---
references:
- id: AchlioptasAlgorithmic2008
  accessed:
    - year: 2018
      month: 4
      day: 3
  author:
    - family: Achlioptas
      given: Dimitris
    - family: Coja-Oghlan
      given: Amin
  citation-key: AchlioptasAlgorithmic2008
  container-title: arXiv:0803.2122 [math]
  DOI: 10.1109/FOCS.2008.11
  issued:
    - year: 2008
      month: 10
  page: 793-802
  title: Algorithmic barriers from phase transitions
  type: article-journal
  URL: http://arxiv.org/abs/0803.2122

- id: BaezRenyi2011
  accessed:
    - year: 2017
      month: 11
      day: 26
  author:
    - family: Baez
      given: John C.
  citation-key: BaezRenyi2011
  issued:
    - year: 2011
      month: 2
      day: 10
  title: Renyi Entropy and Free Energy
  type: article-journal
  URL: https://arxiv.org/abs/1102.2098

- id: BahriStatistical2020
  accessed:
    - year: 2021
      month: 1
      day: 6
  author:
    - family: Bahri
      given: Yasaman
    - family: Kadmon
      given: Jonathan
    - family: Pennington
      given: Jeffrey
    - family: Schoenholz
      given: Sam S.
    - family: Sohl-Dickstein
      given: Jascha
    - family: Ganguli
      given: Surya
  citation-key: BahriStatistical2020
  container-title: Annual Review of Condensed Matter Physics
  DOI: 10.1146/annurev-conmatphys-031119-050745
  issue: '1'
  issued:
    - year: 2020
  page: 501-528
  title: Statistical Mechanics of Deep Learning
  type: article-journal
  volume: '11'

- id: BaldassiUnreasonable2016
  author:
    - family: Baldassi
      given: Carlo
    - family: Borgs
      given: Christian
    - family: Chayes
      given: Jennifer T.
    - family: Ingrosso
      given: Alessandro
    - family: Lucibello
      given: Carlo
    - family: Saglietti
      given: Luca
    - family: Zecchina
      given: Riccardo
  citation-key: BaldassiUnreasonable2016
  container-title: Proceedings of the National Academy of Sciences
  container-title-short: PNAS
  DOI: 10.1073/pnas.1608103113
  ISSN: 0027-8424, 1091-6490
  issue: '48'
  issued:
    - year: 2016
      month: 11
      day: 29
  language: en
  page: E7655-E7662
  PMID: '27856745'
  title: >-
    Unreasonable effectiveness of learning neural networks: From accessible
    states and robust ensembles to basic algorithmic schemes
  type: article-journal
  volume: '113'

- id: BarbierPhase2017
  accessed:
    - year: 2018
      month: 1
      day: 15
  author:
    - family: Barbier
      given: Jean
    - family: Krzakala
      given: Florent
    - family: Macris
      given: Nicolas
    - family: Miolane
      given: Léo
    - family: Zdeborová
      given: Lenka
  citation-key: BarbierPhase2017
  container-title: arXiv:1708.03395 [cond-mat, physics:math-ph]
  issued:
    - year: 2017
      month: 8
      day: 10
  title: >-
    Phase Transitions, Optimal Errors and Optimality of Message-Passing in
    Generalized Linear Models
  type: article-journal
  URL: http://arxiv.org/abs/1708.03395

- id: BarbierStatistical2015
  accessed:
    - year: 2015
      month: 12
      day: 2
  author:
    - family: Barbier
      given: Jean
  citation-key: BarbierStatistical2015
  container-title: arXiv:1511.01650 [cs, math]
  issued:
    - year: 2015
      month: 11
      day: 5
  title: >-
    Statistical physics and approximate message-passing algorithms for sparse
    linear estimation problems in signal processing and coding theory
  type: article-journal
  URL: http://arxiv.org/abs/1511.01650

