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Backpropagation is a method to calculate the gradient of the loss function with respect to the weights in an artificial neural network. It is commonly used as a part of algorithms that optimize the performance of the network by adjusting the weights, for example in the gradient descent algorithm. It is also called backward propagation of errors.

The optimization algorithm repeats a two phase cycle, propagation and weight update. When an input vector is presented to the network, it is propagated forward through the network, layer by layer, until it reaches the output layer. The output of the network is then compared to the desired output, using a loss function, and an error value is calculated for each of the neurons in the output layer. The error values are then propagated backwards, starting from the output, until each neuron has an associated error value which roughly represents its contribution to the original output.

Backpropagation uses these error values to calculate the gradient of the loss function. In the second phase, this gradient is fed to the optimization method, which in turn uses it to update the weights, in an attempt to minimize the loss function.

The importance of this process is that, as the network is trained, the neurons in the intermediate layers organize themselves in such a way that the different neurons learn to recognize different characteristics of the total input space. After training, when an arbitrary input pattern is present which contains noise or is incomplete, neurons in the hidden layer of the network will respond with an active output if the new input contains a pattern that resembles a feature that the individual neurons have learned to recognize during their training.

Backpropagation requires a known, desired output for each input value in order to calculate the loss function gradient – it is therefore usually considered to be a supervised learning method; nonetheless, it is also used in some unsupervised networks such as autoencoders. It is a generalization of the delta rule to multi-layered feedforward networks, made possible by using the chain rule to iteratively compute gradients for each layer. Backpropagation requires that the activation function used by the artificial neurons (or "nodes") is differentiable.


The goal of any supervised learning algorithm is to find a function that best maps a set of inputs to its correct output. An example would be a classification task, where the input is an image of an animal, and the correct output would be the name of the animal.

The motivation for developing the backpropagation algorithm was to find a way 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.[1] The goal of backpropagation is to compute the partial derivative, or gradient, of a loss function with respect to any weight in the network.[2]

Loss function[edit]

Sometimes referred to as the cost function or error function (not to be confused with the Gauss error function), the loss function is a function that maps values of one or more variables onto a real number intuitively representing some "cost" associated with the event. For backpropagation, the loss function calculates the difference between the input training example and its expected output, after the example has been propagated through the network.

Assumptions about the loss function[edit]

For backpropagation to work, two assumptions are made about the form of the error function.[2] 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. In practice, training examples are placed in batches, and the error is averaged at the end of the batch, which is then used to update the weights. The second assumption is that it can be written as a function of the outputs from the neural network.

Example loss function[edit]

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 factor of conveniently cancels the exponent when the error function is subsequently differentiated.

The error function over training examples can be written as an average:

And the partial derivative with respect to the outputs:


Let be a neural network with connections, inputs, and outputs.

Below, will denote vectors in , vectors in , and vectors in . These are called inputs, outputs and weights respectively.

The neural network corresponds to a function which, given a weight , maps an input to an output .

The backpropagation algorithm takes as input a sequence of training examples and produces a sequence of weights starting from some initial weight , usually chosen at random.

These weights are computed in turn: first compute using only for . The output of the backpropagation algorithm is then , giving us a new function . The computation is the same in each step, hence only the case is described.

Calculating from is done by considering a variable weight and applying gradient descent to the function to find a local minimum, starting at .

This makes the minimizing weight found by gradient descent.

Algorithm in code[edit]

To implement the algorithm above, explicit formulas are required for the gradient of the function where the function is.

The backpropagation learning algorithm can be divided into two phases: propagation and weight update.

Phase 1: Propagation[edit]

Each propagation involves the following steps:

  1. Forward propagation of a training pattern's input through the neural network in order to generate the network's output value(s).
  2. Backward propagation of the propagation's output activations through the neural network using the training pattern target in order to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons.

Phase 2: Weight update[edit]

For each weight, the following steps must be followed:

  1. The weight's output delta and input activation are multiplied to find the gradient of the weight.
  2. A ratio (percentage) of the weight's gradient is subtracted from the weight.

