---
references:
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      day: 20
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  citation-key: BachBreaking2014
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  issued:
    - year: 2014
      month: 12
      day: 30
  title: Breaking the Curse of Dimensionality with Convex Neural Networks
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- id: BahadoriNeural2017
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  container-title: arXiv:1702.02604 [cs, stat]
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    - year: 2017
      month: 2
      day: 8
  title: Neural Causal Regularization under the Independence of Mechanisms Assumption
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- id: BaLayer2016
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      day: 20
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    - family: Kiros
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  citation-key: BaLayer2016
  DOI: 10.48550/arXiv.1607.06450
  issued:
    - year: 2016
      month: 7
      day: 21
  number: arXiv:1607.06450
  publisher: arXiv
  title: Layer Normalization
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- id: BaldiLearning2016
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      given: Pierre
    - family: Sadowski
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    - family: Lu
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  issued:
    - year: 2016
      month: 12
      day: 8
  title: 'Learning in the Machine: Random Backpropagation and the Learning Channel'
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- id: BardesVICReg2022
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      month: 12
      day: 6
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    - family: Ponce
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    - family: LeCun
      given: Yann
  citation-key: BardesVICReg2022
  DOI: 10.48550/arXiv.2105.04906
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    - year: 2022
      month: 1
      day: 28
  number: arXiv:2105.04906
  publisher: arXiv
  title: >-
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  type: article
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- id: BartlettDeep2021
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      day: 18
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    - family: Montanari
      given: Andrea
    - family: Rakhlin
      given: Alexander
  citation-key: BartlettDeep2021
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  DOI: 10.1017/S0962492921000027
  ISSN: 0962-4929, 1474-0508
  issued:
    - year: 2021
      month: 5
  language: en
  page: 87-201
  publisher: Cambridge University Press
  title: 'Deep learning: a statistical viewpoint'
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- id: BaydinAutomatic2014
  author:
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      given: Atilim Gunes
    - family: Pearlmutter
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  issued:
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      month: 4
      day: 28
  title: Automatic Differentiation of Algorithms for Machine Learning
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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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- id: BelkinReconciling2019
  accessed:
    - year: 2020
      month: 8
      day: 13
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      given: Mikhail
    - family: Hsu
      given: Daniel
    - family: Ma
      given: Siyuan
    - family: Mandal
      given: Soumik
  citation-key: BelkinReconciling2019
  container-title: Proceedings of the National Academy of Sciences
  container-title-short: Proc Natl Acad Sci USA
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  ISSN: 0027-8424, 1091-6490
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      day: 6
  language: en
  page: 15849-15854
  publisher: Proceedings of the National Academy of Sciences
  title: >-
    Reconciling modern machine-learning practice and the classical bias–variance
    trade-off
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- id: BelkinUnderstand2018
  accessed:
    - year: 2019
      month: 6
      day: 3
  author:
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      given: Mikhail
    - family: Ma
      given: Siyuan
    - family: Mandal
      given: Soumik
  citation-key: BelkinUnderstand2018
  container-title: International Conference on Machine Learning
  event-title: International Conference on Machine Learning
  issued:
    - year: 2018
      month: 7
      day: 3
  language: en
  page: 541-549
  title: To Understand Deep Learning We Need to Understand Kernel Learning
  type: paper-conference
  URL: http://arxiv.org/abs/1802.01396

- id: BengioGradientBased2000
  author:
    - family: Bengio
      given: Yoshua
  citation-key: BengioGradientBased2000
  container-title: Neural Computation
  container-title-short: Neural Computation
  DOI: 10.1162/089976600300015187
  ISSN: 0899-7667
  issue: '8'
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      month: 8
      day: 1
  page: 1889-1900
  title: Gradient-Based Optimization of Hyperparameters
  type: article-journal
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- id: DasguptaRegularized2016
  accessed:
    - year: 2016
      month: 10
      day: 12
  author:
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      given: Sakyasingha
    - family: Yoshizumi
      given: Takayuki
    - family: Osogami
      given: Takayuki
  citation-key: DasguptaRegularized2016
  container-title: arXiv:1610.01989 [cs, stat]
  issued:
    - year: 2016
      month: 9
      day: 22
  title: >-
    Regularized Dynamic Boltzmann Machine with Delay Pruning for Unsupervised
    Learning of Temporal Sequences
  type: article-journal
  URL: http://arxiv.org/abs/1610.01989

