Title: Barron Approximation and locally optimal weight densities for shallow neural networks
Primary area: optimization
Keywords: approximation, Barron spaces, mean field theory, optimization

Abstract: Mean field theories provide optimization results for shallow neural networks by analyzing the weight distribution in infinite width limits. Corresponding results for finite sized networks are obtained by particle approximations, for which sharp quantitative bounds are still an open problem. In this paper, we consider a modified mean field loss, which allows a more fine grained control over finite sized networks. We prove convexity and equidistribution properties, which directly lead to Barron type approximation results for networks sampled from  local loss minimizers. We demonstrate that particle approximations of the new loss function naturally lead to gradient descent methods with dropout type regularization.

TL;DR: We provide Barron type approximation bounds for networks sampled from weight distributions that are local optima of a new mean field type loss.