Backpropagation network
Introduction
[edit  edit source]Backpropagation can also refer to the way the result of a playout is propagated up the search tree in Monte Carlo tree search.
As a machinelearning algorithm, backpropagation performs a backward pass to adjust a neural network model's parameters, aiming to minimize the mean squared error (MSE).^{[1]}^{[2]} In a multilayered network, backpropagation uses the following steps:
Propagate training data
[edit  edit source]Propagate training data through the model from input to predicted output by computing the successive hidden layers' outputs and finally the final layer's output (the feedforward step).
Adjust the model weights
[edit  edit source]Adjust the model weights to reduce the error relative to the weights.
 The error is typically the squared difference between prediction and target.
 For each weight, the slope or derivative of the error is found, and the weight adjusted by a negative multiple of this derivative, so as to go downslope toward the minimumerror configuration.
 This derivative is easy to calculate for final layer weights, and possible to calculate for one layer given the next layer's derivatives. Starting at the end, then, the derivatives are calculated layer by layer toward the beginning  thus "backpropagation".
Update Weights
[edit  edit source]Repeatedly update the weights until they converge or the model has undergone enough iterations.
Origine of basic Idea
[edit  edit source]It is an efficient application of the Leibniz chain rule (1673)^{[3]} to such networks.^{[4]} It is also known as the reverse mode of automatic differentiation or reverse accumulation, due to Seppo Linnainmaa (1970).^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]} The term "backpropagating error correction" was introduced in 1962 by Frank Rosenblatt,^{[12]}^{[4]} but he did not know how to implement this, even though Henry J. Kelley had a continuous precursor of backpropagation^{[13]} already in 1960 in the context of control theory.^{[4]}
Gradient Descent
[edit  edit source]Backpropagation computes the gradient of a loss function with respect to the weights of the network for a single input–output example, and does so efficiently, computing the gradient one layer at a time, iterating backward from the last layer to avoid redundant calculations of intermediate terms in the chain rule; this can be derived through dynamic programming.^{[13]}^{[14]}^{[15]} Gradient descent, or variants such as stochastic gradient descent,^{[16]} are commonly used.
Strictly the term backpropagation refers only to the algorithm for computing the gradient, not how the gradient is used; but the term is often used loosely to refer to the entire learning algorithm – including how the gradient is used, such as by stochastic gradient descent.^{[17]} In 1986 David E. Rumelhart et al. published an experimental analysis of the technique.^{[18]} This contributed to the popularization of backpropagation and helped to initiate an active period of research in multilayer perceptrons.
Overview
[edit  edit source]Backpropagation computes the gradient in weight space of a feedforward neural network, with respect to a loss function. Denote:
 : input (vector of features)
 : target output
 For classification, output will be a vector of class probabilities (e.g., , and target output is a specific class, encoded by the onehot/dummy variable (e.g., ).
 : loss function or "cost function"^{[a]}
 For classification, this is usually crossentropy (XC, log loss), while for regression it is usually squared error loss (SEL).
 : the number of layers
 : the weights between layer and , where is the weight between the th node in layer and the th node in layer ^{[b]}
 : activation functions at layer
 For classification the last layer is usually the logistic function for binary classification, and softmax (softargmax) for multiclass classification, while for the hidden layers this was traditionally a sigmoid function (logistic function or others) on each node (coordinate), but today is more varied, with rectifier (ramp, ReLU) being common.
 : activation of the th node in layer .
In the derivation of backpropagation, other intermediate quantities are used by introducing them as needed below. Bias terms are not treated specially since they correspond to a weight with a fixed input of 1. For backpropagation the specific loss function and activation functions do not matter as long as they and their derivatives can be evaluated efficiently. Traditional activation functions include sigmoid, tanh, and ReLU. swish^{[19]} mish,^{[20]} and other activation functions have since been proposed as well.
The overall network is a combination of function composition and matrix multiplication:
For a training set there will be a set of input–output pairs, . For each input–output pair in the training set, the loss of the model on that pair is the cost of the difference between the predicted output and the target output :
Note the distinction: during model evaluation the weights are fixed while the inputs vary (and the target output may be unknown), and the network ends with the output layer (it does not include the loss function). During model training the input–output pair is fixed while the weights vary, and the network ends with the loss function.
