overparameterized shallow neural networks to learn Lipschitz regression functions with and without label noise when trained by Gradient Descent (GD). To avoid the problem that in the presence of noisy labels, neural networks trained to nearly zero training error are inconsistent on this class, we propose an early stopping rule that allows us to show optimal rates. This provides an alternative to the result of Hu et al. (2021) who studied the performance of $\ell 2$ -regularized GD for training shallow networks in nonparametric regression which fully relied on the infinite-width network (Neural Tangent Kernel (NTK)) approximation. Here we present a simpler analysis which is based on a partitioning argument of the input space (as in the case of 1-nearest-neighbor rule) coupled with the fact that trained neural networks are smooth with respect to their inputs when trained by GD. In the noise-free case the proof does not rely on any kernelization and can be regarded as a finite-width result. In the case of label noise, by slightly modifying the proof, the noise is controlled using a technique of Yao, Rosasco, and Caponnetto (2007).