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    - year: 2018
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      day: 11
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    - family: Li
      given: Yuanzhi
    - family: Song
      given: Zhao
  citation-key: Allen-ZhuConvergence2018
  issued:
    - year: 2018
      month: 10
      day: 29
  language: en
  title: On the Convergence Rate of Training Recurrent Neural Networks
  type: article-journal
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- id: AllenZhuConvergence2018
  accessed:
    - year: 2018
      month: 12
      day: 11
  author:
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      given: Zeyuan
    - family: Li
      given: Yuanzhi
    - family: Song
      given: Zhao
  citation-key: AllenZhuConvergence2018
  issued:
    - year: 2018
      month: 11
      day: 9
  language: en
  title: A Convergence Theory for Deep Learning via Over-Parameterization
  type: article-journal
  URL: https://arxiv.org/abs/1811.03962

- id: AndersonHighDimensional2017
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      given: Alexander G.
    - family: Berg
      given: Cory P.
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      month: 5
      day: 19
  title: The High-Dimensional Geometry of Binary Neural Networks
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- id: AroraOptimization2018
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    - year: 2018
      month: 3
      day: 21
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      month: 2
      day: 18
  title: >-
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- id: BaldassiUnreasonable2016
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  citation-key: BaldassiUnreasonable2016
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      day: 29
  language: en
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- id: BarronUniversal1993
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      day: 20
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  container-title: IEEE Transactions on Information Theory
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- id: BartlettDeep2021
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    - family: Montanari
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    - family: Rakhlin
      given: Alexander
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  issued:
    - year: 2021
      month: 5
  language: en
  page: 87-201
  publisher: Cambridge University Press
  title: 'Deep learning: a statistical viewpoint'
  type: article-journal
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- id: BelilovskyGreedy2019
  accessed:
    - year: 2020
      month: 9
      day: 15
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      given: Eugene
    - family: Eickenberg
      given: Michael
    - family: Oyallon
      given: Edouard
  citation-key: BelilovskyGreedy2019
  container-title: International Conference on Machine Learning
  event-title: International Conference on Machine Learning
  ISSN: 2640-3498
  issued:
    - year: 2019
      month: 5
      day: 24
  language: en
  page: 583-593
  publisher: PMLR
  title: Greedy Layerwise Learning Can Scale To ImageNet
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- id: BelkinFit2021
  accessed:
    - year: 2022
      month: 8
      day: 18
  author:
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      given: Mikhail
  citation-key: BelkinFit2021
  container-title: Acta Numerica
  DOI: 10.1017/S0962492921000039
  ISSN: 0962-4929, 1474-0508
  issued:
    - year: 2021
      month: 5
  language: en
  page: 203-248
  publisher: Cambridge University Press
  title: >-
    Fit without fear: remarkable mathematical phenomena of deep learning through
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  type: article-journal
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  volume: '30'

- id: BernerModern2021
  accessed:
    - year: 2023
      month: 9
      day: 7
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      given: Julius
    - family: Grohs
      given: Philipp
    - family: Kutyniok
      given: Gitta
    - family: Petersen
      given: Philipp
  citation-key: BernerModern2021
  DOI: 10.1017/9781009025096.002
  issued:
    - year: 2021
      month: 5
      day: 9
  language: en
  title: The Modern Mathematics of Deep Learning
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- id: BolcskeiOptimal2019
  accessed:
    - year: 2023
      month: 9
      day: 6
  author:
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      given: Helmut
    - family: Grohs
      given: Philipp
    - family: Kutyniok
      given: Gitta
    - family: Petersen
      given: Philipp
  citation-key: BolcskeiOptimal2019
  container-title: SIAM Journal on Mathematics of Data Science
  DOI: 10.1137/18M118709X
  issue: '1'
  issued:
    - year: 2019
      month: 1
  page: 8-45
  publisher: Society for Industrial and Applied Mathematics
  title: Optimal Approximation with Sparsely Connected Deep Neural Networks
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  volume: '1'

