For classification the last layer is usually the logistic function for binary classification, and softmax (softargmax) for multi-class 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.
In the derivation of backpropagation, other intermediate quantities are used; they are introduced as needed below. Bias terms are not treated specially, as they correspond to a weight with a fixed input of 1. For the purpose of 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 but are not limited to sigmoid, tanh, and ReLU. Since, swish,mish, and other activation functions were proposed as well.
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; however, 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 you can compute the previous layer and repeat recursively. This avoids inefficiency in two ways. Firstly, it avoids duplication because when computing the gradient at layer , you do not need to recompute all the derivatives on later layers each time. Secondly, 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 .
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 :
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 isn't 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.
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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 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.
Learning as an optimization problem
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 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 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.
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 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.
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
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
Diagram of an artificial neural network to illustrate the notation used here
This is the reason why backpropagation requires the activation function to be differentiable. (Nevertheless, the ReLU activation function, which is non-differentiable 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.]
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:
Second-order gradient descent
Using a Hessian matrix of second-order derivatives of the error function, the Levenberg-Marquardt algorithm often converges faster than first-order gradient descent, especially when the topology of the error function is complicated. It may also find solutions in smaller node counts for which other methods might not converge. The Hessian can be approximated by the Fisher information matrix.
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.
The mathematical expression of the loss function must fulfill two conditions in order for it to be possibly used in backpropagation. 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
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:
Gradient descent may find a local minimum instead of the global minimum.
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 non-convexity of error functions in neural networks, was long thought to be a major drawback, but Yann LeCunet al. argue that in many practical problems, it is not.
Backpropagation learning does not require normalization of input vectors; however, normalization could improve performance.
Backpropagation requires the derivatives of activation functions to be known at network design time.
Later the Werbos method was rediscovered and described in 1985 by Parker, and in 1986 by Rumelhart, Hinton and Williams. Rumelhart, Hinton and Williams showed experimentally that this method can generate useful internal representations of incoming data in hidden layers of neural networks.Yann LeCun proposed the modern form of the back-propagation learning algorithm for neural networks in his PhD thesis in 1987. In 1993, Eric Wan won an international pattern recognition contest through backpropagation.
^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.
^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 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.
^Goodfellow, Bengio & Courville 2016, p. 200, "Furthermore, back-propagation is often misunderstood as being specific to multi-layer neural networks, but in principle it can compute derivatives of any function"
^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 called reverse mode accumulation."
^ abGoodfellow, Bengio & Courville (2016, p. 221), "Efficient applications of the chain rule based on dynamic programming began to appear in the 1960s and 1970s, mostly for control applications (Kelley, 1960; Bryson and Denham, 1961; Dreyfus, 1962; Bryson and Ho, 1969; Dreyfus, 1973) but also for sensitivity analysis (Linnainmaa, 1976). ... The idea was finally developed in practice after being independently rediscovered in different ways (LeCun, 1985; Parker, 1985; Rumelhart et al., 1986a). The book Parallel Distributed Processing presented the results of some of the first successful experiments with back-propagation in a chapter (Rumelhart et al., 1986b) that contributed greatly to the popularization of back-propagation and initiated a very active period of research in multilayer neural networks."
^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 ..."
^Bryson, 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.
^Russell, Stuart; Norvig, Peter (1995). Artificial Intelligence : A Modern Approach. Englewood Cliffs: Prentice Hall. p. 578. ISBN0-13-103805-2. The most popular method for learning in multilayer networks is called Back-propagation. It was first invented in 1969 by Bryson and Ho, but was more or less ignored until the mid-1980s.
^Bryson, Arthur Earl; Ho, Yu-Chi (1969). Applied optimal control: optimization, estimation, and control. Waltham: Blaisdell. OCLC3801.
^ abGriewank, Andreas (2012). "Who Invented the Reverse Mode of Differentiation?". Optimization Stories. Documenta Matematica, Extra Volume ISMP. pp. 389–400. S2CID15568746.
^ abSeppo Linnainmaa (1970). The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors. Master's Thesis (in Finnish), Univ. Helsinki, 6–7.
^ abThe thesis, and some supplementary information, can be found in his book, 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.
^Parker, D.B. (1985). "Learning Logic". Center for Computational Research in Economics and Management Science. Cambridge MA: Massachusetts Institute of Technology. ((cite journal)): Cite journal requires |journal= (help)
^ abHertz, 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". In Weigend, 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.