Backpropagation (English Wikipedia)

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Goodfellow, Bengio & Courville 2016, p. 200, "The term back-propagation is often misunderstood as meaning the whole learning algorithm for multilayer neural networks. Backpropagation refers only to the method for computing the gradient, while other algorithms, such as stochastic gradient descent, is used to perform learning using this gradient." Goodfellow, Ian; Bengio, Yoshua; Courville, Aaron (2016). "6.5 Back-Propagation and Other Differentiation Algorithms". Deep Learning. MIT Press. pp. 200–220. ISBN 9780262035613.
Goodfellow, Bengio & Courville (2016, p. 217–218), "The back-propagation algorithm described here is only one approach to automatic differentiation. It is a special case of a broader class of techniques called reverse mode accumulation." Goodfellow, Ian; Bengio, Yoshua; Courville, Aaron (2016). "6.5 Back-Propagation and Other Differentiation Algorithms". Deep Learning. MIT Press. pp. 200–220. ISBN 9780262035613.

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Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (1986a). "Learning representations by back-propagating errors". Nature. 323 (6088): 533–536. Bibcode:1986Natur.323..533R. doi:10.1038/323533a0. S2CID 205001834.
Tan, Hong Hui; Lim, King Han (2019). "Review of second-order optimization techniques in artificial neural networks backpropagation". IOP Conference Series: Materials Science and Engineering. 495 (1): 012003. Bibcode:2019MS&E..495a2003T. doi:10.1088/1757-899X/495/1/012003. S2CID 208124487.
LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning" (PDF). Nature. 521 (7553): 436–444. Bibcode:2015Natur.521..436L. doi:10.1038/nature14539. PMID 26017442. S2CID 3074096.
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Rumelhart; Hinton; Williams (1986). "Learning representations by back-propagating errors" (PDF). Nature. 323 (6088): 533–536. Bibcode:1986Natur.323..533R. doi:10.1038/323533a0. S2CID 205001834.
Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. arXiv:1404.7828. doi:10.1016/j.neunet.2014.09.003. PMID 25462637. S2CID 11715509.
Chang, Franklin; Dell, Gary S.; Bock, Kathryn (2006). "Becoming syntactic". Psychological Review. 113 (2): 234–272. doi:10.1037/0033-295x.113.2.234. PMID 16637761.
Janciauskas, Marius; Chang, Franklin (2018). "Input and Age-Dependent Variation in Second Language Learning: A Connectionist Account". Cognitive Science. 42 (Suppl Suppl 2): 519–554. doi:10.1111/cogs.12519. PMC 6001481. PMID 28744901.
Fitz, Hartmut; Chang, Franklin (2019). "Language ERPs reflect learning through prediction error propagation". Cognitive Psychology. 111: 15–52. doi:10.1016/j.cogpsych.2019.03.002. hdl:21.11116/0000-0003-474D-8. PMID 30921626. S2CID 85501792.

hal.science

LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning" (PDF). Nature. 521 (7553): 436–444. Bibcode:2015Natur.521..436L. doi:10.1038/nature14539. PMID 26017442. S2CID 3074096.

handle.net

hdl.handle.net

Fitz, Hartmut; Chang, Franklin (2019). "Language ERPs reflect learning through prediction error propagation". Cognitive Psychology. 111: 15–52. doi:10.1016/j.cogpsych.2019.03.002. hdl:21.11116/0000-0003-474D-8. PMID 30921626. S2CID 85501792.

harvard.edu

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Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (1986a). "Learning representations by back-propagating errors". Nature. 323 (6088): 533–536. Bibcode:1986Natur.323..533R. doi:10.1038/323533a0. S2CID 205001834.
Tan, Hong Hui; Lim, King Han (2019). "Review of second-order optimization techniques in artificial neural networks backpropagation". IOP Conference Series: Materials Science and Engineering. 495 (1): 012003. Bibcode:2019MS&E..495a2003T. doi:10.1088/1757-899X/495/1/012003. S2CID 208124487.
LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning" (PDF). Nature. 521 (7553): 436–444. Bibcode:2015Natur.521..436L. doi:10.1038/nature14539. PMID 26017442. S2CID 3074096.
Dreyfus, Stuart E. (1990). "Artificial Neural Networks, Back Propagation, and the Kelley-Bryson Gradient Procedure". Journal of Guidance, Control, and Dynamics. 13 (5): 926–928. Bibcode:1990JGCD...13..926D. doi:10.2514/3.25422.
Rumelhart; Hinton; Williams (1986). "Learning representations by back-propagating errors" (PDF). Nature. 323 (6088): 533–536. Bibcode:1986Natur.323..533R. doi:10.1038/323533a0. S2CID 205001834.

