barcode generator in vb.net where D is the set of training examples, td is the target output for training example in Software

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where D is the set of training examples, td is the target output for training example
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d , and o is the output of the linear unit for training example d By this definition, d
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E ( 6 ) is simply half the squared difference between the target output t and the d h e a r unit output od, summed over all training examples Here we characterize E as a function of 27, because the linear unit output o depends on this weight vector Of course E also depends on the particular set of training examples, but
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we assume these are fixed during training, so we do not bother to write E as an explicit function of these 6 provides a Bayesian justification for choosing this particular definition of E In particular, there we show that under certain conditions the hypothesis that minimizes E is also the most probable hypothesis in H given the training data
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4431 VISUALIZING THE HYPOTHESIS SPACE
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To understand the gradient descent algorithm, it is helpful to visualize the entire hypothesis space of possible weight vectors and their associated E values, as illustrated in Figure 44 Here the axes wo and w l represent possible values for the two weights of a simple linear unit The wo, w l plane therefore represents the entire hypothesis space The vertical axis indicates the error E relative to some fixed set of training examples The error surface shown in the figure thus summarizes the desirability of every weight vector in the hypothesis space (we desire a hypothesis with minimum error) Given the way in which we chose to define E, for linear units this error surface must always be parabolic with a single global minimum The specific parabola will depend, of course, on the particular set of training examples
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FIGURE 44 Error of different hypotheses For a linear unit with two weights, the hypothesis space H is the wg, w l plane The vertical axis indicates t error of the corresponding weight vector hypothesis, k relative to a fixed set of training examples The arrow shows the negated gradient at one particular point, indicating the direction in the wo, w l plane producing steepest descent along the error surface
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Gradient descent search determines a weight vector that minimizes E by starting with an arbitrary initial weight vector, then repeatedly modifying it in small steps At each step, the weight vector is altered in the direction that produces the steepest descent along the error surface depicted in Figure 44 This process continues until the global minimum error is reached
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4432 DERIVATION OF THE GRADIENT DESCENT RULE
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How can we calculate the direction of steepest descent along the error surface This direction can be found by computing the derivative of E with respect to each component of the vector 2 This vector derivative is called the gradient of E with respect to 221, written ~ ~ ( i i r )
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Notice VE(221)is itself a vector, whose components are the partial derivatives of E with respect to each of the wi When interpreted as a vector in weight space, the gradient specijies the direction that produces the steepest increase in E The negative of this vector therefore gives the direction of steepest decrease For example, the arrow in Figure 44 shows the negated gradient -VE(G) for a particular point in the wo,wl plane Since the gradient specifies the direction of steepest increase of E, the training rule for gradient descent is
where
Here r] is a positive constant called the learning rate, which determines the step size in the gradient descent search The negative sign is present because we want to move the weight vector in the direction that decreases E This training rule can also be written in its component form
where
which makes it clear that steepest descent is achieved by altering each component w , of i in proportion to i To construct a practical algorithm for iteratively updating weights according to Equation ( 4 4 , we need an efficient way of calculating the gradient at each step Fortunately, this is not difficult The vector of derivatives that form the
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