lecture5-annotated - 1 Eric Xing @ CMU, 2006-2008 1 Machine...

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Unformatted text preview: 1 Eric Xing @ CMU, 2006-2008 1 Machine Learning Machine Learning 10 10-701/15 701/15-781, Fall 2008 781, Fall 2008 Logistic Regression Logistic Regression---- generative verses discriminative generative verses discriminative classifier classifier Eric Xing Eric Xing Lecture 5, September 22, 2008 Reading: Chap. 4.3 CB Eric Xing @ CMU, 2006-2008 2 Announcements z Hw 1 due at the end of the class today z My office hour is 11:50-12:30 today (catching a flight in the afternoon) z Steve Hanneke will give the lecture on Wed 2 Eric Xing @ CMU, 2006-2008 3 Generative vs. Discriminative Classifiers z Goal: Wish to learn f: X Y, e.g., P(Y|X) z Generative classifiers (e.g., Nave Bayes): z Assume some functional form for P(X|Y), P(Y) This is a generative model of the data! z Estimate parameters of P(X|Y), P(Y) directly from training data z Use Bayes rule to calculate P(Y|X= x) z Discriminative classifiers: z Directly assume some functional form for P(Y|X) This is a discriminative model of the data! z Estimate parameters of P(Y|X) directly from training data Y i X i Y i X i Eric Xing @ CMU, 2006-2008 4 Discussion: Generative and discriminative classifiers z Generative: z Modeling the joint distribution of all data z Discriminative: z Modeling only points at the boundary z How? Regression! 3 Eric Xing @ CMU, 2006-2008 5 Linear regression z The data: z Both nodes are observed: z X is an input vector z Y is a response vector (we first consider y as a generic continuous response vector, then we consider the special case of classification where y is a discrete indicator) z A regression scheme can be used to model p ( y | x ) directly, rather than p ( x,y ) Y i X i N { } ) , ( , ), , ( ), , ( ), , ( N N y x y x y x y x L 3 3 2 2 1 1 Eric Xing @ CMU, 2006-2008 6 Classification and logistic regression 4 Eric Xing @ CMU, 2006-2008 7 The logistic function Eric Xing @ CMU, 2006-2008 8 Logistic regression (sigmoid classifier) z The condition distribution: a Bernoulli where is a logistic function z We can used the brute-force gradient method as in LR z But we can also apply generic laws by observing the p ( y | x ) is an exponential family function , more specifically, a generalized linear model (see future lectures ) y y x x x y p = 1 1 )) ( ( ) ( ) | ( x T e x + = 1 1 ) ( 5 Eric Xing @ CMU, 2006-2008 9 Training Logistic Regression: MCLE z Estimate parameters =< , 1 , ... m > to maximize the conditional likelihood of training data z Training data z Data likelihood = z Data conditional likelihood = Eric Xing @ CMU, 2006-2008 10 Expressing Conditional Log Likelihood z Recall the logistic function: and conditional likelihood: 6 Eric Xing @ CMU, 2006-2008 11 Maximizing Conditional Log Likelihood z The objective: z Good news: l( ) is concave function of z Bad news: no closed-form solution to maximize l( ) Eric Xing @ CMU, 2006-2008 12 Gradient Ascent z Property of sigmoid function: z The gradient: The gradient ascent algorithm iterate until change < For all i , repeat 7...
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This note was uploaded on 01/26/2010 for the course MACHINE LE 10701 taught by Professor Ericp.xing during the Fall '08 term at Carnegie Mellon.

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lecture5-annotated - 1 Eric Xing @ CMU, 2006-2008 1 Machine...

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