Why Does Deep and Cheap Learning Work So Well?
© 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 inte...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Springer Nature
2021
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| Online Access: | https://hdl.handle.net/1721.1/135715 |
| _version_ | 1811079043050635264 |
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| author | Lin, Henry W Tegmark, Max Rolnick, David |
| author_facet | Lin, Henry W Tegmark, Max Rolnick, David |
| author_sort | Lin, Henry W |
| collection | MIT |
| description | © 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. |
| first_indexed | 2024-09-23T11:09:09Z |
| format | Article |
| id | mit-1721.1/135715 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:09:09Z |
| publishDate | 2021 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | mit-1721.1/1357152022-04-01T14:44:57Z Why Does Deep and Cheap Learning Work So Well? Lin, Henry W Tegmark, Max Rolnick, David © 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. 2021-10-27T20:28:58Z 2021-10-27T20:28:58Z 2017 2019-06-12T12:41:21Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/135715 en 10.1007/S10955-017-1836-5 Journal of Statistical Physics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer Nature arXiv |
| spellingShingle | Lin, Henry W Tegmark, Max Rolnick, David Why Does Deep and Cheap Learning Work So Well? |
| title | Why Does Deep and Cheap Learning Work So Well? |
| title_full | Why Does Deep and Cheap Learning Work So Well? |
| title_fullStr | Why Does Deep and Cheap Learning Work So Well? |
| title_full_unstemmed | Why Does Deep and Cheap Learning Work So Well? |
| title_short | Why Does Deep and Cheap Learning Work So Well? |
| title_sort | why does deep and cheap learning work so well |
| url | https://hdl.handle.net/1721.1/135715 |
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