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Unformatted text preview: 1 MachineLearning CS6375Fall 2010 a Neural Networks Reading: Section 20.5, R&N Section 4.4, Mitchell 2 Linear Regression We have training data X = { x 1 k }, k =1, , N with corresponding output Y = { y k }, k =1, , N We want to find the parameters that predict the output Y from the data X in a linear fashion: y k w o + w 1 x 1 k 3 Linear Regression We have training data X = { x 1 k }, k =1, , N with corresponding output Y = { y k }, k =1, , N We want to find the parameters that predict the output Y from the data X in a linear fashion: y k w o + w 1 x 1 k Notations: Superscript: Index of the data point in the training data set; k = k th training data point Subscript: Coordinate of the data point; x 1 k = coordinate 1 of data point k . 4 Linear Regression It is convenient to define an additional fakeattribute for the input data: x o = 1 We want to find the parameters that predict the output Y from the data X in a linear fashion: y k w o x o k + w 1 x 1 k 5 More Convenient Notations Vector of attributes for each training data point: x k = [ x o k , , x M k ] We seek a vector of parameters: w = [ w o , , w M ] such that we have a linear relation between prediction Y and attributes X : 6 Neural Network: Linear Perceptron 7 Neural Network: Linear Perceptron Note: This input unit corresponds to the fakeattribute x o = 1. Called the bias Output Unit Input Units Connection with weight Neural Network Learning problem: Adjust the connection weights so that the network generates the correct prediction on the training data. 8 A Perceptron: The General Case 9 CommonlyUsed Activation Functions 10 The Perceptron Training Algorithm But how do we update the weights? 11 Linear Regression: Gradient Descent We seek a vector of parameters: w = [ w o , , w M ] that minimizes the error between the prediction Y and and the data X : 12 Linear Regression: Gradient Descent We seek a vector of parameters: w = [ w o ,.., w M ] that minimizes the error between the prediction Y and and the data X : k is the error between the input x and the prediction y at data point k . Graphically, it is the verticaldistance between data point k and the prediction calculated by using the vector of linear parameters w . 13 Gradient Descent The minimum of E is reached when the derivatives with respect to each of the parameters w i is zero: 14 Gradient Descent The minimum of E is reached when the derivatives with respect to each of the parameters w i is zero: Note that the contribution of training data element number k to the overall gradient is  k x i k 15 Update rule: Move in the direction opposite to the gradient direction Gradient Descent Update Rule 16 Perceptron Training Given input training data x k with corresponding value y k 1. Compute error: 2. Update NN weights: 17...
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 Fall '10
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