- id: BarnumEntropy2010
  accessed:
    - year: 2024
      month: 3
      day: 4
  author:
    - family: Barnum
      given: Howard
    - family: Barrett
      given: Jonathan
    - family: Clark
      given: Lisa Orloff
    - family: Leifer
      given: Matthew
    - family: Spekkens
      given: Robert
    - family: Stepanik
      given: Nicholas
    - family: Wilce
      given: Alex
    - family: Wilke
      given: Robin
  citation-key: BarnumEntropy2010
  container-title: New Journal of Physics
  container-title-short: New J. Phys.
  DOI: 10.1088/1367-2630/12/3/033024
  ISSN: 1367-2630
  issue: '3'
  issued:
    - year: 2010
      month: 3
  language: en
  page: '033024'
  title: Entropy and information causality in general probabilistic theories
  type: article-journal
  URL: http://arxiv.org/abs/0909.5075
  volume: '12'

- id: BraunsteinSurvey2002
  accessed:
    - year: 2018
      month: 4
      day: 3
  author:
    - family: Braunstein
      given: A.
    - family: Mezard
      given: M.
    - family: Zecchina
      given: R.
  citation-key: BraunsteinSurvey2002
  container-title: arXiv:cs/0212002
  issued:
    - year: 2002
      month: 12
      day: 4
  title: 'Survey propagation: an algorithm for satisfiability'
  type: article-journal
  URL: http://arxiv.org/abs/cs/0212002

- id: CastellaniSpinGlass2005
  accessed:
    - year: 2018
      month: 3
      day: 19
  author:
    - family: Castellani
      given: Tommaso
    - family: Cavagna
      given: Andrea
  citation-key: CastellaniSpinGlass2005
  container-title: 'Journal of Statistical Mechanics: Theory and Experiment'
  DOI: 10.1088/1742-5468/2005/05/P05012
  ISSN: 1742-5468
  issue: '05'
  issued:
    - year: 2005
      month: 5
      day: 31
  page: P05012
  title: Spin-Glass Theory for Pedestrians
  type: article-journal
  URL: http://arxiv.org/abs/cond-mat/0505032
  volume: '2005'

- id: CatoniPACBayesian2007
  accessed:
    - year: 2023
      month: 4
      day: 27
  author:
    - family: Catoni
      given: Olivier
  citation-key: CatoniPACBayesian2007
  container-title: IMS Lecture Notes Monograph Series
  container-title-short: IMS Lecture Notes Monogr. Ser.
  DOI: 10.1214/074921707000000391
  ISSN: 0749-2170
  issued:
    - year: 2007
  page: 1-163
  title: >-
    PAC-Bayesian Supervised Classification: The Thermodynamics of Statistical
    Learning
  type: article-journal
  URL: http://arxiv.org/abs/0712.0248
  volume: '56'

- id: ChangReversible2018
  accessed:
    - year: 2018
      month: 4
      day: 23
  author:
    - family: Chang
      given: Bo
    - family: Meng
      given: Lili
    - family: Haber
      given: Eldad
    - family: Ruthotto
      given: Lars
    - family: Begert
      given: David
    - family: Holtham
      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
  type: paper-conference
  URL: http://arxiv.org/abs/1709.03698

- id: ChaudhariEntropySGD2017
  accessed:
    - year: 2023
      month: 9
      day: 6
  author:
    - family: Chaudhari
      given: Pratik
    - family: Choromanska
      given: Anna
    - family: Soatto
      given: Stefano
    - family: LeCun
      given: Yann
    - family: Baldassi
      given: Carlo
    - family: Borgs
      given: Christian
    - family: Chayes
      given: Jennifer
    - family: Sagun
      given: Levent
    - family: Zecchina
      given: Riccardo
  citation-key: ChaudhariEntropySGD2017
  DOI: 10.48550/arXiv.1611.01838
  issued:
    - year: 2017
      month: 4
      day: 21
  number: arXiv:1611.01838
  publisher: arXiv
  title: 'Entropy-SGD: Biasing Gradient Descent Into Wide Valleys'
  type: article
  URL: http://arxiv.org/abs/1611.01838
  version: '5'

- id: ChoromanskaLoss2015
  accessed:
    - year: 2015
      month: 6
      day: 17
  author:
    - family: Choromanska
      given: Anna
    - family: Henaff
      given: MIkael
    - family: Mathieu
      given: Michael
    - family: Ben Arous
      given: Gerard
    - family: LeCun
      given: Yann
  citation-key: ChoromanskaLoss2015
  container-title: >-
    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
  type: paper-conference
  URL: http://proceedings.mlr.press/v38/choromanska15.html

