Error Back Propagation Training Algorithm
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Backpropagation
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be an insurmountable problem - how could we tell the hidden units just what to do? This unsolved question was in fact the reason why neural networks fell out of favor after an initial period of high popularity in the 1950s. back propagation explained It took 30 years before the error backpropagation (or in short: backprop) algorithm popularized a
Back Propagation Explanation
way to train hidden units, leading to a new wave of neural network research and applications. (Fig. 1) In principle, backprop provides a
Back Propogation Algo
way to train networks with any number of hidden units arranged in any number of layers. (There are clear practical limits, which we will discuss later.) In fact, the network does not have to be organized in layers - any https://en.wikipedia.org/wiki/Backpropagation pattern of connectivity that permits a partial ordering of the nodes from input to output is allowed. In other words, there must be a way to order the units such that all connections go from "earlier" (closer to the input) to "later" ones (closer to the output). This is equivalent to stating that their connection pattern must not contain any cycles. Networks that respect this constraint are called feedforward networks; their connection pattern forms a directed acyclic graph or dag. https://www.willamette.edu/~gorr/classes/cs449/backprop.html The Algorithm We want to train a multi-layer feedforward network by gradient descent to approximate an unknown function, based on some training data consisting of pairs (x,t). The vector x represents a pattern of input to the network, and the vector t the corresponding target (desired output). As we have seen before, the overall gradient with respect to the entire training set is just the sum of the gradients for each pattern; in what follows we will therefore describe how to compute the gradient for just a single training pattern. As before, we will number the units, and denote the weight from unit j to unit i by wij. Definitions: the error signal for unit j: the (negative) gradient for weight wij: the set of nodes anterior to unit i: the set of nodes posterior to unit j: The gradient. As we did for linear networks before, we expand the gradient into two factors by use of the chain rule: The first factor is the error of unit i. The second is Putting the two together, we get . To compute this gradient, we thus need to know the activity and the error for all relevant nodes in the network. Forward activaction. The activity of the input units is determined by the network's external input x. For all other units, the activity is propagated forward: Note that before the activity of unit i can
network for a given set of input patterns with known classifications. When each entry of the sample set is presented to the network, the network examines its output response to the sample input pattern. The output response is then compared to the http://wwwold.ece.utep.edu/research/webfuzzy/docs/kk-thesis/kk-thesis-html/node22.html known and desired output and the error value is calculated. Based on the error, the connection weights are adjusted. The backpropagation algorithm is based on Widrow-Hoff delta learning rule in which the weight adjustment is done through mean square error of the output response to the sample input [Vel98]. The set of these sample patterns are repeatedly presented to the network until the error value is minimized. Refer to the back propagation figure 2.12 that illustrates the backpropagation multilayer network with layers. represents the number of neurons in th layer. Here, the network is presented the th pattern of training sample set with -dimensional input and -dimensional known output response . The actual response to the input pattern by the network is represented as . Let be the output from the th neuron in layer for th pattern; be the connection back propagation algorithm weight from th neuron in layer to th neuron in layer ; and be the error value associated with the th neuron in layer . Figure 2.12: Backpropagation Neural Network The following is the outline of the backpropagation learning algorithm [BJ91]: Initialize connection weights into small random values. Present the th sample input vector of pattern and the corresponding output target to the network. Pass the input values to the first layer, layer 1. For every input node in layer 0, perform: For every neuron in every layer , from input to output layer, find the output from the neuron: where Obtain output values. For every output node in layer , perform: Calculate error value for every neuron in every layer in backward order , from output to input layer, followed by weight adjustments. For the output layer, the error value is: (2.10) and for hidden layers: (2.11) The weight adjustment can be done for every connection from neuron in layer to every neuron in every layer : (2.12) where represents weight adjustment factor normalized between 0 and 1. The derivation of the equations above will be discussed soon. The actions in steps 2 through 6 will be repeated for every training sample pattern , and re