Approximation Results for Gradient Flow Trained Neural Networks

Abstract

The paper contains approximation guarantees for neural networks that are trained with gradient flow, with error measured in the continuous L 2 (S d−1 )-norm on the d-dimensional unit sphere and targets that are Sobolev smooth. The networks are fully connected of constant depth and increasing width. We show gradient flow convergence based on a neural tangent kernel (NTK) argument for the non-convex optimization of the second but last layer. Unlike standard NTK analysis, the continuous error norm implies an underparametrized regime, possible by the natural smoothness assumption required for approximation. The typical over-parametrization re-enters the results in form of a loss in approximation rate relative to established approximation methods for Sobolev smooth functions.