Bounding the Width of Neural Networks via Coupled Initialization A Worst Case Analysis

Alexander Munteanu, Simon Omlor, Zhao Song, David Woodruff

Proceedings of the 39th International Conference on Machine Learning, PMLR 162:16083-16122, 2022.

Abstract

A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with logistic loss, from roughly $\gamma^{-8}$ [Ji and Telgarsky, ICLR 2020] to $\gamma^{-2}$, where $\gamma$ denotes the separation margin with a Neural Tangent Kernel, as well as in the over-parameterized setting with squared loss, from roughly $n^4$ [Song and Yang, 2019] to $n^2$, implicitly also improving the recent running time bound of [Brand, Peng, Song and Weinstein, ITCS 2021]. For the under-parameterized setting we also prove new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.

Cite this Paper

BibTeX

@InProceedings{pmlr-v162-munteanu22a,
title = {Bounding the Width of Neural Networks via Coupled Initialization A Worst Case Analysis},
author = {Munteanu, Alexander and Omlor, Simon and Song, Zhao and Woodruff, David},
booktitle = {Proceedings of the 39th International Conference on Machine Learning},
pages = {16083--16122},
year = {2022},
editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan},
volume = {162},
series = {Proceedings of Machine Learning Research},
month = {17--23 Jul},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v162/munteanu22a/munteanu22a.pdf},
url = {https://proceedings.mlr.press/v162/munteanu22a.html},
abstract = {A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with logistic loss, from roughly $\gamma^{-8}$ [Ji and Telgarsky, ICLR 2020] to $\gamma^{-2}$, where $\gamma$ denotes the separation margin with a Neural Tangent Kernel, as well as in the over-parameterized setting with squared loss, from roughly $n^4$ [Song and Yang, 2019] to $n^2$, implicitly also improving the recent running time bound of [Brand, Peng, Song and Weinstein, ITCS 2021]. For the under-parameterized setting we also prove new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.}
}

Endnote

%0 Conference Paper
%T Bounding the Width of Neural Networks via Coupled Initialization A Worst Case Analysis
%A Alexander Munteanu
%A Simon Omlor
%A Zhao Song
%A David Woodruff
%B Proceedings of the 39th International Conference on Machine Learning
%C Proceedings of Machine Learning Research
%D 2022
%E Kamalika Chaudhuri
%E Stefanie Jegelka
%E Le Song
%E Csaba Szepesvari
%E Gang Niu
%E Sivan Sabato
%F pmlr-v162-munteanu22a
%I PMLR
%P 16083--16122
%U https://proceedings.mlr.press/v162/munteanu22a.html
%V 162
%X A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with logistic loss, from roughly $\gamma^{-8}$ [Ji and Telgarsky, ICLR 2020] to $\gamma^{-2}$, where $\gamma$ denotes the separation margin with a Neural Tangent Kernel, as well as in the over-parameterized setting with squared loss, from roughly $n^4$ [Song and Yang, 2019] to $n^2$, implicitly also improving the recent running time bound of [Brand, Peng, Song and Weinstein, ITCS 2021]. For the under-parameterized setting we also prove new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.

APA

Munteanu, A., Omlor, S., Song, Z. & Woodruff, D.. (2022). Bounding the Width of Neural Networks via Coupled Initialization A Worst Case Analysis. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:16083-16122 Available from https://proceedings.mlr.press/v162/munteanu22a.html.