Back Error Propagation Simulator
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input layer, hidden or middle layer(s), one in this case, and an output layer (Figure 1). The network is fully connected from one layer to the next, but lacks any connectivity between neurons belonging wifi propagation simulator to the same layer, or back to previous layers. Figure 1. The
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Backpropagation network architecture is made of an input layer connected to a hidden layer that is then connected error propagation division to an output layer. Units are fully connected between layers without any interconnection to other units in the same layer. The BackPropagation algorithm works in two phases, as in Hopfield. First, a training phase
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adjusts network weights and then a running phase matches patterns against those already learned by the network. However, these two phases are not to be confused with a feedforward and abackpropagation modes introduced above. The training phase is made up of a large number of learning cycles, each comprising a forward pass (feedforward mode) and backward pass (backpropagation mode). The running phase is made of a error propagation calculus single forward pass taking a single cycle although sharing the same forward pass equations (feedforward mode) as in the training phase. Feedforward Mode During the feedforward mode, the network reads an input vector which is fed into the input layer. The input layer does not do any computation on the input pattern and simply sends it out to the hidden layer. Both the hidden layer and output layer have their neuron activity (corresponding to the membrane potential in more biologically oriented models) defined as a direct summation of all inputs to the neuron multiplied by their respective weights. In the model, in represents a unit in the input layer, represents a neuron in the hidden layer and a neuron in the output layer. Hidden Layer The membrane potential for a neuron in the hidden layer receives its activation from the input layer multiplied by the respective weights, as described next. is the threshold value. After is computed, an activation function is used to produce the output . where f is a sigmoid function used to compress the range of so that lies between zero and one, and e is the mathematical exponential constant. The sigmoid
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Guide IMPORTANT: See Backprop_8.0_Update for important changes in version 8.0 of emergent. Backpropagation is perhaps the most commonly used neural https://grey.colorado.edu/emergent/index.php/Backpropagation network learning algorithm. Several different "flavors" of backpropagation have been developed over the years, several of which have been implemented in the software, including the https://web.stanford.edu/group/pdplab/pdphandbook/handbookch6.html use of different error functions such as cross-entropy, and recurrent backprop, from the simple recurrent network to the Almeida-Pineda algorithm up to the real-time continuous recurrent error propagation backprop. The implementation allows the user to extend the unit types to use different activation and error functions in a straightforward manner. Note that the simple recurrent networks (SRN, a.k.a. Elman networks) are described in the feedforward backprop section, as they are more like feedforward networks than the fully recurrent ones. back error propagation The basic structure of the backpropagation algorithm consists of two phases, an activation propagation phase, and an error backpropagation phase. In the simplest version of Bp, both of these phases are strictly feed-forward and feed-back, and are computed sequentially layer-by-layer. Thus, the implementation assumes that the layers are organized sequentially in the order that activation flows. In the recurrent versions, both the activation and the error propagation are computed in two steps so that each unit is effectively being updated simultaneously with the other units. This is done in the activation phase by first computing the net input to each unit based on the other units current activation values, and then updating the activation values based on this net input. Similarly, in the error phase, first the derivative of the error with respect to the activation (dEdA) of each unit is computed based on current dEdNet values, and then the
RUNNING THE PROGRAM 5.4 EXERCISES In this chapter, we introduce the back propagation learning procedure for learning internal representations. We begin by describing the history of the ideas and problems that make clear the need for back propagation. We then describe the procedure, focusing on the goal of helping the student gain a clear understanding of gradient descent learning and how it is used in training PDP networks. The exercises are constructed to allow the reader to explore the basic features of the back propagation paradigm. At the end of the chapter, there is a separate section on extensions of the basic paradigm, including three variants we call cascaded back propagation networks, recurrent networks, and sequential networks. Exercises are provided for each type of extension. 5.1 BACKGROUND The pattern associator described in the previous chapter has been known since the late 1950s, when variants of what we have called the delta rule were first proposed. In one version, in which output units were linear threshold units, it was known as the perceptron (cf. Rosenblatt, 1959, 1962). In another version, in which the output units were purely linear, it was known as the LMS or least mean square associator (cf. Widrow and Hoff, 1960). Important theorems were proved about both of these versions. In the case of the perceptron, there was the so-called perceptron convergence theorem. In this theorem, the major paradigm is pattern classification. There is a set of binary input vectors, each of which can be said to belong to one of two classes. The system is to learn a set of connection strengths and a threshold value so that it can correctly classify each of the input vectors. The basic structure of the perceptron is illustrated in Figure 5.1. The perceptron learning procedure is the following: An input vector is presented to the system (i.e., the input units are given an activation of 1 if the corresponding value of the input vector is 1 and are given 0 otherwise). The net input to the output unit is computed: net = ∑ iwiii. If net is greater than the threshold θ, the unit is turned on, otherw