This ratio (percentage) influences the speed and quality of learning; it is called the learning rate. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurate the training is. The sign of the gradient of a weight indicates whether the error varies directly with, or inversely to, the weight. Therefore, the weight must be updated in the opposite direction, "descending" the gradient.

Phases 1 and 2 are repeated until the performance of the network is satisfactory.


The following is pseudocode for a stochastic gradient descent algorithm for training a three-layer network (only one hidden layer):

  initialize network weights (often small random values)
     forEach training example named ex
        prediction = neural-net-output(network, ex)  // forward pass
        actual = teacher-output(ex)
        compute error (prediction - actual) at the output units
        compute  for all weights from hidden layer to output layer  // backward pass
        compute  for all weights from input layer to hidden layer   // backward pass continued
        update network weights // input layer not modified by error estimate
  until all examples classified correctly or another stopping criterion satisfied
  return the network

The lines labeled "backward pass" can be implemented using the backpropagation algorithm, which calculates the gradient of the error of the network regarding the network's modifiable weights.[3] Often the term "backpropagation" is used to refer to the entire procedure encompassing both the calculation of the gradient and its use in stochastic gradient descent, but backpropagation properties can be used with any gradient-based optimizer, such as L-BFGS or truncated Newton.

Backpropagation networks are necessarily multilayer perceptrons (usually with one input, multiple hidden, and one output layer). In order for the hidden layer to serve any useful function, multilayer networks must have non-linear activation functions for the multiple layers: a multilayer network using only linear activation functions is equivalent to some single layer, linear network. Non-linear activation functions that are commonly used include the rectifier, logistic function, the softmax function, and the gaussian function.

The backpropagation algorithm for calculating a gradient has been rediscovered a number of times, and is a special case of a more general technique called automatic differentiation in the reverse accumulation mode.

It is also closely related to the Gauss–Newton algorithm, and is also part of continuing research in neural backpropagation.


Learning as an optimization problem[edit]

Before showing the mathematical derivation of the backpropagation algorithm, it helps to develop some intuitions about the relationship between the actual output of a neuron and the correct output for a particular training case. Consider a simple neural network with two input units, one output unit and no hidden units. Each neuron uses a linear output[note 1] that is the weighted sum of its input.

A simple neural network with two input units and one output unit

Initially, before training, the weights will be set randomly. Then the neuron learns from training examples, which in this case consists of a set of tuples (, , ) where and are the inputs to the network and is the correct output (the output the network should eventually produce given the identical inputs). The network given and will compute an output which very likely differs from (since the weights are initially random). A common method for measuring the discrepancy between the expected output and the actual output is using the squared error measure:

where is the discrepancy or error.

As an example, consider the network on a single training case: , thus the input and are 1 and 1 respectively and the correct output, is 0. Now if the actual output is plotted on the horizontal axis against the error on the vertical axis, the result is a parabola. The minimum of the parabola corresponds to the output which minimizes the error . 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 that exactly matches the expected output . 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. If each weight is plotted on a separate horizontal axis and the error on the vertical axis, the result is a parabolic bowl. For a neuron with weights, the same plot would require an elliptic paraboloid of dimensions.

Error surface of a linear neuron with two input weights

The backpropagation algorithm aims to find the set of weights that minimizes the error. There are several methods for finding the minima of a parabola or any function in any dimension. One way is analytically by solving systems of equations, however this relies on the network being a linear system, and the goal is to be able to also train multi-layer, non-linear networks (since a multi-layered linear network is equivalent to a single-layer network). The method used in backpropagation is gradient descent.