- id: DomingosEvery2020
  accessed:
    - year: 2020
      month: 12
      day: 12
  author:
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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
  type: article-journal
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- id: FinlayHow
  author:
    - family: Finlay
      given: Chris
    - family: Jacobsen
      given: Jörn-Henrik
    - family: Nurbekyan
      given: Levon
    - family: Oberman
      given: Adam M
  citation-key: FinlayHow
  container-title: ICML
  language: en
  page: '14'
  title: >-
    How to Train Your Neural ODE: the World of Jacobian and Kinetic
    Regularization
  type: paper-conference

- id: GalTheoretically2016
  accessed:
    - year: 2017
      month: 6
      day: 21
  author:
    - family: Gal
      given: Yarin
    - family: Ghahramani
      given: Zoubin
  citation-key: GalTheoretically2016
  container-title: arXiv:1512.05287 [stat]
  event-title: NIPS
  issued:
    - year: 2016
  title: A Theoretically Grounded Application of Dropout in Recurrent Neural Networks
  type: paper-conference
  URL: http://arxiv.org/abs/1512.05287

- id: GolowichSizeIndependent2017
  accessed:
    - year: 2018
      month: 1
      day: 26
  author:
    - family: Golowich
      given: Noah
    - family: Rakhlin
      given: Alexander
    - family: Shamir
      given: Ohad
  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: GravesPractical2011
  accessed:
    - year: 2017
      month: 9
      day: 1
  author:
    - family: Graves
      given: Alex
  citation-key: GravesPractical2011
  collection-title: NIPS'11
  container-title: >-
    Proceedings of the 24th International Conference on Neural Information
    Processing Systems
  event-place: USA
  ISBN: 978-1-61839-599-3
  issued:
    - year: 2011
  page: 2348–2356
  publisher: Curran Associates Inc.
  publisher-place: USA
  title: Practical Variational Inference for Neural Networks
  type: paper-conference
  URL: >-
    https://papers.nips.cc/paper/4329-practical-variational-inference-for-neural-networks.pdf

- id: HardtTrain2015
  accessed:
    - year: 2015
      month: 9
      day: 5
  author:
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      given: Moritz
    - family: Recht
      given: Benjamin
    - family: Singer
      given: Yoram
  citation-key: HardtTrain2015
  container-title: arXiv:1509.01240 [cs, math, stat]
  issued:
    - year: 2015
      month: 9
      day: 3
  title: 'Train faster, generalize better: Stability of stochastic gradient descent'
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- id: ImEmpirical2016
  author:
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      given: Daniel Jiwoong
    - family: Tao
      given: Michael
    - family: Branson
      given: Kristin
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  container-title: arXiv:1612.04010 [cs]
  issued:
    - year: 2016
      month: 12
      day: 12
  title: An empirical analysis of the optimization of deep network loss surfaces
  type: article-journal
  URL: http://arxiv.org/abs/1612.04010

- id: ImmerScalable2021
  accessed:
    - year: 2023
      month: 1
      day: 3
  author:
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      given: Alexander
    - family: Bauer
      given: Matthias
    - family: Fortuin
      given: Vincent
    - family: Rätsch
      given: Gunnar
    - family: Emtiyaz
      given: Khan Mohammad
  citation-key: ImmerScalable2021
  container-title: Proceedings of the 38th International Conference on Machine Learning
  event-title: International Conference on Machine Learning
  ISSN: 2640-3498
  issued:
    - year: 2021
      month: 7
      day: 1
  language: en
  page: 4563-4573
  publisher: PMLR
  title: Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning
  type: paper-conference
  URL: http://arxiv.org/abs/2104.04975