- id: FeldmanBrief2002
  accessed:
    - year: 2023
      month: 9
      day: 4
  author:
    - family: Feldman
      given: David P
  citation-key: FeldmanBrief2002
  issued:
    - year: 2002
  title: >-
    A Brief Introduction to: Information Theory, Excess Entropy and
    Computational Mechanics
  type: article
  URL: http://hornacek.coa.edu/dave/Tutorial/notes.pdf

- id: FengInverse2021
  accessed:
    - year: 2023
      month: 9
      day: 6
  author:
    - family: Feng
      given: Yu
    - family: Tu
      given: Yuhai
  citation-key: FengInverse2021
  container-title: Proceedings of the National Academy of Sciences
  DOI: 10.1073/pnas.2015617118
  issue: '9'
  issued:
    - year: 2021
      month: 3
      day: 2
  page: e2015617118
  publisher: Proceedings of the National Academy of Sciences
  title: >-
    The inverse variance–flatness relation in stochastic gradient descent is
    critical for finding flat minima
  type: article-journal
  volume: '118'

- id: GershenfeldAligning2011
  accessed:
    - year: 2023
      month: 11
      day: 16
  author:
    - family: Gershenfeld
      given: Neil
  citation-key: GershenfeldAligning2011
  container-title: Computing
  container-title-short: Computing
  DOI: 10.1007/s00607-011-0160-1
  ISSN: 0010-485X, 1436-5057
  issue: 2-4
  issued:
    - year: 2011
      month: 12
  language: en
  page: 91-102
  title: >-
    Aligning the representation and reality of computation with asynchronous
    logic automata
  type: article-journal
  URL: http://cba.mit.edu/docs/papers/11.12.Computing.pdf
  volume: '93'

- id: GershenfeldSignal1996
  accessed:
    - year: 2023
      month: 11
      day: 16
  author:
    - family: Gershenfeld
      given: N.
  citation-key: GershenfeldSignal1996
  container-title: IBM Systems Journal
  container-title-short: IBM Syst. J.
  DOI: 10.1147/sj.353.0577
  ISSN: 0018-8670
  issue: '3.4'
  issued:
    - year: 1996
  language: en
  page: 577-586
  title: Signal entropy and the thermodynamics of computation
  type: article-journal
  URL: http://cba.mit.edu/docs/papers/96.isj.ent.pdf
  volume: '35'

- id: GoldtStochastic2017
  accessed:
    - year: 2023
      month: 8
      day: 30
  author:
    - family: Goldt
      given: Sebastian
    - family: Seifert
      given: Udo
  citation-key: GoldtStochastic2017
  container-title: Physical Review Letters
  container-title-short: Phys. Rev. Lett.
  DOI: 10.1103/PhysRevLett.118.010601
  ISSN: 0031-9007, 1079-7114
  issue: '1'
  issued:
    - year: 2017
      month: 1
      day: 6
  page: '010601'
  title: Stochastic Thermodynamics of Learning
  type: article-journal
  URL: http://arxiv.org/abs/1611.09428
  volume: '118'

- id: HaberStable2018
  author:
    - family: Haber
      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: HarperReplicator2009
  accessed:
    - year: 2017
      month: 11
      day: 26
  author:
    - family: Harper
      given: Marc
  citation-key: HarperReplicator2009
  issued:
    - year: 2009
      month: 11
      day: 9
  title: The Replicator Equation as an Inference Dynamic
  type: article-journal
  URL: https://arxiv.org/abs/0911.1763

- id: HasegawaUncertainty2019
  accessed:
    - year: 2021
      month: 4
      day: 19
  author:
    - family: Hasegawa
      given: Yoshihiko
    - family: Van Vu
      given: Tan
  citation-key: HasegawaUncertainty2019
  container-title: Physical Review E
  container-title-short: Phys. Rev. E
  DOI: 10.1103/PhysRevE.99.062126
  issue: '6'
  issued:
    - year: 2019
      month: 6
      day: 24
  page: '062126'
  publisher: American Physical Society
  title: >-
    Uncertainty relations in stochastic processes: An information inequality
    approach
  type: article-journal
  URL: http://arxiv.org/abs/1809.03292
  volume: '99'

