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Unformatted text preview: Lecture 2: Classifcation. Perceptron. Sigmoid classifers. • Classifcation problems. Error Functions • Perceptron • Sigmoid classifers September 10, 2007 1 COMP652 Lecture 2 Classifcation • Given a data set D ⊂ X × Y where Y is a discrete set (usually with a smallish number oF values), fnd a hypothesis h ∈ H which predicts “well” the existing data • IF Y has two possible values, e.g. Y = { 1 , 1 } or Y = { , 1 } , this is called binary classifcation. • Can we develop methods For classifcation as we did For regression? • What does it take to develop a learning algorithm? September 10, 2007 2 COMP652 Lecture 2 Recall: Three decisions • What should be the error function? • What should be the hypothesis class? • How are we going to Fnd the best hypothesis in the class (the one that minimizes the error function)? September 10, 2007 3 COMP652 Lecture 2 Error functions for binary classiFcation • One worthy goal is to minimize the number of misclassified examples • Suppose Y = { 1 , 1 } and the hypotheses h w ∈ H also output a +1 or 1 • An example x , y is misclassiFed if yh w ( x ) is negative. • So a reasonable error function is just counting the number of examples correctly classiFed: J ( w ) = X i ∈ MisclassiFed y i h w ( x i ) This is called 01 loss • This function is not differentiable, so often we will still use the meansquared error. September 10, 2007 4 COMP652 Lecture 2 Choosing the hypothesis class • For regression, we used linear hypotheses (simple, nice) • Is there an analogue for classi¡cation? • What about linear hypotheses? September 10, 2007 5 COMP652 Lecture 2 Example: Wisconsin data 10 15 20 25 30 0.2 0.4 0.6 0.8 1 tumor size (mm?) non ! recurring (0) / recurring (1) What is the meaning of the output in this case? September 10, 2007 6 COMP652 Lecture 2 Output of a classiFer • Useful predictions could be: – The predicted class – The probability that the example belongs to a given class • Just applying linear regression as is gives us neither September 10, 2007 7 COMP652 Lecture 2 Perceptron w 1 w 2 w n w x 1 x 2 x n x =1 . . . ! ! w i x i n i =0 1 if > 01 otherwise { o = ! w i x i n i =0 • We can take a linear combination and threshold it: h w ( x ) = sgn ( w T x ) = 8 < : +1 if w T x > 1 otherwise This is called a perceptron ....
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 Fall '07
 PREICUP
 Machine Learning, Optimization, Gradient descent, Error function

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