Backpropagation computes the gradient for a fixed input–output pair , where the weights can vary. Each individual component of the gradient, can be computed by the chain rule; but doing this separately for each weight is inefficient. Backpropagation efficiently computes the gradient by avoiding duplicate calculations and not computing unnecessary intermediate values, by computing the gradient of each layer – specifically the gradient of the weighted input of each layer, denoted by – from back to front.
Informally, the key point is that since the only way a weight in affects the loss is through its effect on the next layer, and it does so linearly, are the only data you need to compute the gradients of the weights at layer , and then the previous layer can be computed and repeated recursively. This avoids inefficiency in two ways. First, it avoids duplication because when computing the gradient at layer – it is unnecessary to recompute all derivatives on later layers each time. Second, it avoids unnecessary intermediate calculations, because at each stage it directly computes the gradient of the weights with respect to the ultimate output (the loss), rather than unnecessarily computing the derivatives of the values of hidden layers with respect to changes in weights .
Backpropagation can be expressed for simple feedforward networks in terms of matrix multiplication, or more generally in terms of the adjoint graph.
Matrix multiplication
[edit  edit source]For the basic case of a feedforward network, where nodes in each layer are connected only to nodes in the immediate next layer (without skipping any layers), and there is a loss function that computes a scalar loss for the final output, backpropagation can be understood simply by matrix multiplication.^{[c]} Essentially, backpropagation evaluates the expression for the derivative of the cost function as a product of derivatives between each layer from right to left – "backwards" – with the gradient of the weights between each layer being a simple modification of the partial products (the "backwards propagated error").
Given an input–output pair , the loss is:
To compute this, one starts with the input and works forward; denote the weighted input of each hidden layer as and the output of hidden layer as the activation . For backpropagation, the activation as well as the derivatives (evaluated at ) must be cached for use during the backwards pass.
The derivative of the loss in terms of the inputs is given by the chain rule; note that each term is a total derivative, evaluated at the value of the network (at each node) on the input :
where is a diagonal matrix.
These terms are: the derivative of the loss function;^{[d]} the derivatives of the activation functions;^{[e]} and the matrices of weights:^{[f]}
The gradient is the transpose of the derivative of the output in terms of the input, so the matrices are transposed and the order of multiplication is reversed, but the entries are the same:
Backpropagation then consists essentially of evaluating this expression from right to left (equivalently, multiplying the previous expression for the derivative from left to right), computing the gradient at each layer on the way; there is an added step, because the gradient of the weights is not just a subexpression: there's an extra multiplication.
Introducing the auxiliary quantity for the partial products (multiplying from right to left), interpreted as the "error at level " and defined as the gradient of the input values at level :
Note that is a vector, of length equal to the number of nodes in level ; each component is interpreted as the "cost attributable to (the value of) that node".
The gradient of the weights in layer is then:
The factor of is because the weights between level and affect level proportionally to the inputs (activations): the inputs are fixed, the weights vary.
The can easily be computed recursively, going from right to left, as:
The gradients of the weights can thus be computed using a few matrix multiplications for each level; this is backpropagation.
Compared with naively computing forwards (using the for illustration):
there are two key differences with backpropagation:
 Computing in terms of avoids the obvious duplicate multiplication of layers and beyond.
 Multiplying starting from – propagating the error backwards – means that each step simply multiplies a vector () by the matrices of weights and derivatives of activations . By contrast, multiplying forwards, starting from the changes at an earlier layer, means that each multiplication multiplies a matrix by a matrix. This is much more expensive, and corresponds to tracking every possible path of a change in one layer forward to changes in the layer (for multiplying by , with additional multiplications for the derivatives of the activations), which unnecessarily computes the intermediate quantities of how weight changes affect the values of hidden nodes.
Adjoint graph
[edit  edit source]You can help expand this section. 
For more general graphs, and other advanced variations, backpropagation can be understood in terms of automatic differentiation, where backpropagation is a special case of reverse accumulation (or "reverse mode").^{[11]}
Intuition
[edit  edit source]Motivation
[edit  edit source]The goal of any supervised learning algorithm is to find a function that best maps a set of inputs to their correct output. The motivation for backpropagation is to train a multilayered neural network such that it can learn the appropriate internal representations to allow it to learn any arbitrary mapping of input to output.^{[21]}
Learning as an optimization problem
[edit  edit source]To understand the mathematical derivation of the backpropagation algorithm, it helps to first develop some intuition about the relationship between the actual output of a neuron and the correct output for a particular training example. Consider a simple neural network with two input units, one output unit and no hidden units, and in which each neuron uses a linear output (unlike most work on neural networks, in which mapping from inputs to outputs is nonlinear)^{[g]} that is the weighted sum of its input.
