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Backpropagation, or propagation of error, is a common method of teaching artificial neural networks how to perform a given task. It was first described by Arthur E. Bryson and Yu-Chi Ho in 1969,[1][2] but it wasn't until 1986, through the work of David E. Rumelhart, Geoffrey E. Hinton and Ronald J. Williams, that it gained recognition, and it led to a “renaissance” in the field of artificial neural network research.

It is a supervised learning method, and is an implementation of the Delta rule. It requires a teacher that knows, or can calculate, the desired output for any given input. It is most useful for feed-forward networks (networks that have no feedback, or simply, that have no connections that loop). The term is an abbreviation for "backwards propagation of errors". Backpropagation requires that the activation function used by the artificial neurons (or "nodes") is differentiable.



Summary of the backpropagation technique:

  1. Present a training sample to the neural network.
  2. Compare the network's output to the desired output from that sample. Calculate the error in each output neuron.
  3. For each neuron, calculate what the output should have been, and a scaling factor, how much lower or higher the output must be adjusted to match the desired output. This is the local error.
  4. Adjust the weights of each neuron to lower the local error.
  5. Assign "blame" for the local error to neurons at the previous level, giving greater responsibility to neurons connected by stronger weights.
  6. Repeat from step 3 on the neurons at the previous level, using each one's "blame" as its error.


Actual algorithm for a 3-layer network (only one hidden layer):

  Initialize the weights in the network (often randomly)
         For each example e in the training set
              O = neural-net-output(network, e) ; forward pass
              T = teacher output for e
              Calculate error (T - O) at the output units
              Compute delta_wh for all weights from hidden layer to output layer ; backward pass
              Compute delta_wi for all weights from input layer to hidden layer ; backward pass continued
              Update the weights in the network
  Until all examples classified correctly or stopping criterion satisfied
  Return the network

As the algorithm's name implies, the errors (and therefore the learning) propagate backwards from the output nodes to the inner nodes. So technically speaking, backpropagation is used to calculate the gradient of the error of the network with respect to the network's modifiable weights. This gradient is almost always then used in a simple stochastic gradient descent algorithm to find weights that minimize the error. Often the term "backpropagation" is used in a more general sense, to refer to the entire procedure encompassing both the calculation of the gradient and its use in stochastic gradient descent. Backpropagation usually allows quick convergence on satisfactory local minima for error in the kind of networks to which it is suited.

Backpropagation networks are necessarily multilayer perceptrons (usually with one input, one 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 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.


  1. ^ Stuart Russell and Peter Norvig. Artificial Intelligence A Modern Approach. p. 578. "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." 
  2. ^ Arthur Earl Bryson, Yu-Chi Ho (1969). Applied optimal control: optimization, estimation, and control. Blaisdell Publishing Company or Xerox College Publishing. pp. 481. 

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