- id: HayouImpact2019
  accessed:
    - year: 2022
      month: 10
      day: 30
  author:
    - family: Hayou
      given: Soufiane
    - family: Doucet
      given: Arnaud
    - family: Rousseau
      given: Judith
  citation-key: HayouImpact2019
  container-title: Proceedings of the 36th International Conference on Machine Learning
  DOI: 10.48550/arXiv.1902.06853
  event-title: International Conference on Machine Learning
  ISSN: 2640-3498
  issued:
    - year: 2019
      month: 5
      day: 24
  language: en
  page: 2672-2680
  publisher: PMLR
  title: On the Impact of the Activation function on Deep Neural Networks Training
  type: paper-conference
  URL: http://arxiv.org/abs/1902.06853

- id: KrishnamurthyTheory2022
  author:
    - family: Krishnamurthy
      given: Kamesh
    - family: Can
      given: Tankut
    - family: Schwab
      given: David J.
  citation-key: KrishnamurthyTheory2022
  container-title: Physical Review. X
  container-title-short: Phys Rev X
  DOI: 10.1103/physrevx.12.011011
  ISSN: 2160-3308
  issue: '1'
  issued:
    - year: 2022
  language: eng
  page: '011011'
  PMCID: PMC9762509
  PMID: '36545030'
  title: Theory of Gating in Recurrent Neural Networks
  type: article-journal
  URL: http://arxiv.org/abs/2007.14823
  volume: '12'

- id: KrzakalaStatistical
  author:
    - family: Krzakala
      given: Florent
    - family: Zdeborova
      given: Lenka
    - family: Angelini
      given: Maria Chiara
    - family: Caltagirone
      given: Francesco
  citation-key: KrzakalaStatistical
  language: en
  page: '44'
  title: Statistical Physics of Inference and Bayesian Estimation
  type: article-journal
  URL: >-
    http://boulderschool.yale.edu/sites/default/files/files/Lecture_notes_Statistical_inference,_Optimization_and_Machine_Learning.pdf

- id: LangThermodynamics2014
  accessed:
    - year: 2015
      month: 1
      day: 23
  author:
    - family: Lang
      given: Alex H.
    - family: Fisher
      given: Charles K.
    - family: Mora
      given: Thierry
    - family: Mehta
      given: Pankaj
  citation-key: LangThermodynamics2014
  container-title: Physical Review Letters
  DOI: 10.1103/PhysRevLett.113.148103
  ISSN: 0031-9007, 1079-7114
  issue: '14'
  issued:
    - year: 2014
      month: 10
      day: 3
  title: Thermodynamics of statistical inference by cells
  type: article-journal
  URL: http://arxiv.org/abs/1405.4001
  volume: '113'

- id: LaurentRecurrent2016
  author:
    - family: Laurent
      given: Thomas
    - family: Brecht
      given: James
      non-dropping-particle: von
  citation-key: LaurentRecurrent2016
  container-title: arXiv:1612.06212 [cs]
  issued:
    - year: 2016
      month: 12
      day: 19
  title: A recurrent neural network without chaos
  type: article-journal
  URL: http://arxiv.org/abs/1612.06212

- 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:
    - family: Lin
      given: Henry W.
    - family: Tegmark
      given: Max
  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: MandelbrotRole1962
  accessed:
    - year: 2014
      month: 8
      day: 6
  author:
    - family: Mandelbrot
      given: Benoit
  citation-key: MandelbrotRole1962
  container-title: The Annals of Mathematical Statistics
  container-title-short: Ann. Math. Statist.
  DOI: 10.1214/aoms/1177704470
  ISSN: 0003-4851, 2168-8990
  issue: '3'
  issued:
    - year: 1962
      month: 9
  language: EN
  page: 1021-1038
  title: The Role of Sufficiency and of Estimation in Thermodynamics
  type: article-journal
  volume: '33'

