ECE 5984: Introduction to Machine Learning
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1 ECE 5984: Introduction to Machine Learning Topics: Classification: Logistic Regression NB & LR connections Readings: Barber 17.4 Dhruv Batra Virginia Tech
2 Administrativia HW2 Due: Friday 3/6, 3/15, 11:55pm Implement linear regression, Naïve Bayes, Logistic Regression Need a couple of catch-up lectures How about 4-6pm? (C) Dhruv Batra 2
3 Recap of last time (C) Dhruv Batra 3
4 Naïve Bayes (your first probabilistic classifier) x Classification y Discrete (C) Dhruv Batra 4
5 Learn: h:x! Y X features Y target classes Classification Suppose you know P(Y X) exactly, how should you classify? Bayes classifier: Why? Slide Credit: Carlos Guestrin
6 Error Decomposition Approximation/Modeling Error You approximated reality with model Estimation Error You tried to learn model with finite data Optimization Error You were lazy and couldn t/didn t optimize to completion Bayes Error Reality just sucks (C) Dhruv Batra 6
7 Generative vs. Discriminative Generative Approach (Naïve Bayes) Estimate p(x y) and p(y) Use Bayes Rule to predict y Discriminative Approach Estimate p(y x) directly (Logistic Regression) Learn discriminant function h(x) (Support Vector Machine) (C) Dhruv Batra 7
8 The Naïve Bayes assumption Naïve Bayes assumption: Features are independent given class: More generally: d How many parameters now? Suppose X is composed of d binary features (C) Dhruv Batra Slide Credit: Carlos Guestrin 8
9 Generative vs. Discriminative Generative Approach (Naïve Bayes) Estimate p(x y) and p(y) Use Bayes Rule to predict y Discriminative Approach Estimate p(y x) directly (Logistic Regression) Learn discriminant function h(x) (Support Vector Machine) (C) Dhruv Batra 9
10 Today: Logistic Regression Main idea Think about a 2 class problem {,1} Can we regress to P(Y=1 X=x)? Meet the Logistic or Sigmoid function Crunches real numbers down to -1 Model In regression: y ~ N(w x, λ 2 ) Logistic Regression: y ~ Bernoulli(σ(w x)) (C) Dhruv Batra 1
11 Understanding the sigmoid (w + X i w i x i )= 1 1+e w P i w ix i w =2, w 1 =1 w =, w 1 =1 w =, w 1 = (C) Dhruv Batra Slide Credit: Carlos Guestrin 11
12 Logistic Regression a Linear classifier Demo (C) Dhruv Batra 12
13 Visualization W=(1,4) 5 w2 W = ( 2, 3 ) W=(,2) x W = ( 2, 1 ) x1 x1 1 1 x x x2.5 W=(3,) 1 x1 1 1 W=(1,).5 W=(5,1) 1 x2 1 1 x2 W=(2,2).5 x1 1 x x2 1 W=(5,4) 1 x1 1 x2 x1 W = ( 2, 2 ) x2 1 x2 w1 1 1 x x x2 3 3 (C) Dhruv Batra Slide Credit: Kevin Murphy
14 Expressing Conditional Log Likelihood (C) Dhruv Batra Slide Credit: Carlos Guestrin 14
15 Maximizing Conditional Log Likelihood d d Bad news: no closed-form solution to maximize l(w) Good news: l(w) is concave function of w! (C) Dhruv Batra Slide Credit: Carlos Guestrin 15
16 (C) Dhruv Batra Slide Credit: Greg Shakhnarovich 16
17 Careful about step-size (C) Dhruv Batra Slide Credit: Nicolas Le Roux 17
18 (C) Dhruv Batra Slide Credit: Fei Sha 18
19 When does it work? (C) Dhruv Batra 19
20 (C) Dhruv Batra Slide Credit: Fei Sha 2
21 (C) Dhruv Batra Slide Credit: Greg Shakhnarovich 21
22 (C) Dhruv Batra Slide Credit: Fei Sha 22
23 (C) Dhruv Batra Slide Credit: Fei Sha 23
24 (C) Dhruv Batra Slide Credit: Fei Sha 24
25 (C) Dhruv Batra Slide Credit: Fei Sha 25
26 Optimizing concave function Gradient ascent Conditional likelihood for Logistic Regression is concave à Find optimum with gradient ascent Gradient: Learning rate, η> Update rule: (C) Dhruv Batra Slide Credit: Carlos Guestrin 26
27 Maximize Conditional Log Likelihood: Gradient ascent d d (C) Dhruv Batra Slide Credit: Carlos Guestrin 27
28 Gradient Ascent for LR Gradient ascent algorithm: iterate until change < ε For i=1,,n, repeat (C) Dhruv Batra Slide Credit: Carlos Guestrin 28
29 (C) Dhruv Batra 29
30 That s all M(C)LE. How about M(C)AP? One common approach is to define priors on w Normal distribution, zero mean, identity covariance Pushes parameters towards zero Corresponds to Regularization Helps avoid very large weights and overfitting More on this later in the semester MAP estimate (C) Dhruv Batra Slide Credit: Carlos Guestrin 3