- id: IoffeBatch2015
  accessed:
    - year: 2023
      month: 4
      day: 20
  author:
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      given: Sergey
    - family: Szegedy
      given: Christian
  citation-key: IoffeBatch2015
  DOI: 10.48550/arXiv.1502.03167
  issued:
    - year: 2015
      month: 3
      day: 2
  number: arXiv:1502.03167
  publisher: arXiv
  title: >-
    Batch Normalization: Accelerating Deep Network Training by Reducing Internal
    Covariate Shift
  type: article
  URL: http://arxiv.org/abs/1502.03167

- id: IzmailovAveraging2018
  accessed:
    - year: 2021
      month: 9
      day: 24
  author:
    - family: Izmailov
      given: Pavel
    - family: Podoprikhin
      given: Dmitrii
    - family: Garipov
      given: Timur
    - family: Vetrov
      given: Dmitry
    - family: Wilson
      given: Andrew Gordon
  citation-key: IzmailovAveraging2018
  issued:
    - year: 2018
      month: 3
      day: 14
  language: en
  title: Averaging Weights Leads to Wider Optima and Better Generalization
  type: article-journal
  URL: https://arxiv.org/abs/1803.05407v3

- id: KawaguchiGeneralization2017
  accessed:
    - year: 2017
      month: 11
      day: 6
  author:
    - family: Kawaguchi
      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
  type: article-journal
  URL: http://arxiv.org/abs/1710.05468

- id: KellyLearning2020
  accessed:
    - year: 2020
      month: 12
      day: 8
  author:
    - family: Kelly
      given: Jacob
    - family: Bettencourt
      given: Jesse
    - family: Johnson
      given: Matthew James
    - family: Duvenaud
      given: David
  citation-key: KellyLearning2020
  issued:
    - year: 2020
      month: 7
      day: 9
  language: en
  title: Learning Differential Equations that are Easy to Solve
  type: paper-conference
  URL: https://arxiv.org/abs/2007.04504v2

- id: KlambauerSelfNormalizing2017
  accessed:
    - year: 2017
      month: 6
      day: 20
  author:
    - family: Klambauer
      given: Günter
    - family: Unterthiner
      given: Thomas
    - family: Mayr
      given: Andreas
    - family: Hochreiter
      given: Sepp
  citation-key: KlambauerSelfNormalizing2017
  container-title: >-
    Proceedings of the 31st International Conference on Neural Information
    Processing Systems
  event-place: Red Hook, NY, USA
  issued:
    - year: 2017
      month: 12
      day: 4
  language: en
  page: 972–981
  publisher: Curran Associates Inc.
  publisher-place: Red Hook, NY, USA
  title: Self-Normalizing Neural Networks
  type: paper-conference
  URL: http://arxiv.org/abs/1706.02515

- id: KochSparse2016
  author:
    - family: Koch
      given: Parker
    - family: Corso
      given: Jason J.
  citation-key: KochSparse2016
  container-title: arXiv:1612.04468 [cs, stat]
  issued:
    - year: 2016
      month: 12
      day: 13
  title: Sparse Factorization Layers for Neural Networks with Limited Supervision
  type: article-journal
  URL: http://arxiv.org/abs/1612.04468

- id: LeeWide2019
  accessed:
    - year: 2020
      month: 12
      day: 1
  author:
    - family: Lee
      given: Jaehoon
    - family: Xiao
      given: Lechao
    - family: Schoenholz
      given: Samuel S.
    - family: Bahri
      given: Yasaman
    - family: Novak
      given: Roman
    - family: Sohl-Dickstein
      given: Jascha
    - family: Pennington
      given: Jeffrey
  citation-key: LeeWide2019
  container-title: Advances in neural information processing systems
  issued:
    - year: 2019
      month: 12
      day: 8
  page: 8570–8581
  title: >-
    Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient
    Descent
  type: paper-conference
  URL: http://arxiv.org/abs/1902.06720