- id: MarslandLimits2018
  accessed:
    - year: 2023
      month: 5
      day: 18
  author:
    - family: Marsland
      given: Robert
    - family: England
      given: Jeremy
  citation-key: MarslandLimits2018
  container-title: Reports on Progress in Physics
  container-title-short: Rep. Prog. Phys.
  DOI: 10.1088/1361-6633/aa9101
  ISSN: 0034-4885, 1361-6633
  issue: '1'
  issued:
    - year: 2018
      month: 1
      day: 1
  language: en
  page: '016601'
  title: 'Limits of predictions in thermodynamic systems: a review'
  type: article-journal
  URL: >-
    https://www.englandlab.com/uploads/7/8/0/3/7803054/marsland_2018_rep._prog._phys._81_016601.pdf
  volume: '81'

- id: MehtaExact2014
  accessed:
    - year: 2015
      month: 2
      day: 1
  author:
    - family: Mehta
      given: Pankaj
    - family: Schwab
      given: David J.
  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
    Learning
  type: article
  URL: http://arxiv.org/abs/1410.3831

- id: MooreComputer2017
  accessed:
    - year: 2018
      month: 3
      day: 19
  author:
    - family: Moore
      given: Cristopher
  citation-key: MooreComputer2017
  container-title: Bulletin of the EATCS
  issued:
    - year: 2017
      month: 2
      day: 1
  title: >-
    The Computer Science and Physics of Community Detection: Landscapes, Phase
    Transitions, and Hardness
  type: article-journal
  URL: http://arxiv.org/abs/1702.00467

- id: NandaProgress2023
  accessed:
    - year: 2023
      month: 6
      day: 2
  author:
    - family: Nanda
      given: Neel
    - family: Chan
      given: Lawrence
    - family: Lieberum
      given: Tom
    - family: Smith
      given: Jess
    - family: Steinhardt
      given: Jacob
  citation-key: NandaProgress2023
  DOI: 10.48550/arXiv.2301.05217
  issued:
    - year: 2023
      month: 1
      day: 13
  number: arXiv:2301.05217
  publisher: arXiv
  title: Progress measures for grokking via mechanistic interpretability
  type: article
  URL: http://arxiv.org/abs/2301.05217

- id: OymakUniversality2015
  accessed:
    - year: 2015
      month: 12
      day: 3
  author:
    - family: Oymak
      given: Samet
    - family: Tropp
      given: Joel A.
  citation-key: OymakUniversality2015
  container-title: arXiv:1511.09433 [cs, math, stat]
  issued:
    - year: 2015
      month: 11
      day: 30
  title: Universality laws for randomized dimension reduction, with applications
  type: article-journal
  URL: http://arxiv.org/abs/1511.09433

- id: PavonStochastic1989
  accessed:
    - year: 2022
      month: 7
      day: 15
  author:
    - family: Pavon
      given: Michele
  citation-key: PavonStochastic1989
  container-title: Applied Mathematics and Optimization
  container-title-short: Appl Math Optim
  DOI: 10.1007/BF01448198
  ISSN: 1432-0606
  issue: '1'
  issued:
    - year: 1989
      month: 1
      day: 1
  language: en
  page: 187-202
  title: Stochastic control and nonequilibrium thermodynamical systems
  type: article-journal
  volume: '19'

- id: PooleExponential2016
  accessed:
    - year: 2016
      month: 12
      day: 7
  author:
    - family: Poole
      given: Ben
    - family: Lahiri
      given: Subhaneil
    - family: Raghu
      given: Maithreyi
    - family: Sohl-Dickstein
      given: Jascha
    - family: Ganguli
      given: Surya
  citation-key: PooleExponential2016
  container-title: Advances in Neural Information Processing Systems 29
  editor:
    - family: Lee
      given: D. D.
    - family: Sugiyama
      given: M.
    - family: Luxburg
      given: U. V.
    - family: Guyon
      given: I.
    - family: Garnett
      given: R.
  issued:
    - year: 2016
  page: 3360–3368
  publisher: Curran Associates, Inc.
  title: Exponential expressivity in deep neural networks through transient chaos
  type: paper-conference
  URL: >-
    http://papers.nips.cc/paper/6322-exponential-expressivity-in-deep-neural-networks-through-transient-chaos.pdf