An analogy for understanding gradient descent[edit]

The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. A person is stuck in the mountains and is trying to get down (i.e. trying to find the minima). There is heavy fog such that visibility is extremely low. Therefore, the path down the mountain is not visible, so he must use local information to find the minima. He can use the method of gradient descent, which involves looking at the steepness of the hill at his current position, then proceeding in the direction with the steepest descent (i.e. downhill). If he was trying to find the top of the mountain (i.e. the maxima), then he would proceed in the direction steepest ascent (i.e. uphill). Using this method, he would eventually find his way down the mountain. However, assume also that the steepness of the hill is not immediately obvious with simple observation, but rather it requires a sophisticated instrument to measure, which the person happens to have at the moment. It takes quite some time to measure the steepness of the hill with the instrument, thus he should minimize his use of the instrument if he wanted to get down the mountain before sunset. The difficulty then is choosing the frequency at which he should measure the steepness of the hill so not to go off track.

In this analogy, the person represents the backpropagation algorithm, and the path taken down the mountain represents the sequence of parameter settings that the algorithm will explore. The steepness of the hill represents the slope of the error surface at that point. The instrument used to measure steepness is differentiation (the slope of the error surface can be calculated by taking the derivative of the squared error function at that point). The direction he chooses to travel in aligns with the gradient of the error surface at that point. The amount of time he travels before taking another measurement is the learning rate of the algorithm. See the limitation section for a discussion of the limitations of this type of "hill climbing" algorithm.


Since backpropagation uses the gradient descent method, one needs to calculate the derivative of the squared error function with respect to the weights of the network. Assuming one output neuron,[note 2] the squared error function is:


is the squared error,
is the target output for a training sample, and
is the actual output of the output neuron.

The factor of is included to cancel the exponent when differentiating. Later, the expression will be multiplied with an arbitrary learning rate, so that it doesn't matter if a constant coefficient is introduced now.

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 neurons and .

The activation function is in general non-linear and differentiable. A commonly used activation function is the logistic function:

which has a convenient derivative of:

Finding the derivative of the error[edit]

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 right-hand 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 (assuming here that the logistic function is used):

This is the reason why backpropagation requires the activation function to be differentiable.

The first factor is straightforward to evaluate if the neuron is in the output layer, because then and

However, if is in an arbitrary inner layer of the network, finding the derivative with respect to is less obvious.

Considering as a function of the inputs of 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 one closer to the output neuron – are known.

Putting it all together:


To update the weight using gradient descent, one must choose a learning rate, . The change in weight, which is added to the old weight, is equal to the product of the learning rate and the gradient, multiplied by :

The is required in order to update in the direction of a minimum, not a maximum, of the error function.

For a single-layer network, this expression becomes the Delta Rule. To better understand how backpropagation works, here is an example to illustrate it: The Back Propagation Algorithm, page 20.


The choice of learning rate is important for the method, since a high value can cause too strong a change, causing the minimum to be missed, while a too low learning rate slows the training unnecessarily.

Various optimizations of backpropagation, such as Quickprop, are primarily aimed at speeding up the error minimization; other improvements mainly try to increase reliability.

Backpropagation with adaptive learning rate[edit]

In order to avoid oscillation inside the network, such as alternating connection weights, and to improve the rate of convergence, there are refinements of this algorithm that use an adaptive learning rate.[4]

Backpropagation with inertia[edit]

By using a variable inertia term (Momentum) the gradient and the last change can be weighted such that the weight adjustment additionally depends on the previous change. If the Momentum is equal to 0, the change depends solely on the gradient, and a value of 1 will only depend on the last change.

Similar to a ball rolling down a mountain, whose current speed is determined not only by the current slope of the mountain but also by its own inertia, inertia can be added to backpropagation:


is the change in weight in the connection of neuron to neuron at time
a learning rate (
the error signal of neuron and
the output of neuron , which is also an input of the current neuron (neuron )
the influence of the inertial term (in ). This corresponds to the weight change at the previous point in time.

This will depend on the current weight change both from the current gradient of the error function (slope of the mountain, 1st summand), as well as from the weight change from the previous point in time (inertia, 2nd summand).

