The Boosting Approach to. Machine Learning. Maria-Florina Balcan 10/31/2016

Transcription

1 The Boosting Approach to Machine Learning Maria-Florina Balcan 10/31/2016

2 Boosting General method for improving the accuracy of any given learning algorithm. Works by creating a series of challenge datasets s.t. even modest performance on these can be used to produce an overall high-accuracy predictor. Works amazingly well in practice --- Adaboost and its variations one of the top 10 algorithms. Backed up by solid foundations.

3 Readings: The Boosting Approach to Machine Learning: An Overview. Rob Schapire, 2001 Theory and Applications of Boosting. NIPS tutorial. Plan for today: Motivation. A bit of history. Adaboost: algo, guarantees, discussion. Focus on supervised classification.

4 An Example: Spam Detection E.g., classify which s are spam and which are important. Not spam spam Key observation/motivation: Easy to find rules of thumb that are often correct. E.g., If buy now in the message, then predict spam. E.g., If say good-bye to debt in the message, then predict spam. Harder to find single rule that is very highly accurate.

5 s An Example: Spam Detection Boosting: meta-procedure that takes in an algo for finding rules of thumb (weak learner). Produces a highly accurate rule, by calling the weak learner repeatedly on cleverly chosen datasets. h 1 h 2 h 3 h T apply weak learner to a subset of s, obtain rule of thumb apply to 2nd subset of s, obtain 2nd rule of thumb apply to 3 rd subset of s, obtain 3rd rule of thumb repeat T times; combine weak rules into a single highly accurate rule.

6 Boosting: Important Aspects How to choose examples on each round? Typically, concentrate on hardest examples (those most often misclassified by previous rules of thumb) How to combine rules of thumb into single prediction rule? take (weighted) majority vote of rules of thumb

7 Historically.

8 Weak Learning vs Strong Learning [Kearns & Valiant 88]: defined weak learning: being able to predict better than random guessing (error 1 2 γ), consistently. Posed an open pb: Does there exist a boosting algo that turns a weak learner into a strong learner (that can produce arbitrarily accurate hypotheses)? Informally, given weak learning algo that can consistently find classifiers of error 1 γ, a boosting algo would 2 provably construct a single classifier with error ε.

9 Surprisingly. Weak Learning =Strong Learning Original Construction [Schapire 89]: poly-time boosting algo, exploits that we can learn a little on every distribution. A modest booster obtained via calling the weak learning algorithm on 3 distributions. Error = β < 1 2 γ error 3β2 2β 3 Then amplifies the modest boost of accuracy by running this somehow recursively. Cool conceptually and technically, not very practical.

10 An explosion of subsequent work

11 Adaboost (Adaptive Boosting) A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting [Freund-Schapire, JCSS 97] Godel Prize winner 2003

12 Informal Description Adaboost Boosting: turns a weak algo into a strong learner. Input: S={(x 1, y 1 ),,(x m, y m )}; For t=1,2,,t Construct D t on {x 1,, x m } x i X, y i Y = { 1,1} weak learning algo A (e.g., Naïve Bayes, decision stumps) Run A on D t producing h t : X { 1,1} (weak classifier) ϵ t = P xi ~D t (h t x i y i ) error of h t over D t Output H final x = sign σ t=1 α t h t x h t Roughly speaking D t+1 increases weight on x i if h t incorrect on x i ; decreases it on x i if h t correct.

13 Adaboost (Adaptive Boosting) Weak learning algorithm A. For t=1,2,,t Construct D t on {x 1,, x m } Run A on D t producing h t Constructing D t D [i.e., D 1 i = 1 1 uniform on {x 1,, x m } Given D t and h t set D t+1 i = D t i e α t if y i = h t x i D t+1 i = D t i e α t if y i h t x i α t = 1 2 ln 1 ε t ε t > 0 Final hyp: H final x = sign σ t α t h t x m ] D t+1 i = D t i e α ty i h t x i = D t i e α ty i h t x i i D t+1 puts half of weight on examples x i where h t is incorrect & half on examples where h t is correct

14 Adaboost: A toy example Weak classifiers: vertical or horizontal half-planes (a.k.a. decision stumps)

15 Adaboost: A toy example

16 Adaboost: A toy example

17 Adaboost (Adaptive Boosting) Weak learning algorithm A. For t=1,2,,t Construct D t on {x 1,, x m } Run A on D t producing h t Constructing D t D [i.e., D 1 i = 1 1 uniform on {x 1,, x m } Given D t and h t set m ] D t+1 i = D t i e α t if y i = h t x i D t+1 i = D t i e α t if y i h t x i D t+1 i = D t i e α ty i h t x i α t = 1 2 ln 1 ε t ε t > 0 Final hyp: H final x = sign σ t α t h t x D t+1 puts half of weight on examples x i where h t is incorrect & half on examples where h t is correct

18 Nice Features of Adaboost Very general: a meta-procedure, it can use any weak learning algorithm!!! (e.g., Naïve Bayes, decision stumps) Very fast (single pass through data each round) & simple to code, no parameters to tune. Shift in mindset: goal is now just to find classifiers a bit better than random guessing. Grounded in rich theory.

