Ensembles. Léon Bottou COS 424 4/8/2010

Transcription

1 Ensembles Léon Bottou COS 424 4/8/2010

2 Readings T. G. Dietterich (2000) Ensemble Methods in Machine Learning. R. E. Schapire (2003): The Boosting Approach to Machine Learning. Sections 1,2,3,4,6. Léon Bottou 2/33 COS 424 4/8/2010

3 Summary 1. Why ensembles? 2. Combining outputs. 3. Constructing ensembles. 4. Boosting. Léon Bottou 3/33 COS 424 4/8/2010

4 I. Ensembles Léon Bottou 4/33 COS 424 4/8/2010

5 Ensemble of classifiers Ensemble of classifiers Consider a set of classifiers h 1, h 2,..., h L. Construct a classifier by combining their individual decisions. For example by voting their outputs. Accuracy The ensemble works if the classifiers have low error rates. Diversity No gain if all classifiers make the same mistakes. What if classifiers make different mistakes? Léon Bottou 5/33 COS 424 4/8/2010

6 Uncorrelated classifiers Assume r s Cov [ 1I{h r (x) = y}, 1I{h s (x) = y} ] = 0 The tally of classifier votes follows a binomial distribution. Example Twenty-one uncorrelated classifiers with 30% error rate. Léon Bottou 6/33 COS 424 4/8/2010

7 Statistical motivation blue : classifiers that work well on the training set(s) f : best classifier. Léon Bottou 7/33 COS 424 4/8/2010

8 Computational motivation blue : classifier search may reach local optima f : best classifier. Léon Bottou 8/33 COS 424 4/8/2010

9 Representational motivation blue : classifier space may not contain best classifier f : best classifier. Léon Bottou 9/33 COS 424 4/8/2010

10 Practical success Recommendation system Netflix movies you may like. Customers sometimes rate movies they rent. Input: (movie, customer) Output: rating Netflix competition 1M$ for the first team to do 10% better than their system. Winner: BellKor team and friends Ensemble of more than 800 rating systems. Runner-up: everybody else Ensemble of all the rating systems built by the other teams. Léon Bottou 10/33 COS 424 4/8/2010

11 Bayesian ensembles Let D represent the training data. Enumerating all the classifiers P (y x, D) = h P (y, h x, D) = h P (h x, D) P (y h, x, D) = h P (h D) P (y x, h) P (h D) : how well does h match the training data. P (y x, h) : what h predicts for pattern x. Note that this is a weighted average. Léon Bottou 11/33 COS 424 4/8/2010

12 II. Combining Outputs Léon Bottou 12/33 COS 424 4/8/2010

13 Simple averaging Léon Bottou 13/33 COS 424 4/8/2010

14 Weighted averaging a priori Weights derived from the training errors, e.g. exp( β T rainingerror(h t )). Approximate Bayesian ensemble. Léon Bottou 14/33 COS 424 4/8/2010

15 Weighted averaging with trained weights Train weights on the validation set. Training weights on the training set overfits easily. You need another validation set to estimate the performance! Léon Bottou 15/33 COS 424 4/8/2010

16 Stacked classifiers Second tier classifier trained on the validation set. You need another validation set to estimate the performance! Léon Bottou 16/33 COS 424 4/8/2010

17 III. Constructing Ensembles Léon Bottou 17/33 COS 424 4/8/2010

18 Diversification Cause of the mistake Pattern was difficult. Overfitting ( ) Some features were noisy Multiclass decisions were inconsistent Diversification strategy hopeless vary the training sets vary the set of input features vary the class encoding Léon Bottou 18/33 COS 424 4/8/2010

19 Manipulating the training examples Bootstrap replication simulates training set selection Given a training set of size n, construct a new training set by sampling n examples with replacement. About 30% of the examples are excluded. Bagging Create bootstrap replicates of the training set. Build a decision tree for each replicate. Estimate tree performance using out-of-bootstrap data. Average the outputs of all decision trees. Boosting See part IV. Léon Bottou 19/33 COS 424 4/8/2010

