It is an efficient application of thechain rule to neural networks. Backpropagation computes the gradient of aloss function with respect to theweights of the network for a single input–output example, and does soefficiently, 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 throughdynamic programming.[1][2][3]
Strictly speaking, the termbackpropagation refers only to an algorithm for efficiently computing the gradient, not how the gradient is used; but the term is often used loosely to refer to the entire learning algorithm. This includes changing model parameters in the negative direction of the gradient, such as bystochastic gradient descent, or as an intermediate step in a more complicated optimizer, such asAdaptive Moment Estimation.[4]
Backpropagation had multiple discoveries and partial discoveries, with a tangled history and terminology. See thehistory section for details. Some other names for the technique include "reverse mode ofautomatic differentiation" or "reverse accumulation".[5]
Backpropagation computes the gradient inweight space of a feedforward neural network, with respect to aloss 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 theone-hot/dummy variable (e.g.,).
For classification the last layer is usually thelogistic function for binary classification, andsoftmax (softargmax) for multi-class classification, while for the hidden layers this was traditionally asigmoid function (logistic function or others) on each node (coordinate), but today is more varied, withrectifier (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, andReLU.Swish,[6]Mish,[7] and many others.
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 afixed 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 weightedinput 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 thenext layer, and it does solinearly, are the only data you need to compute the gradients of the weights at layer, and then the gradients of weights of previous layer can be computed by 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 ofmatrix multiplication, or more generally in terms of theadjoint graph.
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 layerfrom 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 atotal derivative, evaluated at the value of the network (at each node) on the input:
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 thetranspose 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 errorbackwards – 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 amatrix by amatrix. 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.
For more general graphs, and other advanced variations, backpropagation can be understood in terms ofautomatic differentiation, where backpropagation is a special case ofreverse accumulation (or "reverse mode").[5]
The goal of anysupervised 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 multi-layered neural network such that it can learn the appropriate internal representations to allow it to learn any arbitrary mapping of input to output.[8]
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 alinear output (unlike most work on neural networks, in which mapping from inputs to outputs is non-linear)[g] that is the weighted sum of its input.
A simple neural network with two input units (each with a single input) and one output unit (with two inputs)
Initially, before training, the weights will be set randomly. Then the neuron learns fromtraining examples, which in this case consist of a set oftuples where and are the inputs to the network andt 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 outputy that likely differs fromt (given random weights). Aloss function is used for measuring the discrepancy between the target outputt and the computed outputy. Forregression analysis problems the squared error can be used as a loss function, forclassification thecategorical cross-entropy can be used.
As an example consider a regression problem using the square error as a loss:
whereE 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 outputy on the horizontal axis and the errorE on the vertical axis, the result is a parabola. Theminimum of theparabola corresponds to the outputy which minimizes the errorE. 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 outputy that exactly matches the target outputt. Therefore, the problem of mapping inputs to outputs can be reduced to anoptimization problem of finding a function that will produce the minimal error.
Error surface of a linear neuron for a single training case
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 isgradient 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.
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.
In this section, the order of the weight indexes are reversed relative to the prior section: is weight from theth to theth unit.[i] For each neuron, its output is defined as
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.
This is the reason why backpropagation requires that the activation function bedifferentiable. (Nevertheless, theReLU activation function, which is non-differentiable at 0, has become quite popular, e.g. inAlexNet)
The first factor is straightforward to evaluate if the neuron is in the output layer, because then and
Eq. 4
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 thetotal derivative with respect to, a recursive expression for the derivative is obtained:
Eq. 5
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.]
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:
Using aHessian matrix of second-order derivatives of the error function, theLevenberg–Marquardt algorithm often converges faster than first-order gradient descent, especially when the topology of the error function is complicated.[9][10] It may also find solutions in smaller node counts for which other methods might not converge.[10] The Hessian can be approximated by theFisher information matrix.[11]
As an example, consider a simple feedforward network. At the-th layer, we havewhere are the pre-activations, are the activations, and is the weight matrix. Given a loss function, the first-order backpropagation states thatand the second-order backpropagation states thatwhere is theDirac delta symbol.