- id: LobachevaBayesian2017
  accessed:
    - year: 2017
      month: 8
      day: 12
  author:
    - family: Lobacheva
      given: Ekaterina
    - family: Chirkova
      given: Nadezhda
    - family: Vetrov
      given: Dmitry
  citation-key: LobachevaBayesian2017
  container-title: Workshop on Learning to Generate Natural Language
  event-title: ICML
  issued:
    - year: 2017
      month: 7
      day: 31
  title: Bayesian Sparsification of Recurrent Neural Networks
  type: paper-conference
  URL: http://arxiv.org/abs/1708.00077

- id: LoogBrief2020
  accessed:
    - year: 2020
      month: 8
      day: 13
  author:
    - family: Loog
      given: Marco
    - family: Viering
      given: Tom
    - family: Mey
      given: Alexander
    - family: Krijthe
      given: Jesse H.
    - family: Tax
      given: David M. J.
  citation-key: LoogBrief2020
  container-title: Proceedings of the National Academy of Sciences
  container-title-short: PNAS
  DOI: 10.1073/pnas.2001875117
  ISSN: 0027-8424, 1091-6490
  issue: '20'
  issued:
    - year: 2020
      month: 5
      day: 19
  language: en
  page: 10625-10626
  PMID: '32371495'
  publisher: National Academy of Sciences
  section: Letter
  title: A brief prehistory of double descent
  type: article-journal
  volume: '117'

- id: MaclaurinGradientbased2015
  accessed:
    - year: 2022
      month: 4
      day: 23
  author:
    - family: Maclaurin
      given: Dougal
    - family: Duvenaud
      given: David
    - family: Adams
      given: Ryan
  citation-key: MaclaurinGradientbased2015
  container-title: Proceedings of the 32nd International Conference on Machine Learning
  event-title: International Conference on Machine Learning
  ISSN: 1938-7228
  issued:
    - year: 2015
      month: 6
      day: 1
  language: en
  page: 2113-2122
  publisher: PMLR
  title: Gradient-based Hyperparameter Optimization through Reversible Learning
  type: paper-conference
  URL: https://proceedings.mlr.press/v37/maclaurin15.html

- id: MolchanovVariational2017
  author:
    - family: Molchanov
      given: Dmitry
    - family: Ashukha
      given: Arsenii
    - family: Vetrov
      given: Dmitry
  citation-key: MolchanovVariational2017
  container-title: Proceedings of ICML
  event-title: ICML
  issued:
    - year: 2017
      month: 1
      day: 19
  title: Variational Dropout Sparsifies Deep Neural Networks
  type: paper-conference
  URL: http://arxiv.org/abs/1701.05369

- id: NguyenXuanVinhTraining2016
  accessed:
    - year: 2018
      month: 5
      day: 29
  author:
    - family: Xuan Vinh
      given: Nguyen
    - family: Erfani
      given: Sarah
    - family: Paisitkriangkrai
      given: Sakrapee
    - family: Bailey
      given: James
    - family: Leckie
      given: Christopher
    - family: Ramamohanarao
      given: Kotagiri
  citation-key: NguyenXuanVinhTraining2016
  DOI: 10.1109/ICPR.2016.7899688
  ISBN: 978-1-5090-4847-2
  issued:
    - year: 2016
      month: 12
  language: en
  page: 531-536
  publisher: IEEE
  title: Training robust models using Random Projection
  type: paper-conference

- id: NoklandDirect2016
  author:
    - family: Nøkland
      given: Arild
  citation-key: NoklandDirect2016
  container-title: Advances In Neural Information Processing Systems
  event-title: NIPS
  issued:
    - year: 2016
      month: 9
      day: 6
  title: Direct Feedback Alignment Provides Learning in Deep Neural Networks
  type: paper-conference
  URL: http://arxiv.org/abs/1609.01596

