@inproceedings{4813961d6898491195ac8647c20d3b76,
title = "Minimax lower bounds for ridge combinations including neural nets",
abstract = "Estimation of functions of d variables is considered using ridge combinations of the form Σmk=1 c1, kφ(Σdj=1c0, j, kxj-bk) where the activation function φ is a function with bounded value and derivative. These include single-hidden layer neural networks, polynomials, and sinusoidal models. From a sample of size n of possibly noisy values at random sites X B = [-1, 1]d, the minimax mean square error is examined for functions in the closure of the ℓ1 hull of ridge functions with activation φ. It is shown to be of order d/n to a fractional power (when d is of smaller order than n), and to be of order (log d)/n to a fractional power (when d is of larger order than n). Dependence on constraints v0 and v1 on the ℓ1 norms of inner parameter co and outer parameter c1, respectively, is also examined. Also, lower and upper bounds on the fractional power are given. The heart of the analysis is development of information-theoretic packing numbers for these classes of functions.",
keywords = "Constant weight codes, Generalization error, Greedy algorithms, High-dimensional data analysis, Learning theory, Machine learning, Metric entropy, Neural nets, Nonlinear regression, Nonparametric regression, Packing sets, Penalization, Polynomial nets, Sinusoidal nets",
author = "Klusowski, {Jason M.} and Barron, {Andrew R.}",
note = "Publisher Copyright: {\textcopyright} 2017 IEEE.; 2017 IEEE International Symposium on Information Theory, ISIT 2017 ; Conference date: 25-06-2017 Through 30-06-2017",
year = "2017",
month = aug,
day = "9",
doi = "10.1109/ISIT.2017.8006754",
language = "English (US)",
series = "IEEE International Symposium on Information Theory - Proceedings",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "1376--1380",
booktitle = "2017 IEEE International Symposium on Information Theory, ISIT 2017",
address = "United States",
}