barcode generator in vb.net The BACKPROPAGATION Algorithm in Software

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452 The BACKPROPAGATION Algorithm
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The BACKPROPAGATION algorithm learns the weights for a multilayer network, given a network with a fixed set of units and interconnections It employs gradient descent to attempt to minimize the squared error between the network output values and the target values for these outputs This section presents the BACKPROPAGATION algorithm, and the following section gives the derivation for the gradient descent weight update rule used by BACKPROPAGATION Because we are considering networks with multiple output units rather than single units as before, we begin by redefining E to sum the errors over all of the network output units
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where outputs is the set of output units in the network, and tkd and OM are the target and output values associated with the kth output unit and training example d The learning problem faced by BACKPROPAGATION is to search a large hypothesis space defined by all possible weight values for all the units in the network The situation can be visualized in terms of an error surface similar to that shown for linear units in Figure 44 The error in that diagram is replaced by our new definition of E, and the other dimensions of the space correspond now to all of the weights associated with all of the units in the network As in the case of training a single unit, gradient descent can be used to attempt to find a hypothesis to minimize E
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B ~ c ~ ~ ~ o ~ ~ G A T I O ~ ( t r a i n i n g a x ani,p,~ e s,nhidden) q, m no,, ,
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Each training example is a pair of the form (2,i ), where x' is the vector of network input values, and is the vector of target network output values q is the learning rate (eg, O5) ni, is the number of network inputs, nhidden the number of units in the hidden layer, and no,, the number of output units The inputfiom unit i into unit j is denoted xji, and the weightfrom unit i to unit j is denoted wji a Create a feed-forward network with ni, inputs, m i d d e n hidden units, and nour output units a Initialize all network weights to small random numbers (eg, between -05 and 5 0) r Until the termination condition is met, Do
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a For each (2,i ) in trainingaxamples, Do
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Propagate the input forward through the network:
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1, Input the instance x' to the network and compute the output o, of every unit the network
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Propagate the errors backward through the network:
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2 For each network output unit k, calculate its error term Sk
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6k 4-
ok(l - ok)(tk - 0 k )
3 For each hidden unit h, calculate its error term 6h
4 Update each network weight
where
Aw J l
I 1 1
TABLE 42 The stochastic gradient descent version of the BACKPROPAGATION algorithm for feedforward networks containing two layers of sigmoid units
One major difference in the case of multilayer networks is that the error surface can have multiple local minima, in contrast to the single-minimum parabolic error surface shown in Figure 44 Unfortunately, this means that gradient descent is guaranteed only to converge toward some local minimum, and not necessarily the global minimum error Despite this obstacle, in practice BACKPROPAGATION has been found to produce excellent results in many real-world applications algorithm is presented in Table 42 The algorithm as The BACKPROPAGATION described here applies to layered feedforward networks containing two layers of sigmoid units, with units at each layer connected to all units from the preceding layer This is the incremental, or stochastic, gradient descent version of BACKPROPAGATION The notation used here is the same as that used in earlier sections, with the following extensions:
CHAPTER 4 ARTIFICIAL NEURAL NETWORKS
An index (eg, an integer) is assigned to each node in the network,where a "node" is either an input to the network or the output of some unit in the network xji denotes the input from node i to unit j , and wji denotes the corresponding weight 6, denotes the error term associated with unit n It plays a role analogous to the quantity (t - o ) in our earlier discussion of the delta training rule As we shall see later, 6, = -
Notice the algorithm in Table 42 begins by constructing a network with the desired number of hidden and output units and initializing all network weights to small random values Given this fixed network structure, the main loop of the algorithm then repeatedly iterates over the training examples For each training example, it applies the network to the example, calculates the error of the network output for this example, computes the gradient with respect to the error on this example, then updates all weights in the network This gradient descent step is iterated (often thousands of times, using the same training examples multiple times) until the network performs acceptably well The gradient descent weight-update rule (Equation [T45] in Table 42) is similar to the delta training rule (Equation [410]) Like the delta rule, it updates each weight in proportion to the learning rate r ] , the input value xji to which the weight is applied, and the error in the output of the unit The only difference is that the error ( t - o ) in the delta rule is replaced by a more complex error term, aj The exact form of aj follows from the derivation of the weighttuning rule given in Section 453 To understand it intuitively, first consider how ak is computed for each network output unit k (Equation [T43] in the algorithm) ak is simply the familiar (tk - ok) from the delta rule, multiplied by the factor o k ( l - ok), which is the derivative of the sigmoid squashing function The ah value for each hidden unit h has a similar form (Equation [T44] in the algorithm) However, since training examples provide target values tk only for network outputs, no target values are directly available to indicate the error of hidden units' values Instead, the error term for hidden unit h is calculated by summing the error terms Jk for each output unit influenced by h, weighting each of the ak's by wkh,the weight from hidden unit h to output unit k This weight characterizes the degree to which hidden unit h is "responsible for" the error in output unit k The algorithm in Table 42 updates weights incrementally, following the Presentation of each training example This corresponds to a stochastic approximation to gradient descent To obtain the true gradient of E one would sum the 6, x,, values over all training examples before altering weight values The weight-update loop in BACKPROPAGATIONbe iterated thousands of may times in a typical application A variety of termination conditions can be used to halt the procedure One may choose to halt after a fixed number of iterations through the loop, or once the error on the training examples falls below some threshold, or once the error on a separate validation set of examples meets some
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