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Sutton, Richard S.; Barto, Andrew G. (2018). "11.1 TD-Gammon". Reinforcement Learning: An Introduction (2nd ed.). Cambridge, MA: MIT Press.

janbasktraining.com

"Decoding the Power of Backpropagation: A Deep Dive into Advanced Neural Network Techniques". janbasktraining.com. 30 January 2024.

mit.edu

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Anderson, James A.; Rosenfeld, Edward, eds. (2000). Talking Nets: An Oral History of Neural Networks. The MIT Press. doi:10.7551/mitpress/6626.003.0016. ISBN 978-0-262-26715-1.

neuralnetworksanddeeplearning.com

This follows Nielsen (2015), and means (left) multiplication by the matrix $W^{l}$ corresponds to converting output values of layer $l-1$ to input values of layer $l$ : columns correspond to input coordinates, rows correspond to output coordinates. Nielsen, Michael A. (2015). "How the backpropagation algorithm works". Neural Networks and Deep Learning. Determination Press.
This section largely follows and summarizes Nielsen (2015). Nielsen, Michael A. (2015). "How the backpropagation algorithm works". Neural Networks and Deep Learning. Determination Press.
Nielsen (2015), "[W]hat assumptions do we need to make about our cost function ... in order that backpropagation can be applied? The first assumption we need is that the cost function can be written as an average ... over cost functions ... for individual training examples ... The second assumption we make about the cost is that it can be written as a function of the outputs from the neural network ..." Nielsen, Michael A. (2015). "How the backpropagation algorithm works". Neural Networks and Deep Learning. Determination Press.

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LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning" (PDF). Nature. 521 (7553): 436–444. Bibcode:2015Natur.521..436L. doi:10.1038/nature14539. PMID 26017442. S2CID 3074096.
Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. arXiv:1404.7828. doi:10.1016/j.neunet.2014.09.003. PMID 25462637. S2CID 11715509.
Chang, Franklin; Dell, Gary S.; Bock, Kathryn (2006). "Becoming syntactic". Psychological Review. 113 (2): 234–272. doi:10.1037/0033-295x.113.2.234. PMID 16637761.
Janciauskas, Marius; Chang, Franklin (2018). "Input and Age-Dependent Variation in Second Language Learning: A Connectionist Account". Cognitive Science. 42 (Suppl Suppl 2): 519–554. doi:10.1111/cogs.12519. PMC 6001481. PMID 28744901.
Fitz, Hartmut; Chang, Franklin (2019). "Language ERPs reflect learning through prediction error propagation". Cognitive Psychology. 111: 15–52. doi:10.1016/j.cogpsych.2019.03.002. hdl:21.11116/0000-0003-474D-8. PMID 30921626. S2CID 85501792.

ncbi.nlm.nih.gov

Janciauskas, Marius; Chang, Franklin (2018). "Input and Age-Dependent Variation in Second Language Learning: A Connectionist Account". Cognitive Science. 42 (Suppl Suppl 2): 519–554. doi:10.1111/cogs.12519. PMC 6001481. PMID 28744901.

semanticscholar.org

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Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (1986a). "Learning representations by back-propagating errors". Nature. 323 (6088): 533–536. Bibcode:1986Natur.323..533R. doi:10.1038/323533a0. S2CID 205001834.
Tan, Hong Hui; Lim, King Han (2019). "Review of second-order optimization techniques in artificial neural networks backpropagation". IOP Conference Series: Materials Science and Engineering. 495 (1): 012003. Bibcode:2019MS&E..495a2003T. doi:10.1088/1757-899X/495/1/012003. S2CID 208124487.
LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning" (PDF). Nature. 521 (7553): 436–444. Bibcode:2015Natur.521..436L. doi:10.1038/nature14539. PMID 26017442. S2CID 3074096.
Rodríguez, Omar Hernández; López Fernández, Jorge M. (2010). "A Semiotic Reflection on the Didactics of the Chain Rule". The Mathematics Enthusiast. 7 (2): 321–332. doi:10.54870/1551-3440.1191. S2CID 29739148. Retrieved 2019-08-04.
Linnainmaa, Seppo (1976). "Taylor expansion of the accumulated rounding error". BIT Numerical Mathematics. 16 (2): 146–160. doi:10.1007/bf01931367. S2CID 122357351.
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Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. arXiv:1404.7828. doi:10.1016/j.neunet.2014.09.003. PMID 25462637. S2CID 11715509.
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Fitz, Hartmut; Chang, Franklin (2019). "Language ERPs reflect learning through prediction error propagation". Cognitive Psychology. 111: 15–52. doi:10.1016/j.cogpsych.2019.03.002. hdl:21.11116/0000-0003-474D-8. PMID 30921626. S2CID 85501792.

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Rumelhart; Hinton; Williams (1986). "Learning representations by back-propagating errors" (PDF). Nature. 323 (6088): 533–536. Bibcode:1986Natur.323..533R. doi:10.1038/323533a0. S2CID 205001834.

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Rodríguez, Omar Hernández; López Fernández, Jorge M. (2010). "A Semiotic Reflection on the Didactics of the Chain Rule". The Mathematics Enthusiast. 7 (2): 321–332. doi:10.54870/1551-3440.1191. S2CID 29739148. Retrieved 2019-08-04.

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Werbos, Paul (1982). "Applications of advances in nonlinear sensitivity analysis" (PDF). System modeling and optimization. Springer. pp. 762–770. Archived (PDF) from the original on 14 April 2016. Retrieved 2 July 2017.

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Mizutani, Eiji; Dreyfus, Stuart; Nishio, Kenichi (July 2000). "On derivation of MLP backpropagation from the Kelley-Bryson optimal-control gradient formula and its application" (PDF). Proceedings of the IEEE International Joint Conference on Neural Networks.