Initially, before training, the weights will be set randomly. Then the neuron learns from training examples, which in this case consist of a set of tuples where and are the inputs to the network and t is the correct output (the output the network should produce given those inputs, when it has been trained). The initial network, given and , will compute an output y that likely differs from t (given random weights). A loss function is used for measuring the discrepancy between the target output t and the computed output y. For regression analysis problems the squared error can be used as a loss function, for classification the categorical crossentropy can be used.
As an example consider a regression problem using the square error as a loss:
where E is the discrepancy or error.
Consider the network on a single training case: . Thus, the input and are 1 and 1 respectively and the correct output, t is 0. Now if the relation is plotted between the network's output y on the horizontal axis and the error E on the vertical axis, the result is a parabola. The minimum of the parabola corresponds to the output y which minimizes the error E. For a single training case, the minimum also touches the horizontal axis, which means the error will be zero and the network can produce an output y that exactly matches the target output t. Therefore, the problem of mapping inputs to outputs can be reduced to an optimization problem of finding a function that will produce the minimal error.
However, the output of a neuron depends on the weighted sum of all its inputs:
where and are the weights on the connection from the input units to the output unit. Therefore, the error also depends on the incoming weights to the neuron, which is ultimately what needs to be changed in the network to enable learning.
In this example, upon injecting the training data , the loss function becomes
Then, the loss function takes the form of a parabolic cylinder with its base directed along . Since all sets of weights that satisfy minimize the loss function, in this case additional constraints are required to converge to a unique solution. Additional constraints could either be generated by setting specific conditions to the weights, or by injecting additional training data.
One commonly used algorithm to find the set of weights that minimizes the error is gradient descent. By backpropagation, the steepest descent direction is calculated of the loss function versus the present synaptic weights. Then, the weights can be modified along the steepest descent direction, and the error is minimized in an efficient way.
Derivation
[edit  edit source]The gradient descent method involves calculating the derivative of the loss function with respect to the weights of the network. This is normally done using backpropagation. Assuming one output neuron,^{[h]} the squared error function is
where
 is the loss for the output and target value ,
 is the target output for a training sample, and
 is the actual output of the output neuron.
For each neuron , its output is defined as
where the activation function is nonlinear and differentiable over the activation region (the ReLU is not differentiable at one point). A historically used activation function is the logistic function:
which has a convenient derivative of:
The input to a neuron is the weighted sum of outputs of previous neurons. If the neuron is in the first layer after the input layer, the of the input layer are simply the inputs to the network. The number of input units to the neuron is . The variable denotes the weight between neuron of the previous layer and neuron of the current layer.
Finding the derivative of the error
[edit  edit source]Calculating the partial derivative of the error with respect to a weight is done using the chain rule twice:

(
)
In the last factor of the righthand side of the above, only one term in the sum depends on , so that

(
)
If the neuron is in the first layer after the input layer, is just .
The derivative of the output of neuron with respect to its input is simply the partial derivative of the activation function:

(
)
which for the logistic activation function
This is the reason why backpropagation requires that the activation function be differentiable. (Nevertheless, the ReLU activation function, which is nondifferentiable at 0, has become quite popular, e.g. in AlexNet)
The first factor is straightforward to evaluate if the neuron is in the output layer, because then and

(
)
If half of the square error is used as loss function we can rewrite it as
However, if is in an arbitrary inner layer of the network, finding the derivative with respect to is less obvious.
Considering as a function with the inputs being all neurons receiving input from neuron ,
and taking the total derivative with respect to , a recursive expression for the derivative is obtained:

(
)
Therefore, the derivative with respect to can be calculated if all the derivatives with respect to the outputs of the next layer – the ones closer to the output neuron – are known. [Note, if any of the neurons in set were not connected to neuron , they would be independent of and the corresponding partial derivative under the summation would vanish to 0.]
