Boos$ng Can we make dumb learners smart?

Transcription

1 Boos$ng Can we make dumb learners smart? Aarti Singh Machine Learning Nov 29, 2011 Slides Courtesy: Carlos Guestrin, Freund & Schapire 1

2 Why boost weak learners? Goal: Automa'cally categorize type of call requested (Collect, Calling card, Person- to- person, etc.) Easy to find rules of thumb that are o>en correct. E.g. If card occurs in uberance, then predict calling card Hard to find single highly accurate predic$on rule. 2

3 Figh$ng the bias- variance tradeoff Simple (a.k.a. weak) learners e.g., naïve Bayes, logis'c regression, decision stumps (or shallow decision trees) Are good J - Low variance, don t usually overfit Are bad L - High bias, can t solve hard learning problems Can we make weak learners always good??? No!!! But o>en yes 3

4 Vo$ng (Ensemble Methods) Instead of learning a single (weak) classifier, learn many weak classifiers that are good at different parts of the input space Output class: (Weighted) vote of each classifier Classifiers that are most sure will vote with more convic'on Classifiers will be most sure about a par'cular part of the space On average, do beber than single classifier! h1(x) h2(x) H: X Y (-1,1) H(X) = h1(x)+h2(x) H(X) = sign( αt ht(x)) t 1-1???? 1-1 weights 4

5 Vo$ng (Ensemble Methods) Instead of learning a single (weak) classifier, learn many weak classifiers that are good at different parts of the input space Output class: (Weighted) vote of each classifier Classifiers that are most sure will vote with more convic'on Classifiers will be most sure about a par'cular part of the space On average, do beber than single classifier! But how do you??? force classifiers h t to learn about different parts of the input space? weigh the votes of different classifiers? α t 5

6 Boos$ng [Schapire 89] Idea: given a weak learner, run it mul'ple 'mes on (reweighted) training data, then let learned classifiers vote On each itera'on t: weight each training example by how incorrectly it was classified Learn a weak hypothesis h t A strength for this hypothesis α t Final classifier: H(X) = sign( αt ht(x)) Prac$cally useful Theore$cally interes$ng 6

7 Learning from weighted data Consider a weighted dataset D(i) weight of i th training example (x i,y i ) Interpreta'ons: i th training example counts as D(i) examples If I were to resample data, I would get more samples of heavier data points Now, in all calcula$ons, whenever used, i th training example counts as D(i) examples e.g., in MLE redefine Count(Y=y) to be weighted count Unweighted data m Count(Y=y) = 1(Y i =y) i =1 Weights D(i) m Count(Y=y) = D(i)1(Y i =y) i =1 7

8 AdaBoost [Freund & Schapire 95] Ini$ally equal weights weak weak Magic (+ve) Naïve bayes, decision stump Increase weight if wrong on pt i yi ht(xi) = - 1 < 0 8

9 AdaBoost [Freund & Schapire 95] Ini$ally equal weights weak weak Magic (+ve) Naïve bayes, decision stump Increase weight if wrong on pt i yi ht(xi) = - 1 < 0 Weights for all pts must sum to 1 Dt+1(i) = 1 t 9

10 AdaBoost [Freund & Schapire 95] Ini$ally equal weights weak weak Magic (+ve) Naïve bayes, decision stump Increase weight if wrong on pt i yi ht(xi) = - 1 < 0 10

11 What α t to choose for hypothesis h t? Weight Update Rule: [Freund & Schapire 95] Weighted training error Does ht get i th point wrong ε t = 0 if h t perfectly classifies all weighted data pts α t = ε t = 1 if h t perfectly wrong => -h t perfectly right α t = - ε t = 0.5 α t = 0 11

12 Boos$ng Example (Decision Stumps) 12

13 Boos$ng Example (Decision Stumps) 13

14 Analyzing training error Analysis reveals: What α t to choose for hypothesis h t? ε t - weighted training error If each weak learner h t is slightly beber than random guessing (ε t < 0.5), then training error of AdaBoost decays exponen'ally fast in number of rounds T. Training Error 14

