Why Does Deep and Cheap Learning Work So Well?
Name
1608.08225.pdf
Description
Accepted version
Size
2.15 MB
Format
Unknown
Checksum (MD5)
5617bf50efc883546c662572fdd86e53
Author(s)
Lin, Henry W.
Tegmark, Max Erik
Rolnick, David S.
Date Issued
2017
Journal
Journal of Statistical Physics
Publisher
Springer Nature
Version
Author's final manuscript
Abstract
© 2017, Springer Science+Business Media, LLC. We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of practical interest can frequently be approximated through “cheap learning” with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We prove various “no-flattening theorems” showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss; for example, we show that n variables cannot be multiplied using fewer than 2 n neurons in a single hidden layer.
MIT Department
Massachusetts Institute of Technology. Department of Physics
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/S10955-017-1836-5