Arbitrary-order derivatives in arbitrary computational graphs can be computed with backpropagation, but with more complex expressions for higher orders.
The loss function is a function that maps values of one or more variables onto areal 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.
The mathematical expression of the loss function must fulfill two conditions in order for it to be possibly used in backpropagation.[12] 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.
Select an error function measuring the difference between two outputs. The standard choice is the square of theEuclidean distance between the vectors and:The error function over training examples can then be written as an average of losses over individual examples:
Gradient descent may find a local minimum instead of the global minimum.
Gradient descent with backpropagation is not guaranteed to find theglobal minimum of the error function, but only a local minimum; also, it has trouble crossingplateaus in the error function landscape. This issue, caused by thenon-convexity of error functions in neural networks, was long thought to be a major drawback, butYann LeCunet al. argue that in many practical problems, it is not.[13]
Backpropagation learning does not require normalization of input vectors; however, normalization could improve performance.[14]
Backpropagation requires the derivatives of activation functions to be known at network design time.
Backpropagation had been derived repeatedly, as it is essentially an efficient application of thechain rule (first written down byGottfried Wilhelm Leibniz in 1676)[15][16] to neural networks.
The terminology "back-propagating error correction" was introduced in 1962 byFrank Rosenblatt, but he did not know how to implement this.[17] In any case, he only studied neurons whose outputs were discrete levels, which only had zero derivatives, making backpropagation impossible.
Precursors to backpropagation appeared inoptimal control theory since 1950s.Yann LeCun et al credits 1950s work byPontryagin and others in optimal control theory, especially theadjoint state method, for being a continuous-time version of backpropagation.[18]Hecht-Nielsen[19] credits theRobbins–Monro algorithm (1951)[20] andArthur Bryson andYu-Chi Ho'sApplied Optimal Control (1969) as presages of backpropagation. Other precursors wereHenry J. Kelley 1960,[1] andArthur E. Bryson (1961).[2] In 1962,Stuart Dreyfus published a simpler derivation based only on thechain rule.[21][22][23] In 1973, he adaptedparameters of controllers in proportion to error gradients.[24] Unlike modern backpropagation, 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.[25]
TheADALINE (1960) learning algorithm was gradient descent with a squared error loss for a single layer. The firstmultilayer perceptron (MLP) with more than one layer trained bystochastic gradient descent[20] was published in 1967 byShun'ichi Amari.[26] The MLP had 5 layers, with 2 learnable layers, and it learned to classify patterns not linearly separable.[25]
In 1982,Paul Werbos applied backpropagation to MLPs in the way that has become standard.[31][32] Werbos described how he developed backpropagation in an interview. In 1971, during his PhD work, he developed backpropagation to mathematicizeFreud's "flow of psychic energy". He faced repeated difficulty in publishing the work, only managing in 1981.[33] He also claimed that "the first practical application of back-propagation was for estimating a dynamic model to predict nationalism and social communications in 1974" by him.[34]
Around 1982,[33]: 376 David E. Rumelhart independently developed[35]: 252 backpropagation and taught the algorithm to others in his research circle. He did not cite previous work as he was unaware of them. He published the algorithm first in a 1985 paper, then in a 1986Nature paper an experimental analysis of the technique.[36] These papers became highly cited, contributed to the popularization of backpropagation, and coincided with the resurging research interest in neural networks during the 1980s.[8][37][38]
In 1985, the method was also described by David Parker.[39][40]Yann LeCun proposed an alternative form of backpropagation for neural networks in his PhD thesis in 1987.[41]
Gradient descent took a considerable amount of time to reach acceptance. Some early objections were: there were no guarantees that gradient descent could reach a global minimum, only local minimum; neurons were "known" by physiologists as making discrete signals (0/1), not continuous ones, and with discrete signals, there is no gradient to take. See the interview withGeoffrey Hinton,[33] who was awarded the 2024Nobel Prize in Physics for his contributions to the field.[42]
Contributing to the acceptance were several applications in training neural networks via backpropagation, sometimes achieving popularity outside the research circles.