Substituting Eq. 2, Eq. 3 Eq.4 and Eq. 5 in Eq. 1 we obtain:
with
if is the logistic function, and the error is the square error:
To update the weight using gradient descent, one must choose a learning rate, . The change in weight needs to reflect the impact on of an increase or decrease in . If , an increase in increases ; conversely, if , an increase in decreases . The new is added to the old weight, and the product of the learning rate and the gradient, multiplied by guarantees that changes in a way that always decreases . In other words, in the equation immediately below, always changes in such a way that is decreased:
Secondorder gradient descent
[edit  edit source]Using a Hessian matrix of secondorder derivatives of the error function, the Levenberg–Marquardt algorithm often converges faster than firstorder gradient descent, especially when the topology of the error function is complicated.^{[22]}^{[23]} It may also find solutions in smaller node counts for which other methods might not converge.^{[23]} The Hessian can be approximated by the Fisher information matrix.^{[24]}
Loss function
[edit  edit source]The loss function is a function that maps values of one or more variables onto a real number intuitively representing some "cost" associated with those values. For backpropagation, the loss function calculates the difference between the network output and its expected output, after a training example has propagated through the network.
Assumptions
[edit  edit source]The mathematical expression of the loss function must fulfill two conditions in order for it to be possibly used in backpropagation.^{[25]} The first is that it can be written as an average over error functions , for individual training examples, . The reason for this assumption is that the backpropagation algorithm calculates the gradient of the error function for a single training example, which needs to be generalized to the overall error function. The second assumption is that it can be written as a function of the outputs from the neural network.
Example loss function
[edit  edit source]Let be vectors in .
Select an error function measuring the difference between two outputs. The standard choice is the square of the Euclidean distance between the vectors and :The error function over training examples can then be written as an average of losses over individual examples:
Limitations
[edit  edit source] Gradient descent with backpropagation is not guaranteed to find the global minimum of the error function, but only a local minimum; also, it has trouble crossing plateaus in the error function landscape. This issue, caused by the nonconvexity of error functions in neural networks, was long thought to be a major drawback, but Yann LeCun et al. argue that in many practical problems, it is not.^{[26]}
 Backpropagation learning does not require normalization of input vectors; however, normalization could improve performance.^{[27]}
 Backpropagation requires the derivatives of activation functions to be known at network design time.
History
[edit  edit source]Modern backpropagation is Seppo Linnainmaa's reverse mode of automatic differentiation (1970) for discrete connected networks of nested differentiable functions.^{[5]}^{[6]}^{[9]}^{[10]}^{[7]}^{[8]} It is an efficient application of the chain rule (derived by Gottfried Wilhelm Leibniz in 1673^{[3]}^{[28]}) to such networks.^{[4]} The terminology "backpropagating error correction" was introduced in 1962 by Frank Rosenblatt,^{[29]}^{[4]} but he did not know how to implement this, although Henry J. Kelley had a continuous precursor of backpropagation^{[13]} already in 1960 in the context of control theory.^{[4]} The first deep learning multilayer perceptron (MLP) trained by stochastic gradient descent^{[16]} was published in 1967 by Shun'ichi Amari.^{[30]}^{[4]} In computer experiments, his five layer MLP with two modifiable layers learned internal representations required to classify nonlinearily separable pattern classes.^{[4]} In 1982, Paul Werbos applied backpropagation to MLPs in the way that has become standard.^{[31]}^{[32]}^{[4]} In 1985, David E. Rumelhart et al. published an experimental analysis of the technique.^{[18]} This contributed to the popularization of backpropagation and helped to initiate an active period of research in multilayer perceptrons.^{[21]}^{[33]}^{[34]}
Kelley (1960)^{[13]} and Arthur E. Bryson (1961)^{[14]} used principles of dynamic programming to derive the abovementioned continuous precursor of the method. In 1962, Stuart Dreyfus published a simpler derivation based only on the chain rule.^{[35]}^{[36]}^{[37]}^{[9]}^{[10]} In 1973, he adapted parameters of controllers in proportion to error gradients.^{[38]} Unlike Linnainmaa's 1970 method,^{[5]}^{[7]} these precursors used "standard Jacobian matrix calculations from one stage to the previous one, neither addressing direct links across several stages nor potential additional efficiency gains due to network sparsity."^{[4]}
In 1985, the method was also described by Parker.^{[39]}^{[40]} Yann LeCun proposed an alternative form of backpropagation for neural networks in his PhD thesis in 1987. In 1993, Eric Wan won an international pattern recognition contest through backpropagation.^{[9]}^{[41]}
During the 2000s it fell out of favourGPUbased computing systems. This has been especially so in speech recognition, machine vision, natural language processing, and language structure learning research (in which it has been used to explain a variety of phenomena related to first^{[42]} and second language learning.^{[43]}).