- id: PowerGrokking2022
  accessed:
    - year: 2023
      month: 6
      day: 2
  author:
    - family: Power
      given: Alethea
    - family: Burda
      given: Yuri
    - family: Edwards
      given: Harri
    - family: Babuschkin
      given: Igor
    - family: Misra
      given: Vedant
  citation-key: PowerGrokking2022
  DOI: 10.48550/arXiv.2201.02177
  issued:
    - year: 2022
      month: 1
      day: 6
  number: arXiv:2201.02177
  publisher: arXiv
  title: 'Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets'
  type: article
  URL: http://arxiv.org/abs/2201.02177

- id: RobertsPrinciples2021
  accessed:
    - year: 2021
      month: 10
      day: 23
  author:
    - family: Roberts
      given: Daniel A.
    - family: Yaida
      given: Sho
    - family: Hanin
      given: Boris
  citation-key: RobertsPrinciples2021
  container-title: arXiv:2106.10165 [hep-th, stat]
  issued:
    - year: 2021
      month: 8
      day: 24
  title: The Principles of Deep Learning Theory
  type: article-journal
  URL: http://arxiv.org/abs/2106.10165

- id: RuthottoDeep2020
  accessed:
    - year: 2023
      month: 5
      day: 22
  author:
    - family: Ruthotto
      given: Lars
    - family: Haber
      given: Eldad
  citation-key: RuthottoDeep2020
  container-title: Journal of Mathematical Imaging and Vision
  container-title-short: J Math Imaging Vis
  DOI: 10.1007/s10851-019-00903-1
  ISSN: 1573-7683
  issue: '3'
  issued:
    - year: 2020
      month: 4
      day: 1
  language: en
  page: 352-364
  title: Deep Neural Networks Motivated by Partial Differential Equations
  type: article-journal
  URL: http://arxiv.org/abs/1804.04272
  volume: '62'

- id: SchoenholzDeep2017
  accessed:
    - year: 2022
      month: 10
      day: 30
  author:
    - family: Schoenholz
      given: Samuel S.
    - family: Gilmer
      given: Justin
    - family: Ganguli
      given: Surya
    - family: Sohl-Dickstein
      given: Jascha
  citation-key: SchoenholzDeep2017
  event-title: International Conference on Learning Representations
  issued:
    - year: 2017
  language: en
  title: Deep Information Propagation
  type: paper-conference
  URL: https://arxiv.org/abs/1611.01232v2

- id: ShaliziDynamics2009
  accessed:
    - year: 2014
      month: 10
      day: 9
  author:
    - family: Shalizi
      given: Cosma Rohilla
  citation-key: ShaliziDynamics2009
  container-title: Electronic Journal of Statistics
  container-title-short: Electron. J. Statist.
  DOI: 10.1214/09-EJS485
  ISSN: 1935-7524
  issued:
    - year: 2009
  language: EN
  page: 1039-1074
  title: Dynamics of Bayesian updating with dependent data and misspecified models
  type: article-journal
  volume: '3'

- id: ShaliziWhat2003
  accessed:
    - year: 2023
      month: 9
      day: 4
  author:
    - family: Shalizi
      given: Cosma Rohilla
    - family: Moore
      given: Cristopher
  citation-key: ShaliziWhat2003
  DOI: 10.48550/arXiv.cond-mat/0303625
  issued:
    - year: 2003
      month: 3
      day: 29
  number: arXiv:cond-mat/0303625
  publisher: arXiv
  title: What Is a Macrostate? Subjective Observations and Objective Dynamics
  type: article
  URL: http://arxiv.org/abs/cond-mat/0303625

- id: Shwartz-ZivOpening2017
  author:
    - family: Shwartz-Ziv
      given: Ravid
    - family: Tishby
      given: Naftali
  citation-key: Shwartz-ZivOpening2017
  container-title: arXiv:1703.00810 [cs]
  issued:
    - year: 2017
      month: 3
      day: 2
  title: Opening the Black Box of Deep Neural Networks via Information
  type: article-journal
  URL: http://arxiv.org/abs/1703.00810