In 1987,NETtalk learned to convert English text into pronunciation. Sejnowski tried training it with both backpropagation and Boltzmann machine, but found the backpropagation significantly faster, so he used it for the final NETtalk.[33]: 324 The NETtalk program became a popular success, appearing on theToday show.[43]
In 1989, Dean A. Pomerleau published ALVINN, a neural network trained todrive autonomously using backpropagation.[44]
TheLeNet was published in 1989 to recognize handwritten zip codes.
In 1992,TD-Gammon achieved top human level play in backgammon. It was a reinforcement learning agent with a neural network with two layers, trained by backpropagation.[45]
In 1993, Eric Wan won an international pattern recognition contest through backpropagation.[46][47]
During the 2000s it fell out of favour[citation needed], but returned in the 2010s, benefiting from cheap, powerfulGPU-based computing systems. This has been especially so inspeech 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[48] and second language learning.[49])[50]
^Use for the loss function to allow to be used for the number of layers
^This followsNielsen (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.
^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 theHadamard 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 multi-layer neural networks use non-linear activation functions, so an example with linear neurons seems obscure. However, even though the error surface of multi-layer 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.
^This order follows (Rumelhart, Hinton & Williams, 1986a):[8] " is the change to be made to the weight from theth to theth unit"
^abBryson, Arthur E. (1962). "A gradient method for optimizing multi-stage allocation processes".Proceedings of the Harvard Univ. Symposium on digital computers and their applications, 3–6 April 1961. Cambridge: Harvard University Press.OCLC498866871.
^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."
^abGoodfellow, 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 calledreverse mode accumulation."
^Ramachandran, Prajit; Zoph, Barret; Le, Quoc V. (2017-10-27). "Searching for Activation Functions".arXiv:1710.05941 [cs.NE].
^Martens, James (August 2020). "New Insights and Perspectives on the Natural Gradient Method".Journal of Machine Learning Research (21).arXiv:1412.1193.
^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 ..."
^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.
^Amari, Shun'ichi (1967). "A theory of adaptive pattern classifier".IEEE Transactions.EC (16):279–307.
^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.
^Griewank, Andreas (2012). "Who Invented the Reverse Mode of Differentiation?".Optimization Stories. Documenta Mathematica, Extra Volume ISMP. pp. 389–400.S2CID15568746.
^Werbos, Paul J. (1994).The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Political Forecasting. New York: John Wiley & Sons.ISBN0-471-59897-6.
^P. J. Werbos, "Backpropagation through time: what it does and how to do it," in Proceedings of the IEEE, vol. 78, no. 10, pp. 1550–1560, Oct. 1990,doi:10.1109/5.58337
^Parker, D.B. (1985). Learning Logic: Casting the Cortex of the Human Brain in Silicon. Center for Computational Research in Economics and Management Science (Report). Cambridge MA: Massachusetts Institute of Technology. Technical Report TR-47.
^Hertz, John (1991).Introduction to the theory of neural computation. Krogh, Anders., Palmer, Richard G. Redwood City, Calif.: Addison-Wesley. p. 8.ISBN0-201-50395-6.OCLC21522159.
^Wan, Eric A. (1994). "Time Series Prediction by Using a Connectionist Network with Internal Delay Lines". InWeigend, Andreas S.;Gershenfeld, Neil A. (eds.).Time Series Prediction: Forecasting the Future and Understanding the Past. Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis. Vol. 15. Reading: Addison-Wesley. pp. 195–217.ISBN0-201-62601-2.S2CID12652643.
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