- id: PanDropNeuron2016
  accessed:
    - year: 2016
      month: 7
      day: 4
  author:
    - family: Pan
      given: Wei
    - family: Dong
      given: Hao
    - family: Guo
      given: Yike
  citation-key: PanDropNeuron2016
  container-title: arXiv:1606.07326 [cs, stat]
  issued:
    - year: 2016
      month: 6
      day: 23
  title: 'DropNeuron: Simplifying the Structure of Deep Neural Networks'
  type: article-journal
  URL: http://arxiv.org/abs/1606.07326

- id: PapyanConvolutional2017
  author:
    - family: Papyan
      given: Vardan
    - family: Romano
      given: Yaniv
    - family: Sulam
      given: Jeremias
    - family: Elad
      given: Michael
  citation-key: PapyanConvolutional2017
  container-title: Proceedings of the IEEE International Conference on Computer Vision
  issued:
    - year: 2017
  page: 5296–5304
  title: Convolutional dictionary learning via local processing
  type: paper-conference

- id: PerezDeep2016
  author:
    - family: Perez
      given: Carlos E.
  citation-key: PerezDeep2016
  container-title: Medium
  issued:
    - year: 2016
      month: 11
      day: 6
  title: 'Deep Learning: The Unreasonable Effectiveness of Randomness'
  type: post-weblog
  URL: >-
    https://medium.com/intuitionmachine/deep-learning-the-unreasonable-effectiveness-of-randomness-14d5aef13f87#.g5sjhxjrn

- id: PrecheltEarly2012
  author:
    - family: Prechelt
      given: Lutz
  citation-key: PrecheltEarly2012
  collection-number: '7700'
  collection-title: Lecture Notes in Computer Science
  container-title: 'Neural Networks: Tricks of the Trade'
  DOI: 10.1007/978-3-642-35289-8_5
  editor:
    - family: Montavon
      given: Grégoire
    - family: Orr
      given: Geneviève B.
    - family: Müller
      given: Klaus-Robert
  ISBN: 978-3-642-35288-1 978-3-642-35289-8
  issued:
    - year: 2012
  language: en
  page: 53-67
  publisher: Springer Berlin Heidelberg
  title: Early Stopping — But When?
  type: chapter
  URL: http://page.mi.fu-berlin.de/~prechelt/Biblio/stop_tricks1997.pdf

- id: SalimansWeight2016
  author:
    - family: Salimans
      given: Tim
    - family: Kingma
      given: Diederik P
  citation-key: SalimansWeight2016
  container-title: Advances in Neural Information Processing Systems 29
  editor:
    - family: Lee
      given: D. D.
    - family: Sugiyama
      given: M.
    - family: Luxburg
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    - family: Guyon
      given: I.
    - family: Garnett
      given: R.
  issued:
    - year: 2016
  page: 901–901
  publisher: Curran Associates, Inc.
  title: >-
    Weight Normalization: A Simple Reparameterization to Accelerate Training of
    Deep Neural Networks
  type: paper-conference
  URL: >-
    http://papers.nips.cc/paper/6114-weight-normalization-a-simple-reparameterization-to-accelerate-training-of-deep-neural-networks.pdf

- id: SanturkarHow2019
  accessed:
    - year: 2020
      month: 6
      day: 2
  author:
    - family: Santurkar
      given: Shibani
    - family: Tsipras
      given: Dimitris
    - family: Ilyas
      given: Andrew
    - family: Madry
      given: Aleksander
  citation-key: SanturkarHow2019
  container-title: arXiv:1805.11604 [cs, stat]
  issued:
    - year: 2019
      month: 4
      day: 14
  title: How Does Batch Normalization Help Optimization?
  type: article-journal
  URL: http://arxiv.org/abs/1805.11604