- id: ChangReversible2018
  accessed:
    - year: 2018
      month: 4
      day: 23
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    - family: Meng
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    - family: Haber
      given: Eldad
    - family: Ruthotto
      given: Lars
    - family: Begert
      given: David
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      given: Elliot
  citation-key: ChangReversible2018
  container-title: arXiv:1709.03698 [cs, stat]
  event-title: AAAI
  issued:
    - year: 2018
  title: Reversible Architectures for Arbitrarily Deep Residual Neural Networks
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- id: ChoromanskaLoss2015
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    - year: 2015
      month: 6
      day: 17
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      given: Anna
    - family: Henaff
      given: MIkael
    - family: Mathieu
      given: Michael
    - family: Ben Arous
      given: Gerard
    - family: LeCun
      given: Yann
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    Proceedings of the Eighteenth International Conference on Artificial
    Intelligence and Statistics
  event-title: Artificial Intelligence and Statistics
  issued:
    - year: 2015
  page: 192–204
  title: The Loss Surfaces of Multilayer Networks
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- id: ChouRobust2023
  accessed:
    - year: 2023
      month: 6
      day: 11
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      given: Hung-Hsu
    - family: Rauhut
      given: Holger
    - family: Ward
      given: Rachel
  citation-key: ChouRobust2023
  DOI: 10.48550/arXiv.2305.05448
  issued:
    - year: 2023
      month: 5
      day: 9
  number: arXiv:2305.05448
  publisher: arXiv
  title: Robust Implicit Regularization via Weight Normalization
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  URL: http://arxiv.org/abs/2305.05448

- id: DalalyanFurther2017
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    - year: 2017
      month: 6
      day: 13
  author:
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      given: Arnak S.
  citation-key: DalalyanFurther2017
  container-title: arXiv:1704.04752 [math, stat]
  issued:
    - year: 2017
      month: 4
      day: 16
  title: >-
    Further and stronger analogy between sampling and optimization: Langevin
    Monte Carlo and gradient descent
  type: article-journal
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- id: DauphinIdentifying2014
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      given: Yann
    - family: Pascanu
      given: Razvan
    - family: Gulcehre
      given: Caglar
    - family: Cho
      given: Kyunghyun
    - family: Ganguli
      given: Surya
    - family: Bengio
      given: Yoshua
  citation-key: DauphinIdentifying2014
  container-title: Advances in Neural Information Processing Systems 27
  issued:
    - year: 2014
      month: 6
      day: 10
  page: 2933–2941
  publisher: Curran Associates, Inc.
  title: >-
    Identifying and attacking the saddle point problem in high-dimensional
    non-convex optimization
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  URL: http://arxiv.org/abs/1406.2572

- id: DomingosEvery2020
  accessed:
    - year: 2020
      month: 12
      day: 12
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      given: Pedro
  citation-key: DomingosEvery2020
  container-title: arXiv:2012.00152 [cs, stat]
  issued:
    - year: 2020
      month: 11
      day: 30
  title: Every Model Learned by Gradient Descent Is Approximately a Kernel Machine
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- id: ElbrachterDeep2021
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      given: Dennis
    - family: Perekrestenko
      given: Dmytro
    - family: Grohs
      given: Philipp
    - family: Bölcskei
      given: Helmut
  citation-key: ElbrachterDeep2021
  container-title: IEEE Transactions on Information Theory
  DOI: 10.1109/TIT.2021.3062161
  ISSN: 1557-9654
  issue: '5'
  issued:
    - year: 2021
      month: 5
  page: 2581-2623
  title: Deep Neural Network Approximation Theory
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- id: GilbertUnderstanding2017
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    - year: 2018
      month: 2
      day: 4
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    - family: Zhang
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    - family: Lee
      given: Kibok
    - family: Zhang
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    - family: Lee
      given: Honglak
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  container-title: arXiv:1705.08664 [cs, stat]
  issued:
    - year: 2017
      month: 5
      day: 24
  title: Towards Understanding the Invertibility of Convolutional Neural Networks
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  URL: http://arxiv.org/abs/1705.08664

- id: GolowichSizeIndependent2017
  accessed:
    - year: 2018
      month: 1
      day: 26
  author:
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      given: Noah
    - family: Rakhlin
      given: Alexander
    - family: Shamir
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  citation-key: GolowichSizeIndependent2017
  container-title: arXiv:1712.06541 [cs, stat]
  issued:
    - year: 2017
      month: 12
      day: 18
  title: Size-Independent Sample Complexity of Neural Networks
  type: article-journal
  URL: http://arxiv.org/abs/1712.06541

- id: HaberLearning2017
  accessed:
    - year: 2018
      month: 4
      day: 23
  author:
    - family: Haber
      given: Eldad
    - family: Ruthotto
      given: Lars
    - family: Holtham
      given: Elliot
    - family: Jun
      given: Seong-Hwan
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  container-title: arXiv:1703.02009 [cs]
  issued:
    - year: 2017
      month: 3
      day: 6
  title: Learning across scales - A multiscale method for Convolution Neural Networks
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  URL: http://arxiv.org/abs/1703.02009

- id: HaberStable2018
  author:
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      given: Eldad
    - family: Ruthotto
      given: Lars
  citation-key: HaberStable2018
  container-title: Inverse Problems
  container-title-short: Inverse Problems
  DOI: 10.1088/1361-6420/aa9a90
  ISSN: 0266-5611
  issue: '1'
  issued:
    - year: 2018
  language: en
  page: '014004'
  title: Stable architectures for deep neural networks
  type: article-journal
  URL: http://arxiv.org/abs/1705.03341
  volume: '34'