20 Manipulating the features Random forests Construct decision trees on bootstrap replicas. Restrict the node decisions to a small subset of features picked randomly for each node. Do not prune the trees. Estimate tree performance using out-of-bootstrap data. Average the outputs of all decision trees. Multiband speech recognition Filter speech to eliminate a random subset of the frequencies. Train speech recognizer on filtered data. Repeat and combine with a second tier classifier. Resulting recognizer is more robust to noise. Léon Bottou 20/33 COS 424 4/8/2010

21 Manipulating the output codes Reducing multiclass problems to binary classification We have seen one versus all. We have seen all versus all. Error correcting codes for multiclass problems Code the class numbers with an error correcting code. Construct a binary classifier for each bit of the code. Run the error correction algorithm on the binary classifier outputs. Léon Bottou 21/33 COS 424 4/8/2010

22 IV. Boosting Léon Bottou 22/33 COS 424 4/8/2010

23 Motivation Easy to come up with rough rules of thumb for classifying data contains more than 50% capital letters. contains expression buy now. Each alone isnt great, but better than random. Boosting converts rough rules of thumb into an accurate classier. Boosting was invented by Prof. Schapire. Léon Bottou 23/33 COS 424 4/8/2010

24 Adaboost Given examples (x 1, y 1 )... (x n, y n ) with y i = ±1. Let D 1 (i) = 1/n for i = 1... n. For t = 1... T do Run weak learner using examples with weights D t. Get weak classifier h t Compute error: ε t = i D t(i) 1I(h t (x i ) y i ) Compute magic coefficient α t = 1 ( 1 εt 2 log ε t Update weights D t+1 (i) = D t(i) e α t y i h t (x i ) Output the final classifier f T (x) = sign Z t ) T α t h t (x) t=1 Léon Bottou 24/33 COS 424 4/8/2010

25 Toy example Weak classifiers: vertical or horizontal half-planes. Léon Bottou 25/33 COS 424 4/8/2010

26 Adaboost round 1 Léon Bottou 26/33 COS 424 4/8/2010

27 Adaboost round 2 Léon Bottou 27/33 COS 424 4/8/2010

28 Adaboost round 3 Léon Bottou 28/33 COS 424 4/8/2010

29 Adaboost final classifier Léon Bottou 29/33 COS 424 4/8/2010

30 From weak learner to strong classifier (1) Preliminary D T +1 (i) = D 1 (i) e α 1 y i h 1 (x i ) Z 1 e α T y i h T (x i) Z T = 1 n e y i f T (x i ) t Z t Bounding the training error 1 1I{f n T (x i ) y i } 1 e y n i f T (x i ) = i i 1 n D T +1 (i) t i Z t = t Z t Idea: make Z t as small as possible. Z t = n D t (i)e α t y i h t (x i ) i=1 1. Pick h t to minimize ε t. 2. Pick α t to minimize Z t. = n (1 ε t ) e α t + n ε t e α t Léon Bottou 30/33 COS 424 4/8/2010

31 From weak learner to strong classifier (2) Pick α t to minimize Z t (the magic coefficient) Z t α t = (1 ε t ) e α t + ε t e α t = 0 = α t = 1 2 log 1 ε t ε t Weak learner assumption: γ t = 2 1 ε t is positive and small. ε 1 ε Z t = (1 ε) 1 ε + ε = 4ε(1 ε) = 1 4γ 2 ε t exp ( 2γt 2 ) T T TrainingError(f T ) Z t exp 2 t=1 t=1 γ 2 t The training error decreases exponentially if inf γ t > 0. But that does not happen beyond a certain point... Léon Bottou 31/33 COS 424 4/8/2010

32 Boosting and exponential loss Proofs are instructive We obtain the bound TrainingError(f T ) 1 n i e y ih(x i ) without saying how D t relates to h t without using the value of α t = T t=1 Z t y y(x) ^ Conclusion Round T chooses the h T and α T that maximize the exponential loss reduction from f T 1 to f T. Exercise Tweak Adaboost to minimize the log loss instead of the exp loss. Léon Bottou 32/33 COS 424 4/8/2010

33 Boosting and margins margin H (x, y) = y H(x) t α t = t α t y h t (x) t α t Remember support vector machines? Léon Bottou 33/33 COS 424 4/8/2010