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Computer Science > Machine Learning

arXiv:1711.01530 (cs)
[Submitted on 5 Nov 2017 (v1), last revised 23 Feb 2019 (this version, v2)]

Title:Fisher-Rao Metric, Geometry, and Complexity of Neural Networks

Authors:Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, James Stokes
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Abstract:We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks.
Comments: To appear in the proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS) 2019
Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML)
Cite as: arXiv:1711.01530 [cs.LG]
  (or arXiv:1711.01530v2 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.1711.01530
Journal reference: The 22nd International Conference on Artificial Intelligence and Statistics 89 (2019) 888-896

Submission history

From: James Stokes [view email]
[v1] Sun, 5 Nov 2017 04:32:59 UTC (212 KB)
[v2] Sat, 23 Feb 2019 21:27:30 UTC (196 KB)
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Tengyuan Liang
Tomaso Poggio
Tomaso A. Poggio
Alexander Rakhlin
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