- id: SinervoRock1996
  accessed:
    - year: 2017
      month: 11
      day: 26
  author:
    - family: Sinervo
      given: B.
    - family: Lively
      given: C. M.
  citation-key: SinervoRock1996
  container-title: Nature
  DOI: 10.1038/380240a0
  ISSN: 1476-4687
  issue: '6571'
  issued:
    - year: 1996
      month: 3
      day: 21
  language: En
  page: '240'
  title: >-
    The rock–paper–scissors game and the evolution of alternative male
    strategies
  type: article-journal
  URL: http://www.indiana.edu/~curtweb/L567/readings/Sinervo&Lively1996.pdf
  volume: '380'

- id: SunScheduling2023
  accessed:
    - year: 2023
      month: 9
      day: 6
  author:
    - family: Sun
      given: Jianhui
    - family: Yang
      given: Ying
    - family: Xun
      given: Guangxu
    - family: Zhang
      given: Aidong
  citation-key: SunScheduling2023
  container-title: ACM Transactions on Knowledge Discovery from Data
  container-title-short: ACM Trans. Knowl. Discov. Data
  DOI: 10.1145/3544782
  ISSN: 1556-4681
  issue: '2'
  issued:
    - year: 2023
      month: 3
      day: 20
  page: 29:1–29:37
  title: >-
    Scheduling Hyperparameters to Improve Generalization: From Centralized SGD
    to Asynchronous SGD
  type: article-journal
  volume: '17'

- id: SzekelyEnergy2017
  accessed:
    - year: 2017
      month: 10
      day: 2
  author:
    - family: Székely
      given: Gábor J.
    - family: Rizzo
      given: Maria L.
  citation-key: SzekelyEnergy2017
  container-title: Annual Review of Statistics and Its Application
  DOI: 10.1146/annurev-statistics-060116-054026
  issue: '1'
  issued:
    - year: 2017
  page: 447-479
  title: The Energy of Data
  type: article-journal
  volume: '4'

- id: VentolaProbabilistic2023
  accessed:
    - year: 2023
      month: 5
      day: 29
  author:
    - family: Ventola
      given: Fabrizio
    - family: Braun
      given: Steven
    - family: Yu
      given: Zhongjie
    - family: Mundt
      given: Martin
    - family: Kersting
      given: Kristian
  citation-key: VentolaProbabilistic2023
  container-title: arXiv.org
  issued:
    - year: 2023
      month: 2
      day: 13
  language: en
  title: Probabilistic Circuits That Know What They Don't Know
  type: webpage
  URL: https://arxiv.org/abs/2302.06544v2

- id: WolpertConstraints2017
  accessed:
    - year: 2021
      month: 4
      day: 19
  author:
    - family: Wolpert
      given: David
  citation-key: WolpertConstraints2017
  issued:
    - year: 2017
      month: 11
      day: 9
  language: en
  title: Constraints on physical reality arising from a formalization of knowledge
  type: article-journal
  URL: https://arxiv.org/abs/1711.03499v3

- id: WolpertInformation2006
  author:
    - family: Wolpert
      given: David H
  citation-key: WolpertInformation2006
  collection-title: Understanding Complex Systems
  container-title: Complex Engineered Systems
  ISBN: 978-3-540-32831-5 978-3-540-32834-6
  issued:
    - year: 2006
  language: en
  page: 262-290
  publisher: Springer Berlin Heidelberg
  title: >-
    Information Theory — The Bridge Connecting Bounded Rational Game Theory and
    Statistical Physics
  type: chapter
  URL: http://arxiv.org/abs/cond-mat/0402508

- id: WolpertPhysical2008
  accessed:
    - year: 2014
      month: 8
      day: 10
  author:
    - family: Wolpert
      given: David H.
  citation-key: WolpertPhysical2008
  collection-title: 'Novel Computing Paradigms: Quo Vadis?'
  container-title: 'Physica D: Nonlinear Phenomena'
  container-title-short: 'Physica D: Nonlinear Phenomena'
  DOI: 10.1016/j.physd.2008.03.040
  ISSN: 0167-2789
  issue: '9'
  issued:
    - year: 2008
      month: 7
      day: 1
  page: 1257-1281
  title: Physical limits of inference
  type: article-journal
  URL: https://arxiv.org/abs/0708.1362
  volume: '237'