- id: HalkoFinding2010
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      month: 10
      day: 22
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      given: Nathan
    - family: Martinsson
      given: Per-Gunnar
    - family: Tropp
      given: Joel A.
  citation-key: HalkoFinding2010
  DOI: 10.48550/arXiv.0909.4061
  issued:
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      month: 12
      day: 14
  number: arXiv:0909.4061
  publisher: arXiv
  title: >-
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  URL: http://arxiv.org/abs/0909.4061

- id: HeRecent2021
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      day: 15
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      given: Fengxiang
    - family: Tao
      given: Dacheng
  citation-key: HeRecent2021
  DOI: 10.48550/arXiv.2012.10931
  issued:
    - year: 2021
      month: 3
      day: 11
  number: arXiv:2012.10931
  publisher: arXiv
  title: Recent advances in deep learning theory
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  URL: http://arxiv.org/abs/2012.10931

- id: HuTraining2022
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      month: 8
      day: 11
  author:
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      given: Hang
    - family: Song
      given: Zhao
    - family: Weinstein
      given: Omri
    - family: Zhuo
      given: Danyang
  citation-key: HuTraining2022
  DOI: 10.48550/arXiv.2208.04508
  issued:
    - year: 2022
      month: 8
      day: 8
  number: arXiv:2208.04508
  publisher: arXiv
  title: Training Overparametrized Neural Networks in Sublinear Time
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  URL: http://arxiv.org/abs/2208.04508

- id: ImEmpirical2016
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      given: Daniel Jiwoong
    - family: Tao
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    - family: Branson
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  container-title: arXiv:1612.04010 [cs]
  issued:
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      month: 12
      day: 12
  title: An empirical analysis of the optimization of deep network loss surfaces
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- id: JentzenMathematical2023
  accessed:
    - year: 2023
      month: 11
      day: 15
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      given: Arnulf
    - family: Kuckuck
      given: Benno
    - family: Wurstemberger
      given: Philippe
      non-dropping-particle: von
  citation-key: JentzenMathematical2023
  DOI: 10.48550/arXiv.2310.20360
  issued:
    - year: 2023
      month: 10
      day: 31
  number: arXiv:2310.20360
  publisher: arXiv
  title: >-
    Mathematical Introduction to Deep Learning: Methods, Implementations, and
    Theory
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  URL: http://arxiv.org/abs/2310.20360

- id: JinHow2017
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      given: Chi
    - family: Ge
      given: Rong
    - family: Netrapalli
      given: Praneeth
    - family: Kakade
      given: Sham M.
    - family: Jordan
      given: Michael I.
  citation-key: JinHow2017
  container-title: PMLR
  event-title: International Conference on Machine Learning
  issued:
    - year: 2017
      month: 7
      day: 17
  language: en
  page: 1724-1732
  title: How to Escape Saddle Points Efficiently
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- id: KawaguchiGeneralization2017
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      month: 11
      day: 6
  author:
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      given: Kenji
    - family: Kaelbling
      given: Leslie Pack
    - family: Bengio
      given: Yoshua
  citation-key: KawaguchiGeneralization2017
  container-title: arXiv:1710.05468 [cs, stat]
  issued:
    - year: 2017
      month: 10
      day: 15
  title: Generalization in Deep Learning
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  URL: http://arxiv.org/abs/1710.05468

- id: KhanBayesian2023
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    - year: 2024
      month: 2
      day: 8
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      given: Mohammad Emtiyaz
    - family: Rue
      given: Håvard
  citation-key: KhanBayesian2023
  DOI: 10.48550/arXiv.2107.04562
  issued:
    - year: 2023
      month: 6
      day: 30
  number: arXiv:2107.04562
  publisher: arXiv
  title: The Bayesian Learning Rule
  type: article
  URL: http://arxiv.org/abs/2107.04562

- id: LeeFirstorder2017
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    - year: 2020
      month: 12
      day: 14
  author:
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      given: Jason D.
    - family: Panageas
      given: Ioannis
    - family: Piliouras
      given: Georgios
    - family: Simchowitz
      given: Max
    - family: Jordan
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    - family: Recht
      given: Benjamin
  citation-key: LeeFirstorder2017
  container-title: arXiv:1710.07406 [cs, math, stat]
  issued:
    - year: 2017
      month: 10
      day: 19
  title: First-order Methods Almost Always Avoid Saddle Points
  type: article-journal
  URL: http://arxiv.org/abs/1710.07406