With inertia, the previous problems of the backpropagation getting stuck in steep ravines and flat plateaus are avoided. Since, for example, the gradient of the error function becomes very small in flat plateaus, it would immediately lead to a "deceleration" of the gradient descent. This "deceleration" is delayed by the addition of the inertia term so that a flat plateau can be overcome more quickly.

Modes of learning[edit]

There are two modes of learning to choose from: stochastic and batch. In stochastic learning, each propagation is followed immediately by a weight update. In batch learning many propagations occur before updating the weights, accumulating errors over the samples within a batch. Stochastic learning introduces "noise" into the gradient descent process, using the local gradient calculated from one data point; this reduces the chance of the network getting stuck in a local minima. Yet batch learning typically yields a faster, more stable descent to a local minima, since each update is performed in the direction of the average error of the batch samples. In modern applications a common compromise choice is to use "mini-batches", meaning batch learning but with a batch of small size and with stochastically selected samples.

Training data collection[edit]

Online learning is used for dynamic environments that provide a continuous stream of new training data patterns. Offline learning makes use of a training set of static patterns.


Gradient descent can find the 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 plateaux 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 in a 2015 review article, Yann LeCun et al. argue that in many practical problems, it is not.[5]
  • Backpropagation learning does not require normalization of input vectors; however, normalization could improve performance.[6]


According to various sources,[7][8][9][10] basics of continuous backpropagation were derived in the context of control theory by Henry J. Kelley[11] in 1960 and by Arthur E. Bryson in 1961,[12] using principles of dynamic programming. In 1962, Stuart Dreyfus published a simpler derivation based only on the chain rule.[13] Vapnik cites reference[14] in his book on Support Vector Machines. Arthur E. Bryson and Yu-Chi Ho described it as a multi-stage dynamic system optimization method in 1969.[15][16]

In 1970, Seppo Linnainmaa finally published the general method for automatic differentiation (AD) of discrete connected networks of nested differentiable functions.[17][18] This corresponds to the modern version of backpropagation which is efficient even when the networks are sparse.[9][10][19][20]

In 1973, Stuart Dreyfus used backpropagation to adapt parameters of controllers in proportion to error gradients.[21] In 1974, Paul Werbos mentioned the possibility of applying this principle to artificial neural networks,[22] and in 1982, he applied Linnainmaa's AD method to neural networks in the way that is widely used today.[10][23]

In 1986, David E. Rumelhart, Geoffrey E. Hinton and Ronald J. Williams showed through computer experiments that this method can generate useful internal representations of incoming data in hidden layers of neural networks.[1] [24] In 1993, Eric A. Wan was the first[9] to win an international pattern recognition contest through backpropagation.[25]

During the 2000s it fell out of favour but has returned again in the 2010s, now able to train much larger networks using huge modern computing power such as GPUs. In the context of this new hardware, backpropogation is sometimes referred to as deep learning,[26] a term used to describe neural networks with more than one layer not dedicated to input or output.[27] For example, in 2014, backpropagation was used to train a deep neural network for state of the art speech recognition.[28]

See also[edit]


  1. ^ 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.
  2. ^ There can be multiple output neurons, in which case the error is the squared norm of the difference vector.


