Theoretical issues in deep networks

Proceedings of the National Academy of Sciences of the United States of America
T PoggioQianli Liao

Abstract

While deep learning is successful in a number of applications, it is not yet well understood theoretically. A theoretical characterization of deep learning should answer questions about their approximation power, the dynamics of optimization, and good out-of-sample performance, despite overparameterization and the absence of explicit regularization. We review our recent results toward this goal. In approximation theory both shallow and deep networks are known to approximate any continuous functions at an exponential cost. However, we proved that for certain types of compositional functions, deep networks of the convolutional type (even without weight sharing) can avoid the curse of dimensionality. In characterizing minimization of the empirical exponential loss we consider the gradient flow of the weight directions rather than the weights themselves, since the relevant function underlying classification corresponds to normalized networks. The dynamics of normalized weights turn out to be equivalent to those of the constrained problem of minimizing the loss subject to a unit norm constraint. In particular, the dynamics of typical gradient descent have the same critical points as the constrained problem. Thus there is implicit re...Continue Reading

References

Aug 2, 2017·Neural Networks : the Official Journal of the International Neural Network Society·Dmitry Yarotsky
Sep 25, 2018·Neural Networks : the Official Journal of the International Neural Network Society·Philipp Petersen, Felix Voigtlaender
Feb 26, 2020·Nature Communications·Tomaso PoggioAndrzej Banburski

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Citations

Nov 25, 2020·Proceedings of the National Academy of Sciences of the United States of America·Richard BaraniukMatan Gavish
Jan 30, 2020·Proceedings of the National Academy of Sciences of the United States of America·Terrence J Sejnowski
May 20, 2021·Physical Review. E·Daniele Musso
Aug 12, 2021·Journal of Chemical Information and Modeling·Junhui LuMinghui Yang
Aug 5, 2021·Annual Review of Vision Science·Alice J O'Toole, Carlos D Castillo
Oct 18, 2021·Neuroscience·Jyotibdha AcharyaXundong Wu
Jan 28, 2022·Nature·Logan G WrightPeter L McMahon

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