Topological Properties of the Set of Functions Generated by Neural Networks of Fixed Size

Author:

Petersen Philipp,Raslan Mones,Voigtlaender Felix

Abstract

AbstractWe analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to $$L^p$$ L p -norms, $$0< p < \infty $$ 0 < p < , for all practically used activation functions, and also not closed with respect to the $$L^\infty $$ L -norm for all practically used activation functions except for the ReLU and the parametric ReLU. Finally, the function that maps a family of weights to the function computed by the associated network is not inverse stable for every practically used activation function. In other words, if $$f_1, f_2$$ f 1 , f 2 are two functions realized by neural networks and if $$f_1, f_2$$ f 1 , f 2 are close in the sense that $$\Vert f_1 - f_2\Vert _{L^\infty } \le \varepsilon $$ f 1 - f 2 L ε for $$\varepsilon > 0$$ ε > 0 , it is, regardless of the size of $$\varepsilon $$ ε , usually not possible to find weights $$w_1, w_2$$ w 1 , w 2 close together such that each $$f_i$$ f i is realized by a neural network with weights $$w_i$$ w i . Overall, our findings identify potential causes for issues in the training procedure of deep learning such as no guaranteed convergence, explosion of parameters, and slow convergence.

Funder

University of Vienna

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Computational Theory and Mathematics,Computational Mathematics,Analysis

Reference73 articles.

1. Z. Allen-Zhu, Y. Li, and Z. Song, A Convergence Theory for Deep Learning via Over-Parameterization, Proceedings of the 36th International Conference on Machine Learning, 2019, pp. 242–252.

2. H. Amann and J. Escher, Analysis III, Birkhäuser Verlag, Basel, 2009.

3. P. M. Anselone and J. Korevaar, Translation Invariant Subspaces of Finite Dimension, Proc. Amer. Math. Soc. 15 (1964), 747–752.

4. M. Anthony and P. L. Bartlett, Neural Network Learning: Theoretical Foundations, Cambridge University Press, Cambridge, 1999.

5. F. Bach, Breaking the Curse of Dimensionality with Convex Neural Networks, J. Mach. Learn. Res. 18 (2017), no. 1, 629–681.

Cited by 28 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3