Convergence rates for shallow neural networks learned by gradient descent

TL;DR

This paper analyzes the convergence rates of shallow neural networks trained with gradient descent, showing that proper initialization and weight adjustment lead to optimal $1/\sqrt{n}$ rates, supported by theoretical proofs and simulations.

Contribution

It provides a theoretical analysis of convergence rates for shallow neural networks with specific initialization and training methods, highlighting the effectiveness of linear least squares for outer weights.

Findings

Achieves $1/\sqrt{n}$ convergence rate under certain conditions.

Proper initialization and weight adjustment are crucial for optimal convergence.

Linear least squares can effectively replace gradient descent for outer weights.

Abstract

In this paper we analyze the $L_2$ error of neural network regression estimates with one hidden layer. Under the assumption that the Fourier transform of the regression function decays suitably fast, we show that an estimate, where all initial weights are chosen according to proper uniform distributions and where the weights are learned by gradient descent, achieves a rate of convergence of $1/\sqrt{n}$ (up to a logarithmic factor). Our statistical analysis implies that the key aspect behind this result is the proper choice of the initial inner weights and the adjustment of the outer weights via gradient descent. This indicates that we can also simply use linear least squares to choose the outer weights. We prove a corresponding theoretical result and compare our new linear least squares neural network estimate with standard neural network estimates via simulated data. Our simulations show that our theoretical considerations lead to an estimate with an improved performance in many cases.