31 Large parameters Overfitting If data is linearly separable, weights go to infinity Leads to overfitting Penalizing high weights can prevent overfitting (C) Dhruv Batra Slide Credit: Carlos Guestrin 31
32 Gradient of M(C)AP (C) Dhruv Batra Slide Credit: Carlos Guestrin 32
33 MLE vs MAP Maximum conditional likelihood estimate Maximum conditional a posteriori estimate (C) Dhruv Batra Slide Credit: Carlos Guestrin 33
34 HW2 Tips Naïve Bayes Train_NB Implement factor_tables -- X i x Y matrices Prior Y x 1 vector Fill entries by counting + smoothing Test_NB argmax_y P(Y=y) P(X i =x i ) TIP: work in log domain Logistic Regression Use small step-size at first Make sure you maximize log-likelihood not minimize it Sanity check: plot objective (C) Dhruv Batra 34
35 Finishing up: Connections between NB & LR (C) Dhruv Batra 35
36 Logistic regression vs Naïve Bayes Consider learning f: X à Y, where X is a vector of real-valued features, <X1 Xd> Y is boolean Gaussian Naïve Bayes classifier assume all X i are conditionally independent given Y model P(X i Y = k) as Gaussian N(µ ik,σ i ) model P(Y) as Bernoulli(θ,1-θ) What does that imply about the form of P(Y X)? Cool!!!! (C) Dhruv Batra P (Y =1 X = x) = 1 1+exp( w Pi w ix i ) Slide Credit: Carlos Guestrin 36
37 Derive form for P(Y X) for continuous X i (C) Dhruv Batra Slide Credit: Carlos Guestrin 37
38 Ratio of class-conditional probabilities (C) Dhruv Batra Slide Credit: Carlos Guestrin 38
39 Derive form for P(Y X) for continuous X i P (Y =1 X = x) = 1 1+exp( w Pi w ix i ) (C) Dhruv Batra Slide Credit: Carlos Guestrin 39
40 Gaussian Naïve Bayes vs Logistic Regression Set of Gaussian Naïve Bayes parameters (feature variance independent of class label) Set of Logistic Regression parameters Not necessarily Representation equivalence But only in a special case!!! (GNB with class-independent variances) But what s the difference??? LR makes no assumptions about P(X Y) in learning!!! Loss function!!! Optimize different functions à Obtain different solutions (C) Dhruv Batra Slide Credit: Carlos Guestrin 4
41 Naïve Bayes vs Logistic Regression Consider Y boolean, Xi continuous, X=<X1... Xd> Number of parameters: NB: 4d +1 (or 3d+1) LR: d+1 Estimation method: NB parameter estimates are uncoupled LR parameter estimates are coupled (C) Dhruv Batra Slide Credit: Carlos Guestrin 41
42 G. Naïve Bayes vs. Logistic Regression 1 Generative and Discriminative classifiers [Ng & Jordan, 22] Asymptotic comparison (# training examples à infinity) when model correct GNB (with class independent variances) and LR produce identical classifiers when model incorrect LR is less biased does not assume conditional independence therefore LR expected to outperform GNB (C) Dhruv Batra Slide Credit: Carlos Guestrin 42
43 G. Naïve Bayes vs. Logistic Regression 2 Generative and Discriminative classifiers [Ng & Jordan, 22] Non-asymptotic analysis convergence rate of parameter estimates, d = # of attributes in X Size of training data to get close to infinite data solution GNB needs O(log d) samples LR needs O(d) samples GNB converges more quickly to its (perhaps less helpful) asymptotic estimates (C) Dhruv Batra Slide Credit: Carlos Guestrin 43
44 Naïve bayes Logistic Regression Some experiments from UCI data sets (C) Dhruv Batra
45 What you should know about LR Gaussian Naïve Bayes with class-independent variances representationally equivalent to LR Solution differs because of objective (loss) function In general, NB and LR make different assumptions NB: Features independent given class assumption on P(X Y) LR: Functional form of P(Y X), no assumption on P(X Y) LR is a linear classifier decision rule is a hyperplane LR optimized by conditional likelihood no closed-form solution Concave à global optimum with gradient ascent Maximum conditional a posteriori corresponds to regularization Convergence rates GNB (usually) needs less data LR (usually) gets to better solutions in the limit (C) Dhruv Batra Slide Credit: Carlos Guestrin 45
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