15 Analyzing training error Training error of final classifier is bounded by: Convex upper bound Where 0/1 loss exp loss If boosting can make upper bound 0, then training error

16 Analyzing training error Training error of final classifier is bounded by: Where Proof: Using Weight Update Rule Wts of all pts add to 1 16

17 Analyzing training error Training error of final classifier is bounded by: Where If Zt < 1, training error decreases exponentially (even though weak learners may not be good ε t ~0.5) Training error Upper bound t 17

18 What α t to choose for hypothesis h t? Training error of final classifier is bounded by: Where If we minimize t Z t, we minimize our training error We can 'ghten this bound greedily, by choosing α t and h t on each itera'on to minimize Z t. 18

19 What α t to choose for hypothesis h t? We can minimize this bound by choosing α t on each itera'on to minimize Z t. For boolean target func'on, this is accomplished by [Freund & Schapire 97]: Proof: 19

20 What α t to choose for hypothesis h t? We can minimize this bound by choosing α t on each itera'on to minimize Z t. For boolean target func'on, this is accomplished by [Freund & Schapire 97]: Proof: 20

21 Dumb classifiers made Smart Training error of final classifier is bounded by: Using 1-x e -x grows as ε t moves away from 1/2 If each classifier is (at least slightly) bemer than random ε t < 0.5 AdaBoost will achieve zero training error exponen$ally fast (in number of rounds T)!! What about test error? 21

22 Boos$ng results Digit recogni$on [Schapire, 1989] Test Error Training Error Boos'ng ohen, Robust to overfijng but not always Test set error decreases even aher training error is zero 22

23 Margin Based Bounds [Schapire, Freund, Bartlett, Lee 98] With high probability Boos'ng increases the margin very aggressively since it concentrates on the hardest examples. If margin is large, more weak learners agree and hence more rounds does not necessarily imply that final classifier is gejng more complex. Bound is independent of number of rounds T! Boos$ng can s$ll overfit if margin is too small or weak learners are too complex 23

24 Okay Train Test Train Test Okay Overfits Okay Overfits Okay Overfits Overfits 24

25 Boos$ng and Logis$c Regression Logis'c regression equivalent to minimizing log loss Boos'ng minimizes similar loss func'on!! Weighted average of weak learners log loss 0/1 loss exp loss 1 Both smooth approxima$ons of 0/1 loss! 0 25

26 Boos$ng and Logis$c Regression Logis'c regression: Minimize log loss Boos'ng: Minimize exp loss Define Define where x j predefined features (linear classifier) where h t (x) defined dynamically to fit data (not a linear classifier) Jointly op'mize over all weights w0, w1, w2 Weights α t learned per itera'on t incrementally 26

27 Hard & So> Decision Weighted average of weak learners Hard Decision/Predicted label: Soh Decision: (based on analogy with logis'c regression) 27

28 Effect of Outliers Good J : Can identify outliers since focuses on examples that are hard to categorize Bad L : Too many outliers can degrade classification performance dramatically increase time to convergence 28

29 Bagging [Breiman, 1996] Related approach to combining classifiers: 1. Run independent weak learners on bootstrap replicates (sample with replacement) of the training set 2. Average/vote over weak hypotheses Bagging vs. Boos$ng Resamples data points Reweights data points (modifies their distribu'on) Weight of each classifier Weight is dependent on is the same classifier s accuracy Only variance reduc'on Both bias and variance reduced learning rule becomes more complex with itera'ons 29

30 Boos$ng Summary Combine weak classifiers to obtain very strong classifier Weak classifier slightly beber than random on training data Resul'ng very strong classifier can eventually provide zero training error AdaBoost algorithm Boos'ng v. Logis'c Regression Similar loss func'ons Single op'miza'on (LR) v. Incrementally improving classifica'on (B) Most popular applica'on of Boos'ng: Boosted decision stumps! Very simple to implement, very effec've classifier 30