, but returned in the 2010s, benefitting from cheap, powerfulError backpropagation has been suggested to explain human brain ERP components like the N400 and P600.^{[44]}
In 2023, a backpropagation algorithm was implemented on a photonic processor by a team at Stanford University.^{[45]}
See also
[edit  edit source] Open Machine Learning
 Artificial neural network
 Neural circuit
 Catastrophic interference
 Ensemble learning
 AdaBoost
 Overfitting
 Neural backpropagation
 Backpropagation through time
Notes
[edit  edit source] ↑ Use for the loss function to allow to be used for the number of layers
 ↑ This follows Nielsen (2015), and means (left) multiplication by the matrix corresponds to converting output values of layer to input values of layer : columns correspond to input coordinates, rows correspond to output coordinates.
 ↑ This section largely follows and summarizes Nielsen (2015).
 ↑ The derivative of the loss function is a covector, since the loss function is a scalarvalued function of several variables.
 ↑ The activation function is applied to each node separately, so the derivative is just the diagonal matrix of the derivative on each node. This is often represented as the Hadamard product with the vector of derivatives, denoted by , which is mathematically identical but better matches the internal representation of the derivatives as a vector, rather than a diagonal matrix.
 ↑ Since matrix multiplication is linear, the derivative of multiplying by a matrix is just the matrix: .
 ↑ One may notice that multilayer neural networks use nonlinear activation functions, so an example with linear neurons seems obscure. However, even though the error surface of multilayer networks are much more complicated, locally they can be approximated by a paraboloid. Therefore, linear neurons are used for simplicity and easier understanding.
 ↑ There can be multiple output neurons, in which case the error is the squared norm of the difference vector.
References
[edit  edit source] ↑ Goodfellow, Bengio & Courville 2016, p. 200, "Furthermore, backpropagation is often misunderstood as being specific to multilayer neural networks, but in principle it can compute derivatives of any function"
 ↑ Graves, Alex; Wayne, Greg; Danihelka, Ivo (2014). "Neural Turing Machines". arXiv:1410.5401 [cs.NE].
 ↑ ^{3.0} ^{3.1} Leibniz, Gottfried Wilhelm Freiherr von (1920). The Early Mathematical Manuscripts of Leibniz: Translated from the Latin Texts Published by Carl Immanuel Gerhardt with Critical and Historical Notes (Leibniz published the chain rule in a 1676 memoir) (in en). Open court publishing Company. ISBN 9780598818461. https://books.google.com/books?id=bOIGAAAAYAAJ&q=leibniz+altered+manuscripts&pg=PA90.
 ↑ ^{4.00} ^{4.01} ^{4.02} ^{4.03} ^{4.04} ^{4.05} ^{4.06} ^{4.07} ^{4.08} ^{4.09} Schmidhuber, Juergen (2022). "Annotated History of Modern AI and Deep Learning". arXiv:2212.11279 [cs.NE].
 ↑ ^{5.0} ^{5.1} ^{5.2} Linnainmaa, Seppo (1970). The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors (Masters) (in Finnish). University of Helsinki. pp. 6–7.
 ↑ ^{6.0} ^{6.1} Linnainmaa, Seppo (1976). "Taylor expansion of the accumulated rounding error". BIT Numerical Mathematics 16 (2): 146–160. doi:10.1007/bf01931367.
 ↑ ^{7.0} ^{7.1} ^{7.2} Griewank, Andreas (2012). "Who Invented the Reverse Mode of Differentiation?". Optimization Stories. Documenta Matematica, Extra Volume ISMP. pp. 389–400.
 ↑ ^{8.0} ^{8.1} Griewank, Andreas; Walther, Andrea (2008). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition. SIAM. ISBN 9780898717761. https://books.google.com/books?id=xoiiLaRxcbEC.
 ↑ ^{9.0} ^{9.1} ^{9.2} ^{9.3} Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks 61: 85–117. doi:10.1016/j.neunet.2014.09.003. PMID 25462637.
 ↑ ^{10.0} ^{10.1} ^{10.2} Schmidhuber, Jürgen (2015). "Deep Learning". Scholarpedia 10 (11): 32832. doi:10.4249/scholarpedia.32832.