- id: WolpertSpacetime2017
  accessed:
    - year: 2021
      month: 4
      day: 17
  author:
    - family: Wolpert
      given: David H.
    - family: Kolchinsky
      given: Artemy
    - family: Owen
      given: Jeremy A.
  citation-key: WolpertSpacetime2017
  DOI: 10.1038/s41467-019-09542-x
  issued:
    - year: 2017
      month: 8
      day: 28
  language: en
  title: >-
    A space-time tradeoff for implementing a function with master equation
    dynamics
  type: article-journal
  URL: https://arxiv.org/abs/1708.08494v4

- id: WolpertStochastic2019
  accessed:
    - year: 2021
      month: 4
      day: 17
  author:
    - family: Wolpert
      given: David H.
  citation-key: WolpertStochastic2019
  DOI: 10.1088/1751-8121/ab0850
  issued:
    - year: 2019
      month: 5
      day: 14
  language: en
  title: Stochastic thermodynamics of computation
  type: article-journal
  URL: https://arxiv.org/abs/1905.05669v1

- id: WolpertTheories2018
  author:
    - family: Wolpert
      given: David H.
  citation-key: WolpertTheories2018
  container-title: >-
    The Map and the Territory: Exploring the Foundations of Science, Thought and
    Reality
  DOI: 10.1007/978-3-319-72478-2_9
  event-place: Cham
  issued:
    - year: 2018
  page: '165'
  publisher: Springer
  publisher-place: Cham
  title: Theories of Knowledge and Theories of Everything
  type: chapter

- id: YarotskyPhase2020a
  accessed:
    - year: 2023
      month: 9
      day: 6
  author:
    - family: Yarotsky
      given: Dmitry
    - family: Zhevnerchuk
      given: Anton
  citation-key: YarotskyPhase2020a
  collection-title: NIPS'20
  container-title: >-
    Proceedings of the 34th International Conference on Neural Information
    Processing Systems
  event-place: Red Hook, NY, USA
  ISBN: 978-1-71382-954-6
  issued:
    - year: 2020
      month: 12
      day: 6
  page: 13005–13015
  publisher: Curran Associates Inc.
  publisher-place: Red Hook, NY, USA
  title: The phase diagram of approximation rates for deep neural networks
  type: paper-conference
  URL: http://arxiv.org/abs/1906.09477
  volume: '33'

- id: ZdeborovaStatistical2016
  accessed:
    - year: 2018
      month: 3
      day: 19
  author:
    - family: Zdeborová
      given: Lenka
    - family: Krzakala
      given: Florent
  citation-key: ZdeborovaStatistical2016
  container-title: Advances in Physics
  DOI: 10.1080/00018732.2016.1211393
  ISSN: 0001-8732, 1460-6976
  issue: '5'
  issued:
    - year: 2016
      month: 9
      day: 2
  page: 453-552
  title: 'Statistical physics of inference: Thresholds and algorithms'
  type: article-journal
  URL: http://arxiv.org/abs/1511.02476
  volume: '65'

- id: ZhangEnergyentropy2018
  accessed:
    - year: 2023
      month: 9
      day: 1
  author:
    - family: Zhang
      given: Yao
    - family: Saxe
      given: Andrew M.
    - family: Advani
      given: Madhu S.
    - family: Lee
      given: Alpha A.
  citation-key: ZhangEnergyentropy2018
  container-title: Molecular Physics
  container-title-short: Molecular Physics
  DOI: 10.1080/00268976.2018.1483535
  ISSN: 0026-8976, 1362-3028
  issue: 21-22
  issued:
    - year: 2018
      month: 11
      day: 17
  page: 3214-3223
  title: >-
    Energy-entropy competition and the effectiveness of stochastic gradient
    descent in machine learning
  type: article-journal
  URL: http://arxiv.org/abs/1803.01927
  volume: '116'

- id: ZurekComplexity1990
  citation-key: ZurekComplexity1990
  contributor:
    - family: Zurek
      given: Wojciech H
  ISBN: 0-201-51506-7
  issued:
    - year: 1990
  publisher: Addison-Wesley Pub. Co.
  title: >-
    Complexity, Entropy and the Physics of Information: The Proceedings of the
    1988 Workshop on Complexity, Entropy, and the Physics of Information
  type: paper-conference
...
