Feedforward backpropagation

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Feedforward backpropagation is an error-driven learning technique popularized in 1986 by David Rumelhart (1942-2011), an American psychologist, Geoffrey Hinton (1947-), a British informatician, and Ronald Williams, an American professor of computer science.[1] It is a supervised learning technique, meaning that the desired outputs are known beforehand, and the task of the network is to learn to generate the desired outputs from the inputs.


File:ArtificialNeuronModel english.png
Model of a neuron. j is the index of the neuron when there is more than one neuron. The activation function for feedforward backpropagation is sigmoidal.

Given a set of k-dimensional inputs with values between 0 and 1 represented as a column vector:


and a nonlinear neuron with (initially random, uniformly distributed between -1 and 1) synaptic weights from the inputs:


then the output of the neuron is defined as follows:


where RBM3.png is a sigmoidal function. We will assume that the sigmoidal function is the simple logistic function:


This function has the useful property that


Feedforward backpropagation is typically applied to multiple layers of neurons, where the inputs are called the input layer, the layer of neurons taking the inputs is called the hidden layer, and the next layer of neurons taking their inputs from the outputs of the hidden layer is called the output layer. There is no direct connectivity between the output layer and the input layer.

If there are FfBp3.png inputs, FfBp4.png hidden neurons, and FfBp5.png output neurons, and the weights from inputs to hidden neurons are FfBp6.png (i being the input index and j being the hidden neuron index), and the weights from hidden neurons to output neurons are FfBp7.png (i being the hidden neuron index and j being the output neuron index), then the equations for the network are as follows:


If the desired outputs for a given input vector are FfBp9.png, then the update rules for the weights are as follows:


where η is some small learning rate, FfBp11.png is an error term for output neuron j and FfBp12.png is a backpropagated error term for hidden neuron j.


We first define an error term which is the cross-entropy of the output and target. We use cross-entropy because, in a sense, each output neuron represents a hypothesis about what the input represents, and the activation of the neuron represents a probability that the hypothesis is correct.


The lower the cross entropy, the more accurately the network represents what needs to be learned.

Next, we determine how the error changes based on changes to an individual weight from hidden neuron to output neuron:


We then want to change FfBp7.png slightly in the direction which reduces E, that is, FfBp15.png. This is called gradient descent.


We do the same thing to find the update rule for the weights between input and hidden neurons:


We then want to change FfBp6.png slightly in the direction which reduces E, that is, FfBp18.png:



While mathematically sound, the feedforward backpropagation algorithm has been called biologically implausible due to its requirements for neural connections to communicate backwards.[2]


  1. Rumelhart, David E.; Hinton, Geoffrey E.; Williams, Ronald J. (October 8, 1986). "Learning representations by back-propagating errors" Nature 323 (6088): 533–536
  2. Backpropagation: Theory, Architectures, and Applications. Chauvin, Yves; Rumelhart, David E. (1995). Lawrence Erlbaum Associates, Inc. ISBN 0805812598