 ↑ ^{11.0} ^{11.1} Goodfellow, Bengio & Courville (2016, p. 217–218), "The backpropagation 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."
 ↑ Rosenblatt, Frank (1962). Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms Cornell Aeronautical Laboratory. Report no. VG1196G8 Report (Cornell Aeronautical Laboratory). Spartan. pp. Page XIII Table of contents, Page 292 "13.3 BackPropagating Error Correction Procedures" , Page 301 "figure 39 BACKPROPAGATING ERRORCORRECTION EXPERIMENTS".
 ↑ ^{13.0} ^{13.1} ^{13.2} ^{13.3} Kelley, Henry J. (1960). "Gradient theory of optimal flight paths". ARS Journal 30 (10): 947–954. doi:10.2514/8.5282.
 ↑ ^{14.0} ^{14.1} Bryson, Arthur E. (1962). "A gradient method for optimizing multistage allocation processes". Proceedings of the Harvard Univ. Symposium on digital computers and their applications, 3–6 April 1961. Cambridge: Harvard University Press. OCLC 498866871.
 ↑ Goodfellow, Bengio & Courville 2016, p. 214, "This tablefilling strategy is sometimes called dynamic programming."
 ↑ ^{16.0} ^{16.1} Robbins, H.; Monro, S. (1951). "A Stochastic Approximation Method". The Annals of Mathematical Statistics 22 (3): 400. doi:10.1214/aoms/1177729586.
 ↑ Goodfellow, Bengio & Courville 2016, p. 200, "The term backpropagation 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."
 ↑ ^{18.0} ^{18.1} Rumelhart; Hinton; Williams (1986). "Learning representations by backpropagating errors". Nature 323 (6088): 533–536. doi:10.1038/323533a0. http://www.cs.toronto.edu/~hinton/absps/naturebp.pdf.
 ↑ Ramachandran, Prajit; Zoph, Barret; Le, Quoc V. (20171027). "Searching for Activation Functions". arXiv:1710.05941 [cs.NE].
 ↑ Misra, Diganta (20190823). "Mish: A Self Regularized NonMonotonic Activation Function". arXiv:1908.08681 [cs.LG].
 ↑ ^{21.0} ^{21.1} Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (1986a). "Learning representations by backpropagating errors". Nature 323 (6088): 533–536. doi:10.1038/323533a0.
 ↑ Tan, Hong Hui; Lim, King Han (2019). "Review of secondorder optimization techniques in artificial neural networks backpropagation". IOP Conference Series: Materials Science and Engineering 495 (1): 012003. doi:10.1088/1757899X/495/1/012003.
 ↑ ^{23.0} ^{23.1} Wiliamowski, Bogdan; Yu, Hao (June 2010). "Improved Computation for Levenberg–Marquardt Training". IEEE Transactions on Neural Networks and Learning Systems 21 (6). https://www.eng.auburn.edu/~wilambm/pap/2010/Improved%20Computation%20for%20LM%20Training.pdf.
 ↑ Martens, James (August 2020). "New Insights and Perspectives on the Natural Gradient Method". Journal of Machine Learning Research (21).
 ↑ 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 ..."
 ↑ LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning". Nature 521 (7553): 436–444. doi:10.1038/nature14539. PMID 26017442.
 ↑ Buckland, Matt; Collins, Mark (2002). AI Techniques for Game Programming. Boston: Premier Press. ISBN 193184108X.
 ↑ 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:10.54870/15513440.1191. https://scholarworks.umt.edu/tme/vol7/iss2/10/. Retrieved 20190804.
 ↑ Rosenblatt, Frank (1962). Principles of Neurodynamics. Spartan, New York.
 ↑ Amari, Shun'ichi (1967). "A theory of adaptive pattern classifier". IEEE Transactions EC (16): 279–307.
 ↑ Werbos, Paul (1982). "Applications of advances in nonlinear sensitivity analysis". System modeling and optimization. Springer. pp. 762–770. http://werbos.com/Neural/SensitivityIFIPSeptember1981.pdf. Retrieved 2 July 2017.
 ↑ Werbos, Paul J. (1994). The Roots of Backpropagation : From Ordered Derivatives to Neural Networks and Political Forecasting. New York: John Wiley & Sons. ISBN 0471598976.