19 Guarantees about the Training Error Theorem ε t = 1/2 γ t (error of h t over D t ) err S H final exp 2 t γ t 2 So, if t, γ t γ > 0, then err S H final exp 2 γ 2 T The training error drops exponentially in T!!! To get err S H final ε, need only T = O 1 γ 2 log 1 ε rounds Adaboost is adaptive Does not need to know γ or T a priori Can exploit γ t γ

20 Understanding the Updates & Normalization Claim: D t+1 puts half of the weight on x i where h t was incorrect and half of the weight on x i where h t was correct. Recall D t+1 i = D t i e α ty i h t x i Probabilities are equal! Pr y i h t x i = D t+1 i:y i h t x i D t i e α t = ε t 1 e α t = ε t 1 ε t ε t = ε t 1 ε t Pr y i = h t x i = D t+1 i:y i =h t x i D t i e α t = 1 ε t e α t = 1 ε t ε t 1 ε t = 1 ε t ε t = D t i e α ty i h t x i i = i:y i =h t x i D t i e αt + i:y i h t x i D t i e αt = 1 ε t e α t + ε t e α t = 2 ε t 1 ε t

21 Generalization Guarantees (informal) Theorem err S H final exp 2 t γ t 2 where ε t = 1/2 γ t How about generalization guarantees? Original analysis [Freund&Schapire 97] Let H be the set of rules that the weak learner can use Let G be the set of weighted majority rules over T elements of H (i.e., the things that AdaBoost might output) Theorem [Freund&Schapire 97] g G, err g err S g + O Td m T= # of rounds d= VC dimension of H

22 Generalization Guarantees Theorem [Freund&Schapire 97] g G, err g err S g + O Td m T= # of rounds d= VC dimension of H error train error generalization error T= # of rounds complexity

23 Generalization Guarantees Experiments with boosting showed that the test error of the generated classifier usually does not increase as its size becomes very large. Experiments showed that continuing to add new weak learners after correct classification of the training set had been achieved could further improve test set performance!!!

24 Generalization Guarantees Experiments with boosting showed that the test error of the generated classifier usually does not increase as its size becomes very large. Experiments showed that continuing to add new weak learners after correct classification of the training set had been achieved could further improve test set performance!!! These results seem to contradict FS 87 bound and Occam s razor (in order achieve good test error the classifier should be as simple as possible)!

25 How can we explain the experiments? R. Schapire, Y. Freund, P. Bartlett, W. S. Lee. present in Boosting the margin: A new explanation for the effectiveness of voting methods a nice theoretical explanation. Key Idea: Training error does not tell the whole story. We need also to consider the classification confidence!!

26 What you should know The difference between weak and strong learners. AdaBoost algorithm and intuition for the distribution update step. The training error bound for Adaboost and overfitting/generalization guarantees aspects.

27 Advanced (Not required) Material

28 Analyzing Training Error: Proof Intuition Theorem ε t = 1/2 γ t (error of h t over D t ) err S H final exp 2 t γ t 2 On round t, we increase weight of x i for which h t is wrong. If H final incorrectly classifies x i, - Then x i incorrectly classified by (wtd) majority of h t s. - Which implies final prob. weight of x i is large. Can show probability 1 m 1 ς t Since sum of prob. = 1, can t have too many of high weight. Can show # incorrectly classified m ς t. And (ς t ) 0.

29 Analyzing Training Error: Proof Math Step 1: unwrapping recurrence: D T+1 i = 1 m where f x i = σ t α t h t x i. exp y i f x i ς t [Unthresholded weighted vote of h i on x i ] Step 2: err S H final ς t. Step 3: ς t = ς t 2 ε t 1 ε t = ς t 1 4γ t 2 e 2 σ t γ t 2

30 Analyzing Training Error: Proof Math Step 1: unwrapping recurrence: D T+1 i = 1 m where f x i = σ t α t h t x i. exp y i f x i ς t Recall D 1 i = 1 m and D t+1 i = D t i exp y i α t h t x i D T+1 i = exp y iα T h T x i Z T = exp y iα T h T x i Z T D T i exp y iα T 1 h T 1 x i Z T 1 D T 1 i. = exp y iα T h T x i Z T exp y iα 1 h 1 x i Z 1 1 m = 1 m exp y i (α 1 h 1 x i + +α T h T x T ) Z 1 Z T = 1 m exp y i f x i ς t

31 Analyzing Training Error: Proof Math Step 1: unwrapping recurrence: D T+1 i = 1 m where f x i = σ t α t h t x i. exp y i f x i ς t Step 2: err S H final ς t. err S H final = 1 m i 1 yi H final x i exp loss 0/1 loss = 1 m i 1 yi f x i m i exp y i f x i = D T+1 i i t = ς t.

32 Analyzing Training Error: Proof Math Step 1: unwrapping recurrence: D T+1 i = 1 m where f x i = σ t α t h t x i. exp y i f x i ς t Step 2: err S H final ς t. Step 3: ς t = ς t 2 ε t 1 ε t = ς t 1 4γ t 2 e 2 σ t γ t 2 Note: recall = 1 ε t e α t + ε t e α t = 2 ε t 1 ε t α t minimizer of α 1 ε t e α + ε t e α