A Study of the Mathematics of Deep Learning Article Swipe
"Deep Learning"/"Deep Neural Nets" is a technological marvel that is now increasingly deployed at the cutting-edge of artificial intelligence tasks. This dramatic success of deep learning in the last few years has been hinged on an enormous amount of heuristics and it has turned out to be a serious mathematical challenge to be able to rigorously explain them. In this thesis, submitted to the Department of Applied Mathematics and Statistics, Johns Hopkins University we take several steps towards building strong theoretical foundations for these new paradigms of deep-learning. In chapter 2 we show new circuit complexity theorems for deep neural functions and prove classification theorems about these function spaces which in turn lead to exact algorithms for empirical risk minimization for depth 2 ReLU nets. We also motivate a measure of complexity of neural functions to constructively establish the existence of high-complexity neural functions. In chapter 3 we give the first algorithm which can train a ReLU gate in the realizable setting in linear time in an almost distribution free set up. In chapter 4 we give rigorous proofs towards explaining the phenomenon of autoencoders being able to do sparse-coding. In chapter 5 we give the first-of-its-kind proofs of convergence for stochastic and deterministic versions of the widely used adaptive gradient deep-learning algorithms, RMSProp and ADAM. This chapter also includes a detailed empirical study on autoencoders of the hyper-parameter values at which modern algorithms have a significant advantage over classical acceleration based methods. In the last chapter 6 we give new and improved PAC-Bayesian bounds for the risk of stochastic neural nets. This chapter also includes an experimental investigation revealing new geometric properties of the paths in weight space that are traced out by the net during the training.
Related Topics
Concepts
Deep learning
Mathematical proof
Heuristics
Artificial intelligence
Artificial neural network
Computer science
Empirical risk minimization
Deep neural networks
Convergence (economics)
Function (biology)
Algorithm
Theoretical computer science
Machine learning
Mathematics
Economic growth
Operating system
Geometry
Biology
Economics
Evolutionary biology
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2104.14033
- https://arxiv.org/pdf/2104.14033
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4287197114
All OpenAlex metadata
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- OpenAlex ID
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https://openalex.org/W4287197114Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2104.14033Digital Object Identifier
- Title
-
A Study of the Mathematics of Deep LearningWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2021Year of publication
- Publication date
-
2021-04-28Full publication date if available
- Authors
-
Anirbit MukherjeeList of authors in order
- Landing page
-
https://arxiv.org/abs/2104.14033Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2104.14033Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2104.14033Direct OA link when available
- Concepts
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Deep learning, Mathematical proof, Heuristics, Artificial intelligence, Artificial neural network, Computer science, Empirical risk minimization, Deep neural networks, Convergence (economics), Function (biology), Algorithm, Theoretical computer science, Machine learning, Mathematics, Economic growth, Operating system, Geometry, Biology, Economics, Evolutionary biologyTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
- Related works (count)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.adaptive | 209 |
| abstract_inverted_index.building | 78 |
| abstract_inverted_index.deployed | 12 |
| abstract_inverted_index.detailed | 221 |
| abstract_inverted_index.dramatic | 21 |
| abstract_inverted_index.enormous | 36 |
| abstract_inverted_index.function | 107 |
| abstract_inverted_index.gradient | 210 |
| abstract_inverted_index.improved | 252 |
| abstract_inverted_index.includes | 219, 265 |
| abstract_inverted_index.learning | 25 |
| abstract_inverted_index.methods. | 242 |
| abstract_inverted_index.motivate | 127 |
| abstract_inverted_index.rigorous | 177 |
| abstract_inverted_index.theorems | 96, 104 |
| abstract_inverted_index.versions | 204 |
| abstract_inverted_index.advantage | 237 |
| abstract_inverted_index.algorithm | 151 |
| abstract_inverted_index.challenge | 50 |
| abstract_inverted_index.classical | 239 |
| abstract_inverted_index.empirical | 117, 222 |
| abstract_inverted_index.establish | 137 |
| abstract_inverted_index.existence | 139 |
| abstract_inverted_index.functions | 100, 134 |
| abstract_inverted_index.geometric | 271 |
| abstract_inverted_index.paradigms | 85 |
| abstract_inverted_index.revealing | 269 |
| abstract_inverted_index.submitted | 61 |
| abstract_inverted_index.training. | 288 |
| abstract_inverted_index.Department | 64 |
| abstract_inverted_index.University | 72 |
| abstract_inverted_index.algorithms | 115, 233 |
| abstract_inverted_index.artificial | 17 |
| abstract_inverted_index.complexity | 95, 131 |
| abstract_inverted_index.explaining | 180 |
| abstract_inverted_index.functions. | 143 |
| abstract_inverted_index.heuristics | 39 |
| abstract_inverted_index.phenomenon | 182 |
| abstract_inverted_index.properties | 272 |
| abstract_inverted_index.realizable | 160 |
| abstract_inverted_index.rigorously | 55 |
| abstract_inverted_index.stochastic | 201, 259 |
| abstract_inverted_index.Mathematics | 67 |
| abstract_inverted_index.Statistics, | 69 |
| abstract_inverted_index.algorithms, | 212 |
| abstract_inverted_index.convergence | 199 |
| abstract_inverted_index.foundations | 81 |
| abstract_inverted_index.significant | 236 |
| abstract_inverted_index.theoretical | 80 |
| abstract_inverted_index.PAC-Bayesian | 253 |
| abstract_inverted_index.acceleration | 240 |
| abstract_inverted_index.autoencoders | 184, 225 |
| abstract_inverted_index.cutting-edge | 15 |
| abstract_inverted_index.distribution | 168 |
| abstract_inverted_index.experimental | 267 |
| abstract_inverted_index.increasingly | 11 |
| abstract_inverted_index.intelligence | 18 |
| abstract_inverted_index.mathematical | 49 |
| abstract_inverted_index.minimization | 119 |
| abstract_inverted_index.deep-learning | 211 |
| abstract_inverted_index.deterministic | 203 |
| abstract_inverted_index.investigation | 268 |
| abstract_inverted_index.technological | 6 |
| abstract_inverted_index.classification | 103 |
| abstract_inverted_index.constructively | 136 |
| abstract_inverted_index.deep-learning. | 87 |
| abstract_inverted_index.sparse-coding. | 189 |
| abstract_inverted_index.Learning"/"Deep | 1 |
| abstract_inverted_index.high-complexity | 141 |
| abstract_inverted_index.hyper-parameter | 228 |
| abstract_inverted_index.first-of-its-kind | 196 |
| cited_by_percentile_year | |
| corresponding_author_ids | https://openalex.org/A5084835559 |
| countries_distinct_count | 0 |
| institutions_distinct_count | 1 |
| citation_normalized_percentile |