 ↑ Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (1986b). "8. Learning Internal Representations by Error Propagation". In Rumelhart, David E.; McClelland, James L.. Parallel Distributed Processing : Explorations in the Microstructure of Cognition. 1 : Foundations. Cambridge: MIT Press. ISBN 0262181207. https://archive.org/details/paralleldistribu00rume.
 ↑ Alpaydin, Ethem (2010). Introduction to Machine Learning. MIT Press. ISBN 9780262012430. https://books.google.com/books?id=4j9GAQAAIAAJ.
 ↑ Dreyfus, Stuart (1962). "The numerical solution of variational problems". Journal of Mathematical Analysis and Applications 5 (1): 30–45. doi:10.1016/0022247x(62)900045.
 ↑ Dreyfus, Stuart E. (1990). "Artificial Neural Networks, Back Propagation, and the KelleyBryson Gradient Procedure". Journal of Guidance, Control, and Dynamics 13 (5): 926–928. doi:10.2514/3.25422.
 ↑ Mizutani, Eiji; Dreyfus, Stuart; Nishio, Kenichi (July 2000). "On derivation of MLP backpropagation from the KelleyBryson optimalcontrol gradient formula and its application" (PDF). Proceedings of the IEEE International Joint Conference on Neural Networks.
 ↑ Dreyfus, Stuart (1973). "The computational solution of optimal control problems with time lag". IEEE Transactions on Automatic Control 18 (4): 383–385. doi:10.1109/tac.1973.1100330.
 ↑ Parker, D.B. (1985). Learning Logic. Cambridge MA: Massachusetts Institute of Technology.
 ↑ Hertz, John (1991). Introduction to the theory of neural computation. Krogh, Anders., Palmer, Richard G.. Redwood City, Calif.: AddisonWesley. pp. 8. ISBN 0201503956. OCLC 21522159.
 ↑ Wan, Eric A. (1994). "Time Series Prediction by Using a Connectionist Network with Internal Delay Lines". In Weigend, Andreas S.; Gershenfeld, Neil A.. Time Series Prediction : Forecasting the Future and Understanding the Past. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis. 15. Reading: AddisonWesley. pp. 195–217. ISBN 0201626012.
 ↑ Chang, Franklin; Dell, Gary S.; Bock, Kathryn (2006). "Becoming syntactic.". Psychological Review 113 (2): 234–272. doi:10.1037/0033295x.113.2.234. PMID 16637761.
 ↑ Janciauskas, Marius; Chang, Franklin (2018). "Input and AgeDependent Variation in Second Language Learning: A Connectionist Account". Cognitive Science 42 (Suppl Suppl 2): 519–554. doi:10.1111/cogs.12519. PMID 28744901. PMC 6001481. //www.ncbi.nlm.nih.gov/pmc/articles/PMC6001481/.
 ↑ 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. PMID 30921626.
 ↑ "Photonic Chips Curb AI Training's Energy Appetite  IEEE Spectrum". spectrum.ieee.org. Retrieved 20230525.
Further reading
[edit  edit source] Goodfellow, Ian; Bengio, Yoshua; Courville, Aaron (2016). Deep Learning. MIT Press. pp. 200–220. ISBN 9780262035613. http://www.deeplearningbook.org.
 Nielsen, Michael A. (2015). "How the backpropagation algorithm works". Neural Networks and Deep Learning. Determination Press. http://neuralnetworksanddeeplearning.com/chap2.html.
 McCaffrey, James (October 2012). "Neural Network BackPropagation for Programmers". MSDN Magazine.
 Rojas, Raúl (1996). "The Backpropagation Algorithm". Neural Networks : A Systematic Introduction. Berlin: Springer. ISBN 3540605053. https://page.mi.fuberlin.de/rojas/neural/chapter/K7.pdf.
External links
[edit  edit source] Backpropagation neural network tutorial at the Wikiversity
 Bernacki, Mariusz; Włodarczyk, Przemysław (2004). "Principles of training multilayer neural network using backpropagation".
 Karpathy, Andrej (2016). "Lecture 4: Backpropagation, Neural Networks 1". CS231n. Stanford University. Archived from the original on 20211212 – via YouTube.Template:Cbignore
 "What is Backpropagation Really Doing?". 3Blue1Brown. November 3, 2017. Archived from the original on 20211212 – via YouTube.Template:Cbignore
 Putta, Sudeep Raja (2022). "Yet Another Derivation of Backpropagation in Matrix Form".
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