- id: ScardapaneGroup2016
  accessed:
    - year: 2016
      month: 7
      day: 25
  author:
    - family: Scardapane
      given: Simone
    - family: Comminiello
      given: Danilo
    - family: Hussain
      given: Amir
    - family: Uncini
      given: Aurelio
  citation-key: ScardapaneGroup2016
  container-title: arXiv:1607.00485 [cs, stat]
  issued:
    - year: 2016
      month: 7
      day: 2
  title: Group Sparse Regularization for Deep Neural Networks
  type: article-journal
  URL: http://arxiv.org/abs/1607.00485

- id: SrinivasGeneralized2016
  author:
    - family: Srinivas
      given: Suraj
    - family: Babu
      given: R. Venkatesh
  citation-key: SrinivasGeneralized2016
  container-title: arXiv:1611.06791 [cs]
  issued:
    - year: 2016
      month: 11
      day: 21
  title: Generalized Dropout
  type: article-journal
  URL: http://arxiv.org/abs/1611.06791

- id: SrivastavaDropout2014
  accessed:
    - year: 2017
      month: 6
      day: 19
  author:
    - family: Srivastava
      given: Nitish
    - family: Hinton
      given: Geoffrey
    - family: Krizhevsky
      given: Alex
    - family: Sutskever
      given: Ilya
    - family: Salakhutdinov
      given: Ruslan
  citation-key: SrivastavaDropout2014
  container-title: The Journal of Machine Learning Research
  issue: '1'
  issued:
    - year: 2014
  page: 1929–1958
  title: 'Dropout: A simple way to prevent neural networks from overfitting'
  type: article-journal
  URL: >-
    http://www.jmlr.org/papers/volume15/srivastava14a.old/source/srivastava14a.pdf
  volume: '15'

- id: TaheriStatistical2020
  accessed:
    - year: 2020
      month: 6
      day: 2
  author:
    - family: Taheri
      given: Mahsa
    - family: Xie
      given: Fang
    - family: Lederer
      given: Johannes
  citation-key: TaheriStatistical2020
  container-title: arXiv:2006.00294 [cs, math, stat]
  issued:
    - year: 2020
      month: 5
      day: 30
  title: Statistical Guarantees for Regularized Neural Networks
  type: article-journal
  URL: http://arxiv.org/abs/2006.00294

- id: XieDiversity2016
  author:
    - family: Xie
      given: Bo
    - family: Liang
      given: Yingyu
    - family: Song
      given: Le
  citation-key: XieDiversity2016
  container-title: arXiv:1611.03131 [cs, stat]
  issued:
    - year: 2016
      month: 11
      day: 9
  title: Diversity Leads to Generalization in Neural Networks
  type: article-journal
  URL: http://arxiv.org/abs/1611.03131

- id: YouAdversarial2018
  accessed:
    - year: 2018
      month: 5
      day: 29
  author:
    - family: You
      given: Zhonghui
    - family: Ye
      given: Jinmian
    - family: Li
      given: Kunming
    - family: Wang
      given: Ping
  citation-key: YouAdversarial2018
  container-title: arXiv:1805.08000 [cs]
  issued:
    - year: 2018
      month: 5
      day: 21
  language: en
  title: 'Adversarial Noise Layer: Regularize Neural Network By Adding Noise'
  type: article-journal
  URL: http://arxiv.org/abs/1805.08000

- id: ZhangUnderstanding2017
  author:
    - family: Zhang
      given: Chiyuan
    - family: Bengio
      given: Samy
    - family: Hardt
      given: Moritz
    - family: Recht
      given: Benjamin
    - family: Vinyals
      given: Oriol
  citation-key: ZhangUnderstanding2017
  container-title: Proceedings of ICLR
  event-title: ICLR
  issued:
    - year: 2017
  title: Understanding deep learning requires rethinking generalization
  type: paper-conference
  URL: http://arxiv.org/abs/1611.03530
...