- id: LeeGradient2016
  accessed:
    - year: 2020
      month: 12
      day: 14
  author:
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      given: Jason D.
    - family: Simchowitz
      given: Max
    - family: Jordan
      given: Michael I.
    - family: Recht
      given: Benjamin
  citation-key: LeeGradient2016
  container-title: arXiv:1602.04915 [cs, math, stat]
  issued:
    - year: 2016
      month: 3
      day: 4
  title: Gradient Descent Converges to Minimizers
  type: article-journal
  URL: http://arxiv.org/abs/1602.04915

- id: LevyPower2016
  author:
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      given: Kfir Y.
  citation-key: LevyPower2016
  container-title: arXiv:1611.04831 [cs, math, stat]
  issued:
    - year: 2016
      month: 11
      day: 15
  title: 'The Power of Normalization: Faster Evasion of Saddle Points'
  type: article-journal
  URL: http://arxiv.org/abs/1611.04831

- id: LinCritical2016
  author:
    - family: Lin
      given: Henry W.
    - family: Tegmark
      given: Max
  citation-key: LinCritical2016
  container-title: arXiv:1606.06737 [cond-mat]
  issued:
    - year: 2016
      month: 6
      day: 21
  title: 'Critical Behavior from Deep Dynamics: A Hidden Dimension in Natural Language'
  type: article-journal
  URL: http://arxiv.org/abs/1606.06737

- id: LinWhy2016
  accessed:
    - year: 2016
      month: 9
      day: 18
  author:
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      given: Henry W.
    - family: Tegmark
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  citation-key: LinWhy2016
  container-title: arXiv:1608.08225 [cond-mat, stat]
  issued:
    - year: 2016
      month: 8
      day: 29
  title: Why does deep and cheap learning work so well?
  type: article-journal
  URL: http://arxiv.org/abs/1608.08225

- id: LiptonStuck2016
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  citation-key: LiptonStuck2016
  container-title: arXiv:1602.07320 [cs]
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      month: 2
      day: 23
  title: Stuck in a What? Adventures in Weight Space
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- id: MandtStochastic2017
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      month: 4
      day: 3
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    - family: Hoffman
      given: Matthew D.
    - family: Blei
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  citation-key: MandtStochastic2017
  container-title: JMLR
  issued:
    - year: 2017
      month: 4
      day: 13
  title: Stochastic Gradient Descent as Approximate Bayesian Inference
  type: article-journal
  URL: http://arxiv.org/abs/1704.04289

- id: MehtaExact2014
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      day: 1
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    - family: Schwab
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  citation-key: MehtaExact2014
  DOI: 10.48550/arXiv.1410.3831
  issued:
    - year: 2014
      month: 10
      day: 14
  number: arXiv:1410.3831
  publisher: arXiv
  title: >-
    An exact mapping between the Variational Renormalization Group and Deep
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- id: NeuInformationTheoretic2021
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      day: 14
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    - family: Dziugaite
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    - family: Haghifam
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  issued:
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      month: 8
      day: 15
  title: Information-Theoretic Generalization Bounds for Stochastic Gradient Descent
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- id: OlahFeature2017
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    - year: 2017
      month: 11
      day: 26
  author:
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      given: Chris
    - family: Mordvintsev
      given: Alexander
    - family: Schubert
      given: Ludwig
  citation-key: OlahFeature2017
  container-title: Distill
  container-title-short: Distill
  DOI: 10.23915/distill.00007
  ISSN: 2476-0757
  issue: '11'
  issued:
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      month: 11
      day: 7
  page: e7
  title: Feature Visualization
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- id: PascanuSaddle2014
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  citation-key: PascanuSaddle2014
  container-title: arXiv:1405.4604 [cs]
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      day: 19
  title: On the saddle point problem for non-convex optimization
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- id: PerezDeep2016
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  citation-key: PerezDeep2016
  container-title: Medium
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  title: 'Deep Learning: The Unreasonable Effectiveness of Randomness'
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- id: PhilippGradients2017
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  container-title: arXiv:1712.05577 [cs]
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      month: 12
      day: 15
  title: Gradients explode - Deep Networks are shallow - ResNet explained
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- id: PrinceUnderstanding2022
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  citation-key: PrinceUnderstanding2022
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  title: Understanding deep learning
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- id: RobertsPrinciples2021
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      day: 24
  title: The Principles of Deep Learning Theory
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- id: RobertsSGD2021
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      month: 4
      day: 10
  number: arXiv:2104.04874
  publisher: arXiv
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- id: RobertsWhy2021
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      day: 31
  number: arXiv:2104.00008
  publisher: arXiv
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- id: RolnickPower2017
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- id: RosenfeldIntriguing2018
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  title: >-
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- id: SongComplexity2017
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- id: XieDiversity2016
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...
