Stability and Generalization Analysis of Gradient Methods for Shallow Neural Networks
This provides incremental theoretical insights into the generalization mystery of overparameterized neural networks for researchers in machine learning theory.
The paper tackles the generalization behavior of shallow neural networks by analyzing gradient descent and stochastic gradient descent using algorithmic stability, resulting in consistent excess risk bounds with a relaxed overparameterization assumption compared to prior work.
While significant theoretical progress has been achieved, unveiling the generalization mystery of overparameterized neural networks still remains largely elusive. In this paper, we study the generalization behavior of shallow neural networks (SNNs) by leveraging the concept of algorithmic stability. We consider gradient descent (GD) and stochastic gradient descent (SGD) to train SNNs, for both of which we develop consistent excess risk bounds by balancing the optimization and generalization via early-stopping. As compared to existing analysis on GD, our new analysis requires a relaxed overparameterization assumption and also applies to SGD. The key for the improvement is a better estimation of the smallest eigenvalues of the Hessian matrices of the empirical risks and the loss function along the trajectories of GD and SGD by providing a refined estimation of their iterates.