  1. ^ a b Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (8 October 1986). "Learning representations by back-propagating errors". Nature. 323 (6088): 533–536. doi:10.1038/323533a0. 
  2. ^ a b A., Nielsen, Michael (2015-01-01). "Neural Networks and Deep Learning". 
  3. ^ Paul J. Werbos (1994). The Roots of Backpropagation. From Ordered Derivatives to Neural Networks and Political Forecasting. New York, NY: John Wiley & Sons, Inc.
  4. ^ Li, Y.; Fu, Y.; Li, H.; Zhang, S. W. (2009-06-01). "The Improved Training Algorithm of Back Propagation Neural Network with Self-adaptive Learning Rate". 2009 International Conference on Computational Intelligence and Natural Computing. 1: 73–76. doi:10.1109/CINC.2009.111. 
  5. ^ LeCun, Yann; Bengio, Yoshua; Hinton, Geoffrey (2015). "Deep learning". Nature. 521: 436–444. PMID 26017442. doi:10.1038/nature14539. 
  6. ^ ISBN 1-931841-08-X,
  7. ^ Stuart Dreyfus (1990). Artificial Neural Networks, Back Propagation and the Kelley-Bryson Gradient Procedure. J. Guidance, Control and Dynamics, 1990.
  8. ^ Eiji Mizutani, Stuart Dreyfus, Kenichi Nishio (2000). On derivation of MLP backpropagation from the Kelley-Bryson optimal-control gradient formula and its application. Proceedings of the IEEE International Joint Conference on Neural Networks (IJCNN 2000), Como Italy, July 2000. Online
  9. ^ a b c Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. doi:10.1016/j.neunet.2014.09.003. 
  10. ^ a b c Schmidhuber, Jürgen (2015). "Deep Learning". Scholarpedia. 10 (11): 32832. doi:10.4249/scholarpedia.32832. 
  11. ^ Kelley, Henry J. (1960). "Gradient theory of optimal flight paths". Ars Journal. 30 (10): 947–954. doi:10.2514/8.5282. 
  12. ^ Arthur E. Bryson (1961, April). A gradient method for optimizing multi-stage allocation processes. In Proceedings of the Harvard Univ. Symposium on digital computers and their applications.
  13. ^ Stuart Dreyfus (1962). The numerical solution of variational problems" Journal of Mathematical Analysis and Applications 5(1), 30–45. Online
  14. ^ Bryson, A.E.; W.F. Denham; S.E. Dreyfus. Optimal programming problems with inequality constraints. I: Necessary conditions for extremal solutions. AIAA J. 1, 11 (1963) 2544–2550
  15. ^ Stuart Russell; Peter Norvig. Artificial Intelligence A Modern Approach. p. 578. The most popular method for learning in multilayer networks is called Back-propagation. 
  16. ^ Arthur Earl Bryson; Yu-Chi Ho (1969). Applied optimal control: optimization, estimation, and control. Blaisdell Publishing Company or Xerox College Publishing. p. 481. 
  17. ^ Seppo 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.
  18. ^ Linnainmaa, Seppo (1976). "Taylor expansion of the accumulated rounding error". BIT Numerical Mathematics. 16 (2): 146–160. doi:10.1007/bf01931367. 
  19. ^ Griewank, Andreas (2012). Who Invented the Reverse Mode of Differentiation?. Optimization Stories, Documenta Matematica, Extra Volume ISMP (2012), 389–400.
  20. ^ Griewank, Andreas and Walther, A.. Principles and Techniques of Algorithmic Differentiation, Second Edition. SIAM, 2008.
  21. ^ 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. 
  22. ^ Paul Werbos (1974). Beyond regression: New tools for prediction and analysis in the behavioral sciences. PhD thesis, Harvard University.
  23. ^ Paul Werbos (1982). Applications of advances in nonlinear sensitivity analysis. In System modeling and optimization (pp. 762–770). Springer Berlin Heidelberg. Online
  24. ^ Alpaydın, Ethem (2010). Introduction to machine learning (2nd ed.). Cambridge, Mass.: MIT Press. p. 250. ISBN 978-0-262-01243-0. 
  25. ^ Eric A. Wan (1993). Time series prediction by using a connectionist network with internal delay lines. In SANTA FE INSTITUTE STUDIES IN THE SCIENCES OF COMPLEXITY-PROCEEDINGS (Vol. 15, pp. 195–195). Addison–Wesley Publishing Co.
  26. ^
  27. ^
  28. ^ Abdel-Hamid, O.; Mohamed, A. r; Jiang, H.; Deng, L.; Penn, G.; Yu, D. (2014-10-01). "Convolutional Neural Networks for Speech Recognition". IEEE/ACM Transactions on Audio, Speech, and Language Processing. 22 (10): 1533–1545. ISSN 2329-9290. doi:10.1109/TASLP.2014.2339736. 

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