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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 5

### Dedicated to the memory of Professor David Shen Ou Cai

*A priori* estimates of the population risk for two-layer neural networks

Pages: 1407 – 1425

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n5.a11

#### Authors

#### Abstract

New estimates for the population risk are established for two-layer neural networks. These estimates are nearly optimal in the sense that the error rates scale in the same way as the Monte Carlo error rates. They are equally effective in the over-parametrized regime when the network size is much larger than the size of the dataset. These new estimates are *a priori* in nature in the sense that the bounds depend only on some norms of the underlying functions to be fitted, not the parameters in the model, in contrast with most existing results which are *a posteriori* in nature. Using these *a priori* estimates, we provide a perspective for understanding why two-layer neural networks perform better than the related kernel methods.

#### Keywords

two-layer neural network, Barron space, population risk, *a priori* estimate, Rademacher complexity

#### 2010 Mathematics Subject Classification

41A46, 41A63, 62J02, 65D05

Received 28 April 2019

Accepted 25